text:
\section{Introduction}
First proposed in \citep{kalman1964linear}, Inverse Optimal Control (IOC) is an inverse problem whose aim is to reconstruct the cost function and predict the closed-loop system's behaviour, based on knowledge of the underlying system dynamics and observations of the system.
The problem can be categorized as a system identification problem where the system is assumed to be governed by an optimal control model with known structure, and hence it is a so-called gray-box system identification problem \cite[p.~13]{ljung1999system}.
IOC problems are of great interest, not least due to the fact that many processes in nature have been observed to be optimal with respect to some criteria \cite{alexander1996optima}. ``Reverse-engineering'' the systems' objective function not only enables us to understand how their decisions are made, but also provides a way to predict and imitate their behaviours.
For example, the IOC problem has found applications in human motion analysis \citep{berret2011evidence, westermann2020inverse}, robot locomotion \citep{mombaur2010human}, robot manipulation \citep{jin2019inverse, menner2019constrained}, systems biology \citep{tsiantis2018optimality} and machine learning \citep{finn2016guided, kopf2017inverse}.

Nevertheless, for some scenarios with multiple individuals, such as human crowds, bacteria, bird flocks, and schools of fish, the observation data is naturally collected by observing all the individuals simultaneously in discrete time, for instance via video cameras.
Due to the similarity between the individuals and the fact that population is often huge, it is often too expensive or simply impossible to track each individual in the group.
Under such conditions, the individuals can be assumed to be homogeneous and ``indistinguishable".
This ``indistinguishablility" means that we can not tell ``which individual system goes where" for consecutive observations. ``Reverse-engineering" the objective function for such individual systems is challenging due to the fact that:
\begin{enumerate}
\item The control input measurement is usually not available.
Moreover, the ``indistinguishable" characteristics means that the individual systems' optimal trajectories are not available. Therefore, the existing IOC methods, such as \citep{zhang2019inverse,  yu2021system, molloy2018finite, jin2019inverse, zhang2019inverseCDC, aswani2018inverse}, that minimizes the difference between the observed and expected optimal trajectories, or minimizes the violation of optimality conditions, can not be applied.
\item The structural identifiability under such ``indistinguishable" observations is not guaranteed.
In particular, given observed data for optimal ``indistinguishable" individual systems, there is a priori no guarantee that there aren't different objective functions or closed-loop systems that corresponds to it.
\item The observations are usually contaminated by noise due to limitations of experiments and measuring methods. The identification methods should be statistically consistent and robust to observation noise in order to provide an accurate estimate of the objective function or the closed-loop system.
\end{enumerate}

On the other hand, the Linear-Quadratic (LQ) optimal control formulation is one of the most commonly used optimal control methods in practice. Due to its simplicity, it is often used to approximate complex control problems and optimal behaviours \citep{fridovich2020efficient, toumi2020tractable, ji2019feedback}.
It is therefore not surprising that the inverse LQ optimal control problem (with distinguishable observations) has been studied in different settings and in various degrees of generality; see, e.g.,
\citep[Sec.~5.6]{anderson2007optimal}, \citep[Sec.~10.5]{boyd1994linear} for the continuous-time infinite-horizon case,
\citep{li2018convex, li2020continuous} for the continuous-time finite-horizon case,
\citep{priess2014solutions} for the discrete-time infinite-horizon case,
and \citep{keshavarz2011imputing, zhang2019inverse, zhang2019inverseCDC, yu2021system} for the discrete-time finite-horizon case, respectively.
Moreover, although the inverse LQ optimal control problem can be seen as a special case of the IOC problem for general nonlinear systems \citep{molloy2018finite,  keshavarz2011imputing, molloy2020online, pauwels2016linear, hatz2012estimating, rouot2017inverse}, the special structure of the LQ formulation allows for precise answers to structural identifiability, and in the finite time horizon case for the identification of a linear time-varying system from a limited amount of data.
Further, to the best of our knowledge, the IOC problem for ``indistinguishable" systems has not been considered in existing literatures.

In this work, we consider the linear-quadratic discrete-time finite-horizon IOC problem  with indistinguishable observations. In particular, we assume that each system is governed by a discrete-time finite-horizon Linear Quadratic Regulator (LQR). Moreover, in this initial work on ``indistinguishable'' systems, we also assume that there is no interaction between the systems. This is a simplifying assumption which we intend to relax in the future. In the current work, we focus on studying how to tackle this ``indistinguishability'' issue in the observations. The goal is to develop an inverse LQ optimal control algorithm for an external observer that can be used to identify the ``microscopic individual's'' objective function using ``macroscopic indistinguishable'' observations.
More precisely, the main contributions are:
\begin{enumerate}[(i)]
\item We show that the corresponding model structure is strictly globally identifiable.
\item In the case of exact measurements, we formulate a convex optimization problem and prove that the unique globally optimal solution is the quadratic cost term of interest.
\item In the case of noisy observations, we formulate an estimator of the sought quadratic cost term as the unique globally optimal solution to a modified convex optimization problem. Moreover, we also show that this estimator is (asymptotically) statistically consistent. The latter means that the estimate converges to the true parameter value as the number of systems tends to infinity.
\end{enumerate}

The article is organized as follows: in Section~\ref{section:problem_formulation_and_identifiability}, we formulate the problem, specify the model structure, and show that the model structure is strictly globally identifiable. Section~\ref{sec:noiseless} investigates the inverse problem in the absence of measurement noise, and we formulate a convex optimization problem whose unique global optimal solution coincides with the parameters of interest. In Section~\ref{sec:noisy}, we adapt the results of the previous section to the case when the observations have measurement noise. In particular, it is shown that the unique global optimal solution of the modified stochastic optimization problem is a statistically consistent estimator. Next, discussions regarding some general numerical difficulties with the discrete-time finite-horizon LQ IOC problem are included in Section~\ref{sec:ill-cond}, and in Section~\ref{sec:numerics}, we present the numerical results. Finally, the article is concluded in Section~\ref{sec:conclusion}.

\textit{Notation:} The following notation is utilized throughout the article: $\mathbb{S}^n$ denotes the set of $n\times n$ symmetric matrices, and $\mathbb{S}^n_+$ denotes the set of symmetric $n \times n$ matrices that are positive semidefinite. For symmetric matrices, $G_1\succeq G_2$ denotes the Loewner partial order of $G_1$ and $G_2$, namely, $G_1-G_2\in\mathbb{S}^n_+$. Moreover, $G_1\succ G_2$ denotes that $G_1-G_2$ is strictly positive definite; see, e.g., \cite[Sec.~7.7]{horn2013matrix}.
$\|\cdot\|_F$ denotes the Frobenius norm, and $\mathbf{1}$ denotes an all-one vector with appropriate length.
Furthermore, we use \textit{\textbf{italic bold font}} to denote stochastic elements.
Finally, we denote $\prod_{k=1}^m A_k = A_m A_{m-1}\ldots A_1$.

\section{Problem formulation, model structure, and identifiability}\label{section:problem_formulation_and_identifiability}
Suppose that each observed system $i$ ($i=1,\ldots,M$) is governed by the following discrete-time finite-horizon LQR:
\begin{subequations}\label{eq:forward_problem}
  \begin{align}
  \min_{x_{1:N}^i, u_{1:N-1}^i}
  & \;\; x_N^{iT} \bar{S} x_N^{i} + \sum_{t = 1}^{N-1} \left( x_t^{iT} \bar{Q} x_t^i + u_t^{iT} \bar{R} u_t^{i} \right) \label{eq:forward_problem_cost} \\
  \text{s.t.}
  & \;\; x_{t+1}^i = Ax_t^i + Bu_t^i, \; t = 1, \ldots, N-1, \label{eq:forward_problem_dynamics} \\
  & \;\; x_1^i = \bar{x}^i. \label{eq:forward_problem_init_cond}
  \end{align}
\end{subequations}
The systems are assumed to be homogeneous, namely, they have the same dynamics as well as the same objective functions.  For the sake of simplicity, as stated in the Introduction, in this work we assume there is no interactions between the systems. We intend to relax this assumption in future work. We will also assume that $\bar{S} = \bar{Q}$ and $\bar{R} = I$ throughout this paper.  It is further assumed that $A$ is invertible, that $B$ has full column rank, and that $(A,B)$ is controllable. The rationality of making the assumption that $A$ is invertible lies in the fact that discrete-time systems are often obtained by sampling of a continuous-time systems. More precisely, if $\hat{A}$ is the system matrix of the continuous-time system and $\Delta t > 0$ is the sample period, then the system matrix $A=\exp(\hat{A}\Delta t)$ of the sampled discrete-time systems is always invertible \citep{zhang2019inverse}.

The optimal control input sequence $\bar{u}_{1:N-1}^{i}$ of \eqref{eq:forward_problem} is given by $\bar{u}_t^{i} = \bar{K}_t x_t^{i}$, $t=1,\ldots,N-1$, where
\begin{align*}
\bar{K}_t = -(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A,
\end{align*}
and $\bar{P}_{2:N}$ is the solution to the following discrete-time Riccati Equation (DRE):
\begin{align}
\bar{P}_t = & \; A^T\bar{P}_{t+1}A+\bar{Q}- \nonumber \\
&\; A^T\bar{P}_{t+1}B(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A, \nonumber \\
& \; t=2:N-1 \nonumber \\
\bar{P}_N = & \; \bar{Q}. \label{eq:DRE}
\end{align}
For the problem of LQ IOC for indistinguishable systems, which is considered in this paper, the goal is to recover the matrix $\bar{Q}$ in the objective function given the knowledge of (possibly noisy) state observations. However, note that since the systems are indistinguishable, this means that the observations at different time instants are \emph{unpaired}.
More precisely,
let $X_t := [x_t^1, \ldots, x_t^M]$ be the state of $M$ systems following the dynamics in \eqref{eq:forward_problem}, then the observations are modelled as
\begin{align*}
\left\{Y_t^\phi \right\}_{t=1}^N=\left\{\left[x_t^{\phi_t(1)},\ldots, x_t^{\phi_t(M)}\right]\right\}_{t=1}^N,
\end{align*}
where $\phi_t:\{1,\ldots,M\} \mapsto \{1,\ldots,M\}$, $t = 1, \ldots, N$, are unknown permutations (the superscript $\phi$ in $Y_t^{\phi}$ is used to emphasizes the fact that the observations are shuffled).
Nevertheless, $\phi_1(\cdot)$ can, without loss of generality, be assumed to be the identity mapping since the ordering of the initial states does not matter. Therefore we will henceforth restrict our attention to the set $\{ \phi_t \}_{t = 2}^N$.

\begin{problem}[IOC for indistinguishable LQ systems]\label{pro:problem_ioc}
Given the unpaired state observations $\{Y_t^\phi\}_{t=1}^N$ of $M$ systems that are governed by \eqref{eq:forward_problem}, recover the corresponding parameter $\bar{Q}$ in the objective function.
\end{problem}

Before we continue, we would like to first discuss the identifiability of such problem.  According to the definition of identifiability in \citep[Def.~4.5, 4.6]{ljung1999system},  \emph{the identifiability is a property of the model structure $\mathcal{M}$ itself and have nothing to do with any concrete data}, where the model structure $\mathcal{M}$ is a parameterized collection of models that describes the relations between the input and the output signal of the system \citep{ljung2013convexity}.  Hence, as a pre-step, we need to first define the underlying model structure $\mathcal{M}$ of Problem \ref{pro:problem_ioc}.

In Problem \ref{pro:problem_ioc}, we see the initial values of the trajectories as the ``input signal", and $\{Y_t^\phi\}_{t=2}^N$ as the ``output signal".
Note that any permutation $\phi_t(\cdot)$ can be uniquely represented by a permutation matrix $\mathscr{P}_t \in \mathcal{P}$, where
\[
\mathcal{P} := \{ \mathscr{P} \in \{0,1\}^{M \times M} \mid \mathscr{P}\mathbf{1} = \mathbf{1}, \;  \mathscr{P}^T\mathbf{1} = \mathbf{1}\},
\]
i.e., the set of $0$-$1$-matrices with exactly one element that is one in each row and in each column. Post-multiplying any matrix with such a permutation matrix results in a matrix with permuted columns. Therefore, $Y^\phi_t$ can be represented as
\begin{align*}
Y^\phi_t = X_t\mathscr{P}_t,\quad t=2,\cdots,N,
\end{align*}
and hence, for the system output it holds that
\begin{align}\label{eq:data_model_noise_free}
Y_t^\phi = \left(\prod_{k=1}^{t-1}A_{cl}(k;Q) X_1\right)\mathscr{P}_t.
\end{align}
where $A_{cl}(k;Q)$ is the closed-loop system matrix at time instant $k$ that is generated by LQR \eqref{eq:forward_problem} using the cost matrix $Q$.
By using the property of the vectorization operator \cite[Lem.~4.3.1]{horn1994topics}, vectorizing the above equation we have that
\begin{align*}
\vect(Y_t^\phi) &= \left[\mathscr{P}_t^T\otimes \prod_{k=1}^{t-1}A_{cl}(k;Q)\right]\vect(X_1).
\end{align*}
Thus, we have the following definition of the model structure $\mathcal{M}$ for Problem \ref{pro:problem_ioc}:
\begin{align*}
\mathcal{M}(Q,\{\mathscr{P}_t\}_{t=2}^N)&:=\left\{\mathcal{M}_t(Q,\mathscr{P}_t)\right\}_{t=2}^N\\
&=\{\mathscr{P}_t^T\otimes \prod_{k=1}^{t-1}A_{cl}(k;Q)\}_{t=2}^N.
\end{align*}
Next, we adopt the definition of identifiability in \citep[Def. ~4.6, 4.7]{ljung1999system}. More precisely in our case:

\begin{definition}[Identifiability]
$\mathcal{M}$ is globally identifiable at $(Q^\prime,\{\mathscr{P}_t^\prime\}_{t=2}^N) \in \mathbb{S}^n_+ \times \mathscr{P}_t^{N-1}$ if it holds that
\begin{equation*}
\begin{aligned}
&\mathcal{M}(Q,\{\mathscr{P}_t\}_{t=2}^N) = \mathcal{M}(Q^\prime,\{\mathscr{P}_t^\prime\}_{t=2}^N),\; Q\in\mathbb{S}^n_+, \mathscr{P}_t\in\mathcal{P}\\
&\implies Q=Q^\prime, \mathscr{P}_t = \mathscr{P}_t^\prime, \; \forall t=2,\ldots,N.
\end{aligned}
\end{equation*}
$\mathcal{M}$ is strictly globally identifiable if it is globally identifiable at all $Q^\prime\in\mathbb{S}^n_+,\mathscr{P}_t^\prime\in\mathcal{P}$.
\end{definition}

\begin{proposition}\label{prop:global_identifiability}
If $N \geq n +1$, then the model structure $\mathcal{M}(Q,\{\mathscr{P}_t\}_{t=2}^N)$ is strictly globally identifiable.
\end{proposition}
\begin{pf}
Let $(Q^\prime, \{\mathscr{P}_t^\prime\}_{t=2}^N) \in \mathbb{S}^n_+ \times \mathcal{P}^{N-1}$, and assume that $\mathcal{M}(Q,\{\mathscr{P}_t\}_{t=2}^N) = \mathcal{M}(Q^\prime,\{\mathscr{P}_t^\prime\}_{t=2}^N)$ for some $(Q, \{\mathscr{P}_t\}_{t=2}^N) \in \mathbb{S}^n_+ \times \mathcal{P}^{N-1}$.
Therefore, $\mathcal{M}_t(Q,\mathscr{P}_t)=\mathcal{M}_t(Q^\prime,\mathscr{P}_t^\prime)$ for $t=2,\ldots,N$. On the other hand, recall that any permutation matrix $\mathscr{P}_t$ is a 0-1 matrix with exactly one element that is one in each row and column. This means that for all $t$, $\mathcal{M}_t(Q,\mathscr{P}_t)$ is composed of all-zero and $\prod_{k=1}^{t-1}A_{cl}(k;Q)$ sub-matrix blocks. Together with the fact that $\mathcal{M}_t(Q,\mathscr{P}_t)=\mathcal{M}_t(Q^\prime,\mathscr{P}_t^\prime)$ for $t=2,\ldots,N$, it implies that $\mathscr{P}_t=\mathscr{P}_t^\prime$ and $\prod_{k=1}^{t-1} A_{cl}(k;Q)=\prod_{k=1}^{t-1} A_{cl}(k;Q^\prime)$ holds for $t=2,\ldots,N$. Hence, by induction $A_{cl}(t;Q)=A_{cl}(t;Q^\prime)$ holds for all $t=1,\ldots,N-1$. Now, if $N \geq n+1$, this implies that $Q=Q^\prime$ \citep[Theorem. ~2.1]{zhang2019inverse}. This gives the global identifiability of $\mathcal{M}(Q^\prime,\{\mathscr{P}_t^\prime\})$.
Finally, since $(Q^\prime, \{ \mathscr{P}_t^\prime\}_{t = 2}^N)$ was arbitrarily chosen from $\mathbb{S}^n_+ \times \mathcal{P}^{N-1}$,  $\mathcal{M}(Q^\prime,\{\mathscr{P}_t^\prime\}_{t=2}^N)$ is strictly globally identifiable.
\end{pf}

As a final note in this section, we observe that by identifying $\bar{Q}$ we implicitly also identify $\{\mathscr{P}_t\}_{t=2}^N$.
More precisely, if $\bar{Q}$ is identified, then by solving the forward problem \eqref{eq:forward_problem}, the estimates of each system's trajectory can be easily obtained; pairing these estimates with the data gives the permutations.
Moreover, as will be shown next, it turns out that $\bar{Q}$ can be identified without explicitly identifying the permutations.

\section{IOC for indistinguishable systems in the noiseless case}
\label{sec:noiseless}

After justifying the identifiability of the problem, we now investigate the IOC problem for indistinguishable systems in the noiseless case, namely, it is assumed that $y_t^i = x_t^{\phi_t(i)}$, $\forall t$ and $i$.
More precisely, we construct the IOC algorithm for indistinguishable observations as a convex optimization problem. However, before presenting the optimization problem, let us first sketch the main idea behind it.
To this end, we first note that the optimal solution to the forward problem \eqref{eq:forward_problem} is characterized by the DRE \eqref{eq:DRE}. However, this is a nonlinear equation in $\bar{P}_{t}$, and in order to tackle to problem we first relax it to a nonlinear inequality, namely
\begin{align*}
\bar{P}_t \preceq & \; A^T\bar{P}_{t+1}A+\bar{Q}- \nonumber \\
&\; A^T\bar{P}_{t+1}B(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A, \nonumber \\
& \; t=2:N-1 \nonumber \\
\bar{P}_N = & \; \bar{Q}.
\end{align*}
In particular, this relaxation means that for any system trajectory $\{x_t^{i}\}_{t=1}^N$, we have that for $t = 1,\ldots, N-1$
\begin{align*}
0 \leq & x_t^{iT}\Big(A^T\bar{P}_{t+1}A+\bar{Q}- \\
& A^T\bar{P}_{t+1}B(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A - \bar{P}_t \Big)x_t^{i}.
\end{align*}
Summing the above inequality, both for $t = 1, \ldots, N~-~1$ and $i = 1, \ldots, M$, taking traces on both sides, and using that the trace operator is invariant under cyclic permutation of the elements, we get the inequality
\begin{align*}
0 \leq & \sum_{t = 1}^{N-1} \trace \Big[\Big(A^T\bar{P}_{t+1}A+\bar{Q}- \\
& A^T\bar{P}_{t+1}B(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A - \bar{P}_t \Big)X_t X_t^{T }\Big].
\end{align*}
In fact, for trajectories that are optimal to \eqref{eq:forward_problem}, the above inequality is an equality.
Therefore, to make the relaxation as tight as possible, we are interested in making the right hand side of the above inequality as small as possible.
Finally, also note that
\begin{equation}\label{eq:YYT_is_XXT}
Y_{t}^{\phi} Y_{t}^{\phi T} = \sum_{i=1}^M x_{t}^{\phi_t(i)} x_{t}^{\phi_t(i) T} = \sum_{i=1}^M x_{t}^i x_{t}^{i T} = X_t^{} X_t^T,
\end{equation}
where the second equality follows since the order of the elements in the sum does not matter. This means that $ X_t^{} X_t^T$ can be computed from the observed, indistinguishable data.

To elaborate on the above idea, we first define the objective function as
\begin{align}\label{eq:def_H}
& H(Q,\{P_t\}) :=  -\trace(P_{1}Y_1^\phi Y_1^{\phi T}) \nonumber \\
& \: + \trace(P_NY_N^\phi Y_N^{\phi T}) +\sum_{t=1}^{N-1} \trace (QY_t^\phi Y_t^{\phi T})
\end{align}
and construct the following optimization problem to reconstruct $Q$:
\begin{subequations}\label{eq:opt_no_noise}
  \begin{align}
    & \underset{\substack{Q\in\mathbb{S}^n_+,\\ \{ P_t \in \mathbb{S}^n_+\} }}{\text{minimize}}
    & & H(Q,\{P_t\}) \label{eq:opt_no_noise_cost}\\
    & \text{subject to}
    & &
    \begin{bmatrix}
    B^TP_{t+1}B+I &B^TP_{t+1}A\\
    A^TP_{t+1}B &A^TP_{t+1}A+Q-P_t
    \end{bmatrix}\succeq 0,\nonumber\\
    & & &t=1,\ldots,N-1,\label{eq:LMI}\\
    & & &P_N=Q.\label{eq:P_N_constr}
  \end{align}
\end{subequations}
This is a semidefinite programming problem, and hence a convex optimization problem, that can be solved using standard numerical solvers, e.g., \citep{lofberg2004yalmip}.
Before we show that solving this optimization problem actually gives the correct estimate $\bar{Q}$ as unique optimal solution, we first present three lemmas that are useful for future analysis.

\begin{lemma}[Persistent excitation]\label{lem:persistent_excitation}
If $Y_1^\phi Y_1^{\phi T}\succ 0$, then it holds that $Y_t^\phi Y_t^{\phi T}\succ 0,\forall t=2,\ldots,N$.
\end{lemma}
\begin{pf}
By using the property $x_{t+1}^{i} = A_{cl}(t; \bar{Q}) x_t^{i}$, as well as \eqref{eq:YYT_is_XXT}, it follows that
\begin{align*}
Y_{t+1}^{\phi} Y_{t+1}^{\phi T} &= \sum_{i=1}^M x_{t+1}^{i} x_{t+1}^{i T}\\
&=A_{cl}(t; \bar{Q}) \left(\sum_{i=1}^M x_t^{i} x_t^{i T}\right) A_{cl}^T(t; \bar{Q}).
\end{align*}
Since for all $Q\in\mathbb{S}^n_+$, $A_{cl}(t;Q)$ is invertible for all $t=1,\ldots,N-1$ \citep{zhang2019inverse}, and since positive definiteness is invariant under congruence \cite[Obs.~7.1.8]{horn2013matrix},
by induction
the statement follows.
\end{pf}

\begin{remark}
The ``persistent excitation" condition $Y_1^\phi Y_1^{\phi T}\succ 0$ is equivalent to that there are $n$ linearly independent initial values $x_1^i$ amongst all $M$ initial values.
\end{remark}

\begin{lemma}\label{lem:H_lower_bound}
For any feasible solution $(Q,\{P_t\}_{t=1}^N)$ to \eqref{eq:opt_no_noise}, it holds that
\begin{align*}
H(Q,\{P_t\})+\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2 \ge 0,
\end{align*}
where $\{\bar{u}_t^{i}\}_{t=1}^{N-1}$ is the optimal control input sequence in \eqref{eq:forward_problem} that corresponds to the ``true" matrix $\bar{Q}$ and the trajectory $\{x_t^{i}\}_{t=1}^N$.
\end{lemma}

In particular, note that $\{\bar{u}_t^{i}\}_{t=1}^{N-1}$ in Lemma~\ref{lem:H_lower_bound} has nothing to do with the decision variable $(Q,\{P_t\})$ of the optimization problem. Therefore, the lemma effectively shows that the objective function $H$ is bounded from below on the feasible set in \eqref{eq:opt_no_noise}.

\begin{pf}
Since $\{P_t\}_{t=1}^N$, $Q$ is feasible, it is clear that $P_t\in\mathbb{S}^n_+$, $\forall t=1,\ldots,N$ and hence $B^TP_{t+1}B+I\succ 0,\forall t=1,\ldots,N-1$. Moreover, since \eqref{eq:LMI} holds, by positive semidefiniteness of the Schur complement \cite[p.~495]{horn2013matrix} it holds that
\begin{align}
&A^TP_{t+1}A-P_t+Q-A^TP_{t+1}B(B^TP_{t+1}B+I)^{-1}\nonumber\\
&\times B^TP_{t+1}A\succeq 0.\label{eq:schur_complement_LMI}
\end{align}
Introducing $K_t := -(B^TP_{t+1}B+I)^{-1}B^TP_{t+1}A$, we can re-write the above matrix inequalities as
\begin{align*}
(A+BK_t)^TP_{t+1}(A+BK_t)-P_t+Q+K_t^TK_t\succeq 0,\\
t=1,\ldots,N-1.
\end{align*}
Rearranging the terms,  we have
\begin{align}
&A^TP_{t+1}A-P_t+Q \succeq -K_t^TB^TP_{t+1}A \nonumber\\
&\quad -A^TP_{t+1}BK_t-K_t^T(B^TP_{t+1}B+I)K_t.\label{eq:expanded_LMI}
\end{align}
On the other hand, for the $i$'th trajectory, consider the following term
\begin{equation}
J_t^{i} = x_{t+1}^{i T}P_{t+1}x_{t+1}^{i} - x_t^{i T} P_t x_t^{i} + x_t^{i T} Q x_t^{i} + \|\bar{u}_t^{i}\|^2,\label{eq:J_def}
\end{equation}
where $\bar{u}_t^{i}$ is the control input that corresponds to $\bar{Q}$ and the trajectory $\{x_t^{i}\}_{t=1}^N$ in the forward problem \eqref{eq:forward_problem}. Using the fact that $x_{t+1}^{i}=(A+B\bar{K}_t)x_t^{i}$ and $\bar{u}_t^{i}=\bar{K}_t x_t^{i}$, it holds for \eqref{eq:J_def} that
\begin{align*}
&J_t^{i} =\nonumber\\
& x_t^{i T}\underbrace{\left[(A+B\bar{K}_t)^TP_{t+1}(A+B\bar{K}_t)-P_t+Q+\bar{K}_t^T\bar{K}_t\right]}_{=: \mathscr{H}_t}x_t^{i}.
\end{align*}
Expanding the terms of $\mathscr{H}_t$, we have
\begin{align*}
&\mathscr{H}_t = A^TP_{t+1}A-P_t+Q+\\
&\bar{K}_t^TB^TP_{t+1}A+A^TP_{t+1}B\bar{K}_t+\bar{K}_t^T(B^TP_{t+1}B+I)\bar{K}_t,
\end{align*}
and using the matrix inequality \eqref{eq:expanded_LMI} we get that
\begin{align}
&\mathscr{H}_t \succeq \label{eq:LMI_H_t}\\
&-K_t^TB^TP_{t+1}A-A^TP_{t+1}BK_t-K_t^T(B^TP_{t+1}B+I)K_t \nonumber \\
&+\bar{K}_t^TB^TP_{t+1}A+A^TP_{t+1}B\bar{K}_t+\bar{K}^T(B^TP_{t+1}B+I)\bar{K}_t. \nonumber
\end{align}
Recalling that $K_t = -(B^TP_{t+1}B+I)^{-1}B^TP_{t+1}A$ and using the fact that
\begin{align*}
&\bar{K}_t^TB^TP_{t+1}A\\
&=\bar{K}_t^T(B^TP_{t+1}B+I)(B^TP_{t+1}B+I)^{-1}B^TP_{t+1}A\\
&=-\bar{K}_t^T(B^TP_{t+1}B+I)K_t,
\end{align*}
and that
\begin{align*}
&K_t^TB^TP_{t+1}A\\
&=K_t^T(B^TP_{t+1}B+I)(B^TP_{t+1}B+I)^{-1}B^TP_{t+1}A\\
&=-K_t^T(B^TP_{t+1}B+I)K_t,
\end{align*}
the matrix inequality \eqref{eq:LMI_H_t} can be further rewritten as
\begin{align*}
&\mathscr{H}_t\succeq 2K_t^T(B^TP_{t+1}B+I)K_t-K_t^T(B^TP_{t+1}B+I)K_t\\
&-\bar{K}_t^T(B^TP_{t+1}B+I)K_t-K_t^T(B^TP_{t+1}B+I)\bar{K}_t\\
&+\bar{K}_t^T(B^TP_{t+1}B+I)\bar{K}_t\\
&=(K_t-\bar{K}_t)^T(B^TP_{t+1}B+I)(K_t-\bar{K}_t).
\end{align*}
Hence it holds that
\begin{align}
&J_t^{i} = x_t^{i T}\mathscr{H}_t x_t^{i}\nonumber\\
&\ge x_t^{i T}(K_t-\bar{K}_t)^T(B^TP_{t+1}B+I)(K_t-\bar{K}_t)x_t^{i}.\label{eq:J_t_inequality}
\end{align}

On the other hand, recall that $Y_t^\phi=\left[x_t^{\phi_t(1)}, \cdots, x_t^{\phi_t(M)}\right]$. Therefore, it follows that
\begin{align}
&\trace(P_tY_t^\phi Y_t^{\phi T}) = \trace(P_t\left[x_t^{\phi_t(1)},\cdots, x_t^{\phi_t(M)}\right]\begin{bmatrix}
x_t^{\phi_t(1) T}\\ \vdots \\ x_t^{\phi_t(M) T}
\end{bmatrix}) \nonumber \\
&=\trace \left(P_t\left(\sum_{i=1}^M x_t^{\phi_t(i)} x_t^{\phi_t(i) T} \right)\right) = \sum_{i=1}^M x_t^{\phi_t(i) T}P_t x_t^{\phi_t(i)} \nonumber \\
&= \sum_{i=1}^M x_t^{i T}P_t x_t^{i},\quad t=1,\ldots,N. \label{eq:trace_P_Y_YT}
\end{align}
Similarly, it also holds that $\trace(Q Y_t^\phi Y_t^{\phi T})=\sum_{i=1}^M x_t^{i T}Q x_t^{i}$, $t=2,\ldots,N$.
Together with \eqref{eq:J_def} and \eqref{eq:J_t_inequality}, we therefore have that
\begin{align}
& H(Q,\{P_t\}) + \sum_{i=1}^M \sum_{t=1}^{N-1} \|\bar{u}_t^{i}\|^2 = \sum_{i=1}^M \sum_{t=1}^{N-1} \|\bar{u}_t^{i}\|^2 + \nonumber \\
& \sum_{t=1}^{N-1} \left\{ \trace(P_{t+1}Y_{t+1}^\phi Y_{t+1}^{\phi T}) - \trace(P_tY_t^\phi Y_t^{\phi T}) + \trace(QY_t^\phi Y_t^{\phi T}) \right\} \nonumber \\
& = \! \sum_{t=1}^{N-1} \left\{ \sum_{i=1}^M \underbrace{x_{t+1}^{i T}P_{t+1}x_t^{i} - x_t^{i T} P_t x_t^{i} + x_t^{i T}Qx_t^{i} + \|\bar{u}_t^{i}\|^2}_{= J_t^{i}} \right\} \nonumber \\
& \ge \! \sum_{t=1}^{N-1} \! \left\{ \! \sum_{i=1}^M \left[x_t^{i T}(K_t-\bar{K}_t)^T(B^TP_{t+1}B+I)(K_t-\bar{K}_t)x_t^{i} \right] \! \right\} \nonumber \\
& \ge 0, \label{eq:H_plus_u_inequality}
\end{align}
which proves the statement.
\end{pf}

\begin{lemma}\label{lem:true_Q_optimal}
Suppose that $\bar{Q}$ is the true parameter that is used to generate the observations $\{Y_t^\phi\}_{t=2}^N$ with unknown permutations,  that  $\{\bar{u}_t^{i}\}_{t=1}^{N-1}$, for $i = 1, \ldots, M$, is the corresponding optimal control signal, and that $\left\{\bar{P}_t\right\}_{t=1}^N$ is the corresponding solution to the DRE \eqref{eq:DRE}. Then $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$ is an optimal solution to \eqref{eq:opt_no_noise}, and $H(\bar{Q},\{\bar{P}_t\}) + \sum_{i=1}^M \sum_{t=1}^{N-1} \|\bar{u}_t^{i}\|^2 = 0$.
\end{lemma}
\begin{pf}
Since $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$ satisfies DRE \eqref{eq:DRE}, \eqref{eq:schur_complement_LMI} as well as \eqref{eq:P_N_constr} holds,  and therefore \eqref{eq:LMI} holds for $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$. Thus $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$ is a feasible solution to \eqref{eq:opt_no_noise}.

Next, since $x_{t+1}^{i}=(A+B\bar{K}_t)x_t^{i}$ and $\bar{u}_t^{i} = \bar{K}_tx_t^{i}$, it follows that
\begin{equation*}
\begin{aligned}
& x_{t+1}^{i T}\bar{P}_{t+1}x_{t+1}^{i} - x_t^{i T}\bar{P}_tx_t^{i}+x_t^{i T}\bar{Q}x_t^{i}+\|\bar{u}_t^{i}\|^2\\
= & x_t^{i  T}\big[(A+B\bar{K}_t)^T\bar{P}_{t+1}(A+B\bar{K}_t)-\bar{P}_t+\bar{Q}\\
& + \bar{K}_t^T\bar{K}_t\big]x_t^{i }\\
= & x_t^{i  T}\big[A^T\bar{P}_{t+1}A - A^T\bar{P}_{t+1}B(B^T\bar{P}_{t+1}B+I)^{-1}B^T\bar{P}_{t+1}A \\
& + \bar{Q} - \bar{P}_{t} \big] x_t^{i } = 0,
\end{aligned}
\end{equation*}
which means that $H(\bar{Q},\{\bar{P}_t\})+\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2=0$.

Finally, by Lemma \ref{lem:H_lower_bound} we know that $H(Q,\{P_t\})+$ $\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2 \geq 0$ for all feasible solutions.
Since $\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2$
is a constant,
and since the lower bound is reached by $(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N)$, the latter is an optimal solution to \eqref{eq:opt_no_noise}.
\end{pf}
Now we are ready to present the main theorem for the noiseless case.
\begin{theorem}\label{thm:main_result_no_noise}
Assume that $N\ge n+1$, $Y_1^\phi Y_1^{\phi T}\succ 0$, and let $(Q^*,\{P_t^*\}_{t=1}^N)$ be an optimal solution to \eqref{eq:opt_no_noise}. Then $Q^*=\bar{Q}$, where $\bar{Q}$ is the ``true" parameter that is used to generate $\{Y_t^\phi\}_{t=2}^N$ with unknown permutations.
\end{theorem}
\begin{pf}
By Lemma \ref{lem:true_Q_optimal}, we know that $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$ is an optimal solution to \eqref{eq:opt_no_noise}. What remains to show is the uniqueness.

Since $\left(Q^*,\{P_t^*\}_{t=1}^N\right)$ is an optimal solution, it must be feasible, thus \eqref{eq:H_plus_u_inequality} also holds for $\left(Q^*,\{P_t^*\}_{t=1}^N\right)$. Moreover, by the assumption that $Y_1^\phi Y_1^{\phi T}\succ 0$ and Lemma~\ref{lem:persistent_excitation}, it holds that $Y_t^\phi Y_t^{\phi T}\succ 0$, $\forall t=2,\ldots,N$, and hence there exists a strictly positive definite matrix $(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}$ such that $Y_t^\phi Y_t^{\phi T}=(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}} (Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}$ \citep[p.~440]{horn2013matrix}.
Letting $K^*_t := -(B^TP_{t+1}^*B+I)^{-1}B^TP_{t+1}^*A$,
by further term manipulation in \eqref{eq:H_plus_u_inequality}, we have
\begin{align*}
& H(Q^*,\{P_t^*\})+\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2\ge\\
& \sum_{t=1}^{N-1} \! \left\{ \! \sum_{i=1}^M \! \left[x_t^{i T}(K_t^*-\bar{K}_t)^T(B^TP_{t+1}^*B+I)(K_t^*-\bar{K}_t)x_t^{i}\right] \! \right\}\\
& = \! \sum_{t=1}^{N-1} \! \left\{ \! \trace \! \left[ (K_t^*-\bar{K}_t)^T(B^T P_{t+1}^*B+I)(K_t^*-\bar{K}_t)Y_t^\phi Y_t^{\phi T} \right] \! \right\} \\
& = \! \sum_{t=1}^{N-1} \! \left\{ \left\|(B^T P_{t+1}^*B+I)^{\frac{1}{2}}(K_t^*-\bar{K}_t)(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}\right\|_F^2 \! \right\} \\
& \ge 0.
\end{align*}
As stated in Lemma \ref{lem:true_Q_optimal},  the lower bound zero in the above inequality is reached by $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$, and it also holds that $H(Q^*,\{P_t^*\}_{t=1}^N)+\sum_{i=1}^M\sum_{t=1}^{N-1}\|\bar{u}_t^{i}\|^2=0$ since $\left(Q^*,\{P_t^*\}_{t=1}^N\right)$ is also an optimal solution. Therefore, since all terms in the above sum are nonnegative, it must hold that
\begin{align*}
&\left\|(B^T P_{t+1}^*B+I)^{\frac{1}{2}}(K_t^*-\bar{K}_t)(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}\right\|_F=0,\\
&\Leftrightarrow (B^T P_{t+1}^*B+I)^{\frac{1}{2}}(K_t^*-\bar{K}_t)(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}=0,\\
&\Leftrightarrow K_t^*-\bar{K}_t=0,\quad t=1,\ldots,N-1
\end{align*}
since $(B^T P_{t+1}^*B+I)^{\frac{1}{2}}$ and $(Y_t^\phi Y_t^{\phi T})^{\frac{1}{2}}$ are invertible.

The above argument shows that $K_t^*=\bar{K}_t$ for all $t=1,\ldots,N-1$. Nevertheless, note that $\{K_t^*\}_{t=1}^{N-1}$ is not the control gain that is generated by $Q^*$ using DRE \eqref{eq:DRE}. Instead, it is defined by $K_t^* = -(B^TP_{t+1}^*B+I)^{-1}B^TP_{t+1}^*A$, for $t=1,\ldots,N-1$, where $(Q^*,\{P_t^*\}_{t=1}^{N-1})$ is an optimizer of \eqref{eq:opt_no_noise}. Hence the result from \citep[Theorem~ 2.1]{zhang2019inverse} can not be directly applied to show that $Q^*=\bar{Q}$.
Nevertheless, to show the latter, first note that we can always write $P_t^* = \bar{P}_t+\Delta P_t,t=1,\ldots,N$, and $Q^* = \bar{Q}+\Delta Q$, for some symmetric matrices $\Delta Q$ and $\Delta P_t$, for $t = 1, \ldots, N$. Since both $(Q^*,\{P_t^*\}_{t=1}^N)$ and $(\bar{Q},\{\bar{P}_t\}_{t=1}^N)$ are optimal solutions to \eqref{eq:opt_no_noise}, they must have the same optimal value, and hence
\begin{align}
&H(Q^*,\{P_t^*\}_{t=1}^N) - H(\bar{Q},\{\bar{P}_t\}_{t=1}^N)=0\nonumber\\
\implies& -\trace(\Delta P_1 Y_1^\phi Y_t^{\phi T})+\trace(\Delta P_N Y_N^\phi Y_N^{\phi T}) \nonumber\\
&+ \sum_{t=1}^{N-1}\trace(\Delta Q Y_t^\phi Y_t^{\phi T})=0\label{eq:Delta_P_1_Delta_P_N_Delta_Q},
\end{align}
where we have used the linearity of the trace operator.
On the other hand, it holds that
\begin{align*}
& \quad K_t^*=-(B^TP_{t+1}^*B+I)^{-1}B^TP_{t+1}^*A\\
\Longleftrightarrow & \quad (B^TP_{t+1}^*B+I)K_t^* = -B^TP_{t+1}^*A\\
\Longleftrightarrow & \quad B^TP_{t+1}^*(A+BK_t^*) = -K_t^*.
\end{align*}
Since $K_t^* = \bar{K}_t,\forall t=1,\ldots,N-1$, $A+BK_t^*=A+B\bar{K}_t = A_{cl}(t; \bar{Q})$, and by the fact that $A_{cl}(t;\bar{Q})$ is invertible for all $t=1,\ldots, N-1$ \citep{zhang2019inverse}, it follows that
\begin{align}
&B^T P_{t+1}^* = -K_t^*A_{cl}(t;\bar{Q})^{-1} = -\bar{K}_tA_{cl}(t;\bar{Q})^{-1}=B^T\bar{P}_{t+1}\nonumber\\
&\implies B^T\Delta P_{t+1} = 0,\forall t=1,\ldots,N-1.\label{eq:B^T_Delta_P}
\end{align}

Next, since $(Q^*,\{P_t^*\}_{t=1}^N)$ is feasible, \eqref{eq:schur_complement_LMI} also holds for $(Q^*,\{P_t^*\}_{t=1}^N)$. This means that
\begin{align}
&A^T(\bar{P}_{t+1}+\Delta P_{t+1})A-(\bar{P}_t+\Delta P_t) + (\bar{Q}+\Delta Q)\nonumber\\
&-A^T(\bar{P}_{t+1}+\Delta P_{t+1})B[B^T(\bar{P}_{t+1}+\Delta P_{t+1})B+I]^{-1}\nonumber\\
&B^T(\bar{P}_{t+1}+\Delta P_{t+1})A\succeq 0,\:t=1,\ldots,N-1\nonumber\\
&\bar{P}_N+\Delta P_N = \bar{Q}+\Delta Q.\nonumber
\end{align}
On the other hand, since $(\bar{Q},\{\bar{P}_t\}_{t=1}^N)$ satisfies \eqref{eq:DRE}, by also using \eqref{eq:B^T_Delta_P} we get that
\begin{align}
&A^T\Delta P_{t+1}A-\Delta P_t + \Delta Q \succeq 0,\;\; t=1,\ldots,N-1,\label{eq:Delta_P_Delta_Q_inequality}\\
&\Delta P_N=\Delta Q.\label{eq:Delta_P_Delta_Q}
\end{align}
Using \eqref{eq:B^T_Delta_P}, we can further manipulate the terms of \eqref{eq:Delta_P_Delta_Q_inequality} and get
\begin{align}
&\underbrace{(A+B\bar{K}_t)^T}_{A_{cl}(t;\bar{Q})}\Delta P_{t+1} \underbrace{(A+B\bar{K}_t)}_{A_{cl}(t;\bar{Q})} - \Delta P_t + \Delta Q\succeq 0,\nonumber\\
&\qquad\forall t=1,\ldots,N-1\nonumber\\
\implies &X_t^TA_{cl}(t;\bar{Q})^T\Delta P_{t+1}A_{cl}(t;\bar{Q})X_t - X_t^T\Delta P_t X_t \nonumber\\
&+ X_t^T\Delta Q X_t\succeq 0,\nonumber\\
\implies &X_{t+1}^T\Delta P_{t+1}X_{t+1}-X_t^T\Delta P_t X_t + X_t^T\Delta Q X_t\succeq 0,\nonumber\\
\implies &\trace(\Delta P_{t+1}X_{t+1}X_{t+1}^T)-\trace(\Delta P_t X_tX_t^T)\nonumber\\
&+\trace(\Delta QX_tX_t^T)\ge 0,t=1,\ldots,N-1, \nonumber\\
\implies &\trace(\Delta P_{t+1}Y_{t+1}^\phi Y_{t+1}^{\phi T})-\trace(\Delta P_t Y_t^\phi Y_t^{\phi T})\nonumber\\
&+\trace(\Delta QY_t^\phi Y_t^{\phi T})\ge 0,t=1,\ldots,N-1, \label{eq:trace_delta_ineq}
\end{align}
where $X_t=[x_t^1,\cdots,x_t^M]$, and where we also used that $\trace(\Delta P_tY_t^\phi Y_t^{\phi T}) = \trace(\Delta P_tX_tX_t^T)$ and that $\trace(\Delta P_{t+1}Y_{t+1}^\phi Y_{t+1}^{\phi T}) = \trace(\Delta P_{t+1}X_{t+1}X_{t+1}^T)$.

Summing \eqref{eq:trace_delta_ineq} from $t=1$ to $N-2$,  we have
\begin{align}
&\trace(\Delta P_{N-1}Y_{N-1}^\phi Y_{N-1}^{\phi T})-\trace(\Delta P_1 Y_1^\phi Y_1^{\phi T})\nonumber\\
&\qquad+\sum_{t=1}^{N-2}\trace(\Delta QY_t^\phi Y_t^{\phi T})\ge 0. \label{eq:trace_delta_ineq_sum}
\end{align}
In view of \eqref{eq:Delta_P_1_Delta_P_N_Delta_Q} and \eqref{eq:trace_delta_ineq_sum}, we have that
\begin{align*}
0=&-\trace(\Delta P_1 Y_1^\phi Y_1^{\phi T})+\trace(\Delta P_N Y_N^\phi Y_N^{\phi T}) \\
&+ \sum_{t=1}^{N-1}\trace(\Delta Q Y_t^\phi Y_t^{\phi T})\\
=&-\trace(\Delta P_1 Y_1^\phi Y_1^{\phi T})+\trace(\Delta P_{N-1} Y_{N-1}^\phi Y_{N-1}^{\phi T})\\
&+ \sum_{t=1}^{N-2}\trace(\Delta Q Y_t^\phi Y_t^{\phi T})-\trace(\Delta P_{N-1}Y_{N-1}^\phi Y_{N-1}^{\phi T})\\
&+\trace(\Delta P_N Y_N^\phi Y_N^{\phi T})+\trace(\Delta QY_{N-1}^\phi Y_{N-1}^{\phi T})\\
\ge&-\trace(\Delta P_{N-1}Y_{N-1}^\phi Y_{N-1}^{\phi T})+\trace(\Delta P_N Y_N^\phi Y_N^{\phi T})\\
&+\trace(\Delta QY_{N-1}^\phi Y_{N-1}^{\phi T})\\
=&-\trace(\Delta P_{N-1}X_{N-1}X_{N-1}^T)+\trace(\Delta P_N X_N X_N^T)\\
&+\trace(\Delta QX_{N-1}X_{N-1}^T)
\end{align*}
Since $X_N=A_{cl}(N-1;\bar{Q})X_{N-1}$ and by also using \eqref{eq:B^T_Delta_P}, from the equation above it follows that
\begin{align*}
0\ge& \trace\Big[ \Big(A_{cl}(N-1;\bar{Q})^T\Delta P_N A_{cl}(N-1;\bar{Q})-\Delta P_{N-1}\\
&+\Delta Q \Big)X_{N-1}X_{N-1}^T\Big]\\
=&\trace\Big[(A^T\Delta P_N A-\Delta P_{N-1}+\Delta Q)X_{N-1}X_{N-1}^T\Big].
\end{align*}
Since $X_{N-1}X_{N-1}^T$ is positive definite (by Lemma~\ref{lem:persistent_excitation}), it follows that $A^T\Delta P_N A-\Delta P_{N-1}+\Delta Q\preceq 0$. Together with \eqref{eq:Delta_P_Delta_Q_inequality}, we can therefore conclude that $A^T\Delta P_N A-\Delta P_{N-1}+\Delta Q= 0$.
Using \eqref{eq:B^T_Delta_P}, we therefore have that $A_{cl}(N-1; \bar{Q})^T\Delta P_N A_{cl}(N-1; \bar{Q})-\Delta P_{N-1}+\Delta Q= 0$, which implies that $-\trace(\Delta P_{N-1}X_{N-1}X_{N-1}^T)+\trace(\Delta P_N X_N X_N^T)+\trace(\Delta QX_{N-1}X_{N-1}^T)=0$. Therefore, summing \eqref{eq:trace_delta_ineq} from $t=1$ to $N-3$ and reiterating the above analysis, we can conclude that $A^T\Delta P_{N-1} A-\Delta P_{N-2}+\Delta Q= 0$. Doing this recursively, we have that
\begin{align}
A^T\Delta P_{t+1} A-\Delta P_t+\Delta Q= 0,\; t=1,\ldots, N-1.\label{eq:A^T_Delta_PA_Delta_P_Delta_Q}
\end{align}
Equipped with \eqref{eq:A^T_Delta_PA_Delta_P_Delta_Q}, we can use the same argument as in the proof of \citep[Theorem 2.1]{zhang2019inverse} to conclude that $\Delta Q=0$. Thus $Q^*=\bar{Q}$,  i.e., the optimal solution of \eqref{eq:opt_no_noise} is unique and hence the theorem follows.
\end{pf}

Before proceeding, we make a few remarks about the formulation in \eqref{eq:opt_no_noise}. As will be seen later, these remarks naturally hold also in the case of noisy observations.

\begin{remark}\label{rem:distinguishable_and_indistinguishable}
The optimization problem \eqref{eq:opt_no_noise} can also be applied to the IOC problem for ``distinguishable" systems, i.e., when the trajectories of the systems are available. Nevertheless, as one can see from \eqref{eq:opt_no_noise}, in this formulation of the problem there is no fundamental difference between IOC problems for ``distinguishable" and ``indistinguishable" systems.
\end{remark}

\begin{remark}\label{rem:scaling_num_agents}
The size of the variables and the number of variables and constraints in the optimization problem \eqref{eq:opt_no_noise} only depends on the size of the state space, $n$, and the length of the time horizon, $N$, respectively.
In particular, the overall size of the problem \eqref{eq:opt_no_noise} is independent of the number of systems, $M$.
The only quantity that scales with $M$ is the computations of the outer products $Y_t^\phi Y_t^{\phi T}$, $t = 1,\ldots, N$. The latter scales linearly in $M$ and can be done before solving the optimization problem.
Therefore, the problem can be efficiently solved for scenarios with a large number of systems.
\end{remark}

\section{IOC for indistinguishable systems in the noisy case}\label{sec:noisy}
Next, we extend the results to the case of noisy observations of the unpaired states. In particular, we show that a modified version of \eqref{eq:opt_no_noise} gives a statistically consistent estimate.
To this end, let $(\Omega, \mathcal{F},\mathbb{P})$ be a probability space which carries random vectors $\bm{x}_1^i\in\mathbb{R}^n$, $\{\bm{v}_t^i \in \mathbb{R}^n\}_{t = 1}^N$, for  $i = 1,\ldots, M$. Based on that, the following assumptions are made for the rest of the paper.

\begin{assumption}[I.I.D.~random variables]\label{ass:noise_iid_and_zero_mean}
The random vectors $\bm{x}_1^i\in\mathbb{R}^n$, $\{\bm{v}_t^i \in \mathbb{R}^n\}_{t = 1}^N$, for  $i = 1,\ldots, M$, are all independent. Moreover,  random vectors  $\bm{x}_1^i\in\mathbb{R}^n$, for  $i = 1,\ldots, M$, are identically distributed, and the random vectors $\bm{v}_t^i \in \mathbb{R}^n$, for $t = 1, \ldots, N$ and $i = 1,\ldots, M$, are  identically distributed. Finally, $\mathbb{E}[\bm{v}_t^{i}]=0$, and $\cov(\bm{v}_t^{i}, \bm{v}_t^{i}):=\Sigma$ is a priori known, where $\|\Sigma\|_F<\infty$.
\end{assumption}

\begin{assumption}[Stochastic persistent excitation]\label{ass:persistent_excitation_noisy}
It holds that $\cov(\bm{x}_1^{i},\bm{x}_1^{i})\succ 0$, where $\mathbb{E}[\|\bm{x}_1^{i}\|^2]<\infty$.
\end{assumption}

\begin{assumption}[Bounded parameter]\label{ass:Q_set_compact}
The ``true" $\bar{Q}$ that governs the systems lives in a compact set $\bar{\mathbb{S}}^n_+(\varphi)=\{Q\in\mathbb{S}^n_+ : \|Q\|_F^2\le \varphi\}$, for some $0 < \varphi < \infty$.
\end{assumption}

Our goal is to seek the ``true" $\bar{Q}$ in $\bar{\mathbb{S}}^n_+(\varphi)$.  Note that in practice, we can set $\varphi$ arbitrarily large if there is no prior knowledge on the norm bound of possible $Q$.

Equipped with the stochastic problem set-up, let the initial value $x_1^{i}$ be a realization of $\bm{x}_1^{i}$. Then the optimal state trajectory and optimal control input of the ``forward" optimal control \eqref{eq:forward_problem} can be seen as mappings from $\Omega$ to $\mathbb{R}^n$ and $\mathbb{R}^m$, respectively.
This means that the states and control signals are in fact stochastic variables, which are parameterized by $Q$ and implicitly determined by
\begin{align}
&\{\bar{\bm{x}}_t^{i}(\omega)\}_{t=2}^N,\{\bar{\bm{u}}_t^{i}(\omega)\}_{t=1}^{N-1} = \label{eq:stochastic_forward_problem} \\
& \quad \left\{
\begin{array}{cl}
\argmin & \; J\left(\{\bm{x}_t^{i}(\omega)\}_{t=2}^N,\{\bm{u}_t^{i}(\omega)\}_{t=1}^{N-1}; \bar{Q}, \bm{x}_1^{i}(\omega) \right) \nonumber \\
\text{subject to} & \; \eqref{eq:forward_problem_dynamics}, \eqref{eq:forward_problem_init_cond}, \nonumber
\end{array}
\right.
\end{align}
for all $\omega\in \Omega$, and where $J$ is the cost function in \eqref{eq:forward_problem_cost}.
From now on, we omit the ``bar'' and simply write $\{\bm{x}_t^{i}\}$ to denote the corresponding optimal state for the sake of brevity.
Next, we assume that the noisy observations of a state is given by  $\tilde{\bm{x}}_t^i = \bm{x}_t^i + \bm{v}_t^i$, and that the observed data thus take the form
$\bm{Y}_t^{\phi} = [\bm{y}_t^{1}, \ldots, \bm{y}_t^{M}] = [\tilde{\bm{x}}_t^{\phi_t(1)}, \ldots, \tilde{\bm{x}}_t^{\phi_t(M)}]$.
Written in a form similar to \eqref{eq:data_model_noise_free}, the measured output can be expressed as
\begin{align*}
\bm{Y}_t^\phi = \left( \bm{X}_t + \bm{V}_t \right)\mathscr{P}_t = \! \left( \! \left( \prod_{k=1}^{t-1}A_{cl}(k; \bar{Q}) \bm{X}_1 \! \right) \! + \! \bm{V}_t \! \right)\mathscr{P}_t,
\end{align*}
where $\bm{X}_t = [\bm{x}_t^1, \ldots, \bm{x}_t^M]$ and $\bm{V}_t = [\bm{v}_t^1, \ldots, \bm{v}_t^M]$.
This together with Assumption~\ref{ass:noise_iid_and_zero_mean} implies that the columns of $\bm{Y}_t^\phi$ are I.I.D.
A calculation similar to \eqref{eq:trace_P_Y_YT} then shows that
\begin{equation*}
\mathbb{E}\left[ \trace \left( Q \bm{Y}_t^\phi \bm{Y}_t^{\phi T} \right)  \right] = \mathbb{E}\left[ \trace \left( Q \bm{X}_t \bm{X}_t^{T} \right) \right] + M\trace \left( Q \Sigma \right)
\end{equation*}
where $\Sigma$ is the covariance of the noise.
In particular, the last equality holds since $\bm{x}_t ^{i}= \prod_{k = 1}^{t-1}A_{cl}(k; \bar{Q}) \bm{x}_1^{i}$, and hence $\bm{x}_t^{i}$ is a stochastic variable that is independent of $\bm{v}_t^{i}$, for $t = 1, \ldots, N$ and $i = 1,\ldots M$.
Similar expressions hold for the other terms in \eqref{eq:def_H}, and based on this we consider the problem
\begin{subequations}\label{eq:opt_noise}
  \begin{align}
    & \underset{\substack{Q\in\bar{\mathbb{S}}^n_+(\varphi),\\ \{P_t\in\mathbb{S}^n_+\}}}{\text{minimize}}
    & & \mathbb{E}\left[ H_S^{\bm{Y}}(Q,\{P_t\}) \right] \label{eq_opt_noise_cost} \\
    & \text{subject to}
    & & \eqref{eq:LMI} \text{ and } \eqref{eq:P_N_constr},
  \end{align}
where
\begin{align}\label{eq:def_H_stochastic}
& H_S^{\bm{Y}}(Q,\{P_t\}) := \frac{1}{M}\Bigg[ -\trace(P_{1}\bm{Y}_1^{\phi } \bm{Y}_1^{\phi T}) \nonumber \\
& + \trace(P_N \bm{Y}_N^{\phi} \bm{Y}_N^{\phi T}) +\sum_{t=1}^{N-1} \trace (Q \bm{Y}_t^{\phi } \bm{Y}_t^{\phi T}) \nonumber \\
& + M\big( \trace \left( P_1\Sigma \right) - \trace(P_N \Sigma) - (N-1)\trace (Q\Sigma) \big) \Bigg].
\end{align}
\end{subequations}
In particular, note that by  a direct calculation it follows that the cost function \eqref{eq_opt_noise_cost} can be written as
\begin{align}
& \mathbb{E}\left[ H_S^{\bm{Y}}(Q,\{P_t\}) \right] = \frac{1}{M}\mathbb{E}\Big[-\trace(P_{1}\bm{X}_1 \bm{X}_1^{T}) \nonumber \\
& + \trace(P_N \bm{X}_N \bm{X}_N^{T}) + \sum_{t=1}^{N-1} \trace (Q \bm{X}_t \bm{X}_t^{T}) \Big].  \label{eq:H_S^y_equivalence}
\end{align}
With this, we can now prove the following result akin Theorem~\ref{thm:main_result_no_noise}.

\begin{proposition}\label{prop:stoc_prob_unique_sol}
Let $\bar{Q}$ be the true parameter in \eqref{eq:stochastic_forward_problem}, and let $\left\{\bar{P}_t\right\}_{t=1}^N$ be the corresponding solution to the DRE \eqref{eq:DRE}. Under Assumptions~\ref{ass:noise_iid_and_zero_mean}, \ref{ass:persistent_excitation_noisy}, and \ref{ass:Q_set_compact}, if $N\ge n+1$, then $\left(\bar{Q},\left\{\bar{P}_t\right\}_{t=1}^N\right)$ is the unique optimal solution to \eqref{eq:opt_noise}.
\end{proposition}

\begin{pf}
Since \eqref{eq:H_S^y_equivalence} holds, by adapting the arguments in the proof of Lemma~\ref{lem:H_lower_bound} it follows that
\begin{align}
&\mathbb{E}[H_S^{\bm{Y}}(Q,\{P_t\})] + \frac{1}{M}\sum_{i = 1}^M \sum_{t=1}^{N-1}\mathbb{E}\left[\|\bar{\bm{u}}_t^{i}\|^2\right]\nonumber\\
&\ge \frac{1}{M}\sum_{i = 1}^M \sum_{t=1}^{N-1} \mathbb{E} \left[\bm{x}_t^{iT}(K_t-\bar{K}_t)^T(B^TP_{t+1}B+I)(K_t-\bar{K}_t)\bm{x}_t^{i}\right]\nonumber\\
&=\frac{1}{M}\sum_{t=1}^{N-1} \trace\left[ (K_t-\bar{K}_t)^T(B^TP_{t+1}B+I)(K_t-\bar{K}_t)\mathbb{E}\left[\bm{X}_t\bm{X}_t^T\right]\right]\nonumber\\
&\ge 0,\label{eq:stochastic_H_lower_bound}
\end{align}
where $\bar{\bm{u}}_t^{i}$ is the stochastic optimal control input signal of the system $i$ at time instant $t$.
Next, by using \eqref{eq:stochastic_H_lower_bound} and following along the lines of the proof of Lemma \ref{lem:true_Q_optimal}, it can be seen that $\left(\bar{Q},\{\bar{P}_t\}\right)$ is an optimal solution to \eqref{eq:opt_noise}, and that
\begin{align}
\mathbb{E} \left[H_S^{\bm{Y}}(\bar{Q},\{\bar{P}_t\})\right] + \frac{1}{M}\sum_{i = 1}^M \sum_{t=1}^{N-1}\mathbb{E}\left[\|\bar{\bm{u}}_t^{i}\|^2\right]=0.\label{eq:lower_bound_attained_noisy}
\end{align}
Therefore, what remains is to show the uniqueness of the optimal solution. To this end, by Assumption \ref{ass:persistent_excitation_noisy} the covariance matrix $\cov(\bm{x}_1^{i},\bm{x}_1^{i})$ is strictly positive definite, and therefore the second-order moment of the initial value $\mathbb{E}[\bm{x}_1^{i} \bm{x}_1^{iT}] = \cov(\bm{x}_1^{i}, \bm{x}_1^{i})+ \mathbb{E}[\bm{x}_1^{i}]\mathbb{E}[\bm{x}_1^{i}]^T\succ 0$. By the fact that $A_{cl}(t;Q)$ is invertible for all $Q\in\mathbb{S}^n_+$, it holds that
\begin{align*}
&\mathbb{E}[\bm{x}_t^{i} \bm{x}_t^{iT}] = \left[\prod_{k=1}^{t-1} A_{cl}(k;Q)\right]\mathbb{E}[\bm{x}_1^{i} \bm{x}_1^{iT}]\left[\prod_{k=t-1}^{1} A_{cl}^T(k;Q)\right]\succ 0,
\end{align*}
cf.~Lemma~\ref{lem:persistent_excitation}, and hence $\mathbb{E}[\bm{X}_t\bm{X}_t^T] \succ 0$.
Now suppose that there exists some other $(Q^*,\{P^*_t\})$ that is also optimal to \eqref{eq:opt_noise}. By \eqref{eq:stochastic_H_lower_bound}, \eqref{eq:lower_bound_attained_noisy},  we have
\begin{align*}
&0 = \mathbb{E}[H_S^{\bm{Y}}(Q^*,\{P^*_t\})] + \frac{1}{M}\sum_{i = 1}^M \sum_{t=1}^{N-1}\mathbb{E}\left[\| \bar{\bm{u}}^i_t \|^2\right]\\
&\ge \frac{1}{M}\sum_{t=1}^{N-1} \trace \big[(K_t^*-\bar{K}_t)^T(B^TP_{t+1}B+I)\\
&\quad \times (K_t^*-\bar{K}_t)\mathbb{E}\left[\bm{X}_t\bm{X}_t^T\right]\big]\\
&=\frac{1}{M}\sum_{t=1}^{N-1}\left\|(B^TP_{t+1}B+I)^{\frac{1}{2}}(K_t^*-\bar{K}_t)\mathbb{E}\left[\bm{X}_t\bm{X}_t^T\right]^{\frac{1}{2}}\right\|_F^2\\
&\ge 0,
\end{align*}
which together with the fact that $\mathbb{E}[\bm{X}_t\bm{X}_t^T]\succ 0$ implies that
\begin{align*}
(B^TP_{t+1}B+I)^{\frac{1}{2}}(K_t^*-\bar{K}_t)\mathbb{E}\left[\bm{X}_t\bm{X}_t^T\right]^{\frac{1}{2}}=0,
\end{align*}
for $t=1,\ldots,N-1$. The latter in turn implies that
\begin{align*}
K_t^*-\bar{K}_t=0,\forall t=1,\ldots,N-1,
\end{align*}
where $K_t^*$ denote the same expression as the one in the proof of Theorem \ref{thm:main_result_no_noise}.
To this end, following the same analysis as in the proof of Theorem \ref{thm:main_result_no_noise}, we can conclude that $Q^*=\bar{Q}$ and hence the statement holds.
\end{pf}

Since the distribution of the initial values $\bm{x}_1^{i}$ and the additive noise $\bm{v}_t^{i}$ are not known, we can not express the expected value of the objective function \eqref{eq:def_H_stochastic} explicitly and hence it is not possible to solve \eqref{eq:opt_noise} directly. Therefore, we derive an empirically estimate of the expectation based on the observations. To this end, first recall that due to Assumption~\ref{ass:noise_iid_and_zero_mean}, $\bm{y}_t^i$ are I.I.D for $i=1,\ldots,M$. This means that we can rewrite the cost \eqref{eq_opt_noise_cost} as
\begin{align}
& \mathbb{E}\left[ H_S^{\bm{Y}}(Q,\{P_t\}) \right] = \nonumber \\
&\mathbb{E}\Big[ -\trace(P_{1} \bm{y}_1 \bm{y}_1^T) + \trace( P_N \bm{y}_N \bm{y}_N^{T}) +\sum_{t=1}^{N-1} \trace (Q \bm{y}_t \bm{y}_t^{T})\Big] \nonumber \\
& + \trace \left( P_1\Sigma \right) - \trace(P_N \Sigma) - (N-1)\trace (Q\Sigma), \label{eq:H_stochastic_rewritten}
\end{align}
where, for $t = 1, \ldots, N$, $\bm{y}_t$ is a random variable with the same distribution as $\bm{y}_t^{i}$ for $i = 1, \ldots, M$.
This means that
an empirical estimate of the expectation of \eqref{eq:def_H_stochastic} can be obtained as
\begin{align*}
& \mathbb{E} \left[\left( -\bm{y}_1^{T}P_1 \bm{y}_1 + \bm{y}_N^{T} P_N \bm{y}_N + \sum_{t= 1}^{N-1} \bm{y}_t^{T} Q \bm{y}_t \right) \right] \\
& \approx \frac{1}{M} \sum_{i = 1}^M \left( -\bm{y}_1^{iT}P_1 \bm{y}_1^{i} + \bm{y}_N^{iT} P_N \bm{y}_N^{i} + \sum_{t= 1}^{N-1} \bm{y}_t^{iT} Q \bm{y}_t^i \right).
\end{align*}

Based on this, we formulate the estimation problem
\begin{subequations}\label{eq:opt_noise_empirical}
  \begin{align}
    & \underset{\substack{Q\in\bar{\mathbb{S}}^n_+(\varphi),\\ \{P_t\in\mathbb{S}^n_+\}}}{\text{minimize}}
    & & H_{E}^{\bm{Y}}(Q,\{P_t\}) \\
    & \text{subject to}
    & & \eqref{eq:LMI} \text{ and } \eqref{eq:P_N_constr},
  \end{align}
where
\begin{align}\label{eq:def_H_stochastic_empirical}
& H_{E}^{\bm{Y}}(Q,\{P_t\}) := \frac{1}{M}\Big[-\trace(P_{1}\bm{Y}_1^\phi \bm{Y}_1^{\phi T}) \nonumber \\
& \; + \trace(P_N \bm{Y}_N^\phi \bm{Y}_N^{\phi T}) +\sum_{t=1}^{N-1} \trace (Q \bm{Y}_t^\phi \bm{Y}_t^{\phi T}) \Big]\nonumber \\
& \; + \big( \trace \left( P_1\Sigma \right) - \trace(P_N \Sigma) - (N-1)\trace (Q\Sigma) \big).
\end{align}
\end{subequations}
The problem \eqref{eq:opt_noise_empirical} defines the estimator, and for a given realization $\{ Y^\phi_t \}_{t=1}^N$ of the stochastic variables $\{ \bm{Y}_t^{\phi} \}_{t = 1}^N$, it can be solved in order to obtain an estimate. In particular, we use the notation $H_{E}^{\bm{Y}}(Q,\{P_t\})|_{\bm{Y}=Y}$ to denote the cost function \eqref{eq:def_H_stochastic_empirical} evaluated at a particular realization. We now want to show that this estimator is in fact (asymptotically) statistically consistent.
However, note that since we approximate the expected value in the objective function by the empirical average, the objective function changes and hence the ``bounded-from-below" argument \eqref{eq:stochastic_H_lower_bound} does not necessarily hold for $H_{E}^{\bm{Y}}(Q,\{P_t\})|_{\bm{Y} = Y}$ on the domain
\begin{equation}\label{eq:domain_D}
\mathscr{D} := \{Q\in\bar{\mathbb{S}}^n_+(\varphi),\{P_t\in\mathbb{S}^n_+\}:\eqref{eq:LMI},\eqref{eq:P_N_constr}\}.
\end{equation}
This issue needs to be addressed in order to make \eqref{eq:opt_noise_empirical} well-posed. This is an important first step towards showing that the estimator is statistically consistent.

\begin{lemma}\label{lem:approx_problem_bounded_from_below}
The domain $\mathscr{D}$ in \eqref{eq:domain_D} is compact, and $H_{E}^{\bm{Y}}(Q,\{P_t\})|_{\bm{Y}=Y}$ is bounded on $\mathscr{D}$.
\end{lemma}
\begin{pf}
Consider the domain $\mathscr{D}$ and recall that, by the property of Schur complement, \eqref{eq:schur_complement_LMI} holds on the feasible domain \eqref{eq:LMI} and \eqref{eq:P_N_constr}.  Since the Frobenius norm is monotone with respect to the Loewner partial order,  it holds that
\begin{align*}
&\|P_t\|_F\le \\
&\|A^TP_{t+1}A+Q-A^TP_{t+1}B(B^TP_{t+1}B+I)^{-1}B^TP_{t+1}A\|_F
\end{align*}
By the Cauchy-Schwarz and the triangular inequality, we have that
\begin{align}
&\|P_t\|_F\le \|A\|_F^2\cdot \|P_{t+1}\|_F+\|Q\|_F \label{eq:norm_ineq_riccati} \\
&\qquad +\|A\|_F^2\cdot\|B\|_F^2\cdot\|P_{t+1}\|_F^2\cdot\|(B^TP_{t+1}B+I)^{-1}\|_F. \nonumber
\end{align}
Next, since $P_{t} \in \mathbb{S}^n_+$, it holds that $B^TP_tB+I \succeq I$, and hence that $ (B^TP_tB+I)^{-1} \preceq I$ \citep[Cor.~7.7.4]{horn2013matrix}. By monotonicity of  the Frobenius norm with respect to the Loewner partial order, it therefore follows that $\|(B^TP_tB+I)^{-1}\|_F \le \|I\|_F = \sqrt{n}$. Now,  since $\|Q\|_F^2\le \varphi$ and $P_{N} = Q$, using this together with \eqref{eq:norm_ineq_riccati}, it follows that $\|P_{N-1}\|_F$ is bounded. Recursively applying this backwards for the time indices $t$, it follows that $\|P_t\|_F$ is bounded for all $t=1,\ldots,N$. This implies that the domain $\mathscr{D}$ is compact. Finally, since $H_{E}^{\bm{Y}}(Q,\{P_t\})|_{\bm{Y}=Y}$ is continuous, it is bounded on $\mathscr{D}$.
\end{pf}
\begin{remark}
Note that Assumption \ref{ass:Q_set_compact} is critical in the proof of Lemma \ref{lem:approx_problem_bounded_from_below}. In fact,  $H_{E}^{\bm{Y}}(Q,\{P_t\})|_{\bm{Y} = Y}$ might not be bounded from below if we only assume $Q\in\mathbb{S}^n_+$. To see this, assume that there exists a realization $\{Y_t^{\phi}\}$ of $\{\bm{Y}_t^{\phi }\}$ such that $Y_1^\phi Y_1^{\phi T} - M\Sigma=0$ and $Y_t^\phi Y_t^{\phi T}-M\Sigma\prec 0$, $t=2,\ldots,N$. Let $(Q,\{P_t\})$ satisfies DRE \eqref{eq:DRE}, and note that then $(\alpha Q,\{\alpha P_t\})$ also satisfies DRE \eqref{eq:DRE} for any positive $\alpha$.  Therefore, $(\alpha Q,\{\alpha P_t\})\in \mathscr{D}^\prime$ for all $\alpha >0$, where $\mathscr{D}^\prime=\left\{Q\in\mathbb{S}^n_+,\{P_t\in\mathbb{S}^n_+\}:\eqref{eq:LMI},\eqref{eq:P_N_constr}\right\}$, and hence the latter is not a bounded set. Moreover, as $\alpha\rightarrow \infty$ it holds that
\begin{align*}
&H_{E}^{\bm{Y}}(\alpha Q,\{\alpha P_t\})|_{\bm{Y}=Y} = \frac{1}{M} \Bigg[ - \trace\left(\alpha P_1(Y_1^\phi Y_1^{\phi T}-M\Sigma)\right)\\
& + \trace\left(\alpha P_N(Y_N^\phi Y_N^{\phi T}-M\Sigma)\right)\\
& + \sum_{t=1}^{N-1} \trace\left(\alpha Q(Y_t^\phi Y_t^{\phi T}-M\Sigma)\right) \Bigg] \rightarrow -\infty,
\end{align*}
hence $H_{E}^{\bm{Y}}(\alpha Q,\{\alpha P_t\})|_{\bm{Y}=Y}$ is not bounded from below on $\mathscr{D}^\prime$.
\end{remark}

Next, we show that the ``Uniform Law of Large Numbers" holds for $H_{E}^{\bm{Y}}(Q,\{P_t\})$.

\begin{lemma}[Uniform law of large numbers]\label{lem:uniform_law_of_large_number}
Under Assumption \ref{ass:noise_iid_and_zero_mean}, \ref{ass:persistent_excitation_noisy} and \ref{ass:Q_set_compact}, it holds that
  \begin{align*}
  \sup_{(Q,\{P_t\})\in\mathscr{D}} \big|H_{E}^{\bm{Y}}(Q,\{P_t\}) - \mathbb{E}\left[H_S^{\bm{Y}}(Q,\{P_t\})\right]\big|\overset{a.s.}{\rightarrow} 0,
  \end{align*}
as $M \to \infty$.
\end{lemma}
\begin{pf}
It is clear that $H_S^{\bm{Y}}(Q,\{P_t\})$ is continuous with respect to $\{ Y^\phi_t \}_{t=1}^N$ and therefore it is a measurable function of $\{ Y^\phi_t \}_{t=1}^N$ for each $Q$ and $\{P_t\}$.
On the other hand, Assumption \ref{ass:persistent_excitation_noisy} implies that $\mathbb{E}[\|\bm{x}_t^{i} \|^2]<\infty$ for $i = 1,\ldots, M$ \citep[cf.~the proof of Theorem 4.1]{zhang2019inverse}. By Assumption \ref{ass:noise_iid_and_zero_mean}, we have $\mathbb{E}[\|\bm{v}_t^{i} \|^2]<\infty$.  Since $\bm{x}_t^{i}$ is independent of $\bm{v}_t^{i}$, and since $\mathbb{E}[\bm{v}_t^{i}]=0$, it follows that
\begin{align*}
\mathbb{E}[\|\bm{y}_t^{i}\|^2]&=\mathbb{E}[(\bm{x}_t^{i}+\bm{v}_t^{i})^T(\bm{x}_t^{i}+\bm{v}_t^{i})]=\mathbb{E}[\|\bm{x}_t^{i}\|^2]+\mathbb{E}[\|\bm{v}_t^{i}\|^2]\\
&< \infty.
\end{align*}
In addition, from Lemma \ref{lem:approx_problem_bounded_from_below}, we know that there exists constants $\{\bar{\varphi}_t\}_{t=1}^N$ such that for all $(Q,\{P_t\})\in\mathscr{D}$ we have that $\|P_t\|_F\le \bar{\varphi}_t$.
Using the form of \eqref{eq_opt_noise_cost} given in \eqref{eq:H_stochastic_rewritten}, by the Cauchy-Schwarz and the triangular inequality it therefore holds that
\begin{align*}
&H_S^{\bm{Y}}(Q,\{P_t\}) \\
&= - \trace(P_1\bm{y}_1\bm{y}_1^T) + \trace(P_N\bm{y}_N\bm{y}_N^T)+\sum_{t=1}^{N-1}\trace(Q\bm{y}_t\bm{y}_t^T)\\
&\quad + \trace(P_1\Sigma)-\trace(P_N\Sigma)-(N-1)\trace(Q\Sigma)\\
&\le \left|\bm{y}_1^TP_1\bm{y}_1\right|+\left|\bm{y}_N^TP_N\bm{y}_N\right|+\sum_{t=1}^{N-1}\left|\bm{y}_t^TQ\bm{y}_t\right|\\
&\quad +\left| \trace(P_1\Sigma)\right|+\left| \trace(P_N\Sigma)\right|+(N-1)\left|\trace(Q\Sigma)\right|\\
&\le \|\bm{y}_1\|^2\|P_1\|_F+\|\bm{y}_N\|^2\|P_N\|_F+\sum_{t=1}^{N-1}\|\bm{y}_t\|^2\|Q\|_F\\
&\quad +\|P_1\|_F\|\Sigma\|_F+\|P_N\|_F\|\Sigma\|_F+(N-1)\|Q\|_F\|\Sigma\|_F\\
&\le \bar{\varphi}_1(\|\bm{y}_1\|^2+\|\Sigma\|_F)+\bar{\varphi}_N(\|\bm{y}_N\|^2+|\Sigma\|_F)\\
&\quad+\varphi\sum_{t=1}^{N-1}(\|\bm{y}_t\|^2+\|\Sigma\|_F):=d(\{\bm{y}_t\}),
\end{align*}
and it is clear that $\mathbb{E}[d(\{\bm{y}_t\})]<\infty$ since $\mathbb{E}[\|\bm{y}_t^{i} \|^2]<\infty$. Therefore, by \citep[Thm.~2]{jennrich1969asymptotic} the result follows.
\end{pf}

We are now ready to prove the main result of this section.

\begin{theorem}[Statistical consistency]\label{thm:statistical_consistency}
Suppose that $(Q_M^*,\{P_{t,M}^*\}_{t=1}^N)$ is an optimal solution to \eqref{eq:opt_noise_empirical} when observing $M$ systems. Then $Q_M^*\overset{p}\rightarrow \bar{Q}$ as $M\rightarrow \infty$, where $\bar{Q}$ is the true parameter used in the objective function of ``forward" problem \eqref{eq:stochastic_forward_problem}.
\end{theorem}

\begin{pf}
The theorem is proved by showing that all the conditions in \cite[Thm.~5.7]{van1998asymptotic} are satisfied.
To this end, the first condition follows from Lemma \ref{lem:uniform_law_of_large_number}, since convergence a.s.~implies convergence in probability \citep[Lem~3.2]{kallenberg1997foundations}.
Next, the second condition holds since by Proposition~\ref{prop:stoc_prob_unique_sol} the optimal solution to \eqref{eq:opt_noise} is unique, together with the fact that $\mathcal{D}$ is compact (see \cite[p.~46]{van1998asymptotic}).
Therefore, all conditions in \cite[Thm.~5.7]{van1998asymptotic} are satisfied, and the statement hence follows.
\end{pf}

\section{On numerical ill-conditioning}\label{sec:ill-cond}
Proposition~\ref{prop:global_identifiability} shows that the model is globally identifiable, and Theorem~\ref{thm:main_result_no_noise} shows that the optimization problem \eqref{eq:opt_no_noise} has a unique optimal solution at the ``true" $\bar{Q}$. Hence, in theory the latter can be recovered by solving the optimization problem.
Nevertheless, recovering this optimal solution turns out to be numerically difficult, in particular for certain problem instances - this will be demonstrated with examples in Section~\ref{sec:numerics}. Here, we argue that this has to do with an intrinsic numerical ill-conditioning of the inverse problem for these problem instances.

To this end, recall that Pontryagin's Maximum Principle (PMP) gives a necessary and sufficient condition for optimality in the forward problem \eqref{eq:forward_problem}.  Namely, $\{x_t\}_{t=1}^N$ and $\{u_t\}_{t=1}^{N-1}$ are the optimal trajectory and control signal of \eqref{eq:forward_problem}, respectively, if and only if there exists adjoint states $\{\lambda_t\}_{t=2}^N$ such that
\begin{subequations}\label{eq:PMP_original}
\begin{align}
&\lambda_t=A^T\lambda_{t+1}+Qx_t,\:t=2,\ldots,N-1,\label{eq:PMP_original_1}\\
&\lambda_N=Qx_N,\label{eq:PMP_original_2}\\
&u_t=-B^T\lambda_{t+1},\:t=1,\ldots,N-1. \label{eq:PMP_original_3}
\end{align}
\end{subequations}
Based on \eqref{eq:PMP_original_1}--\eqref{eq:PMP_original_2}, we can write a linear system of equations for the adjoint variables of the $i$th system, namely
\[
\begin{bmatrix}
I &-A^T\\
  &I &\ddots\\
  &  &\ddots & -A^T\\
  & & &I
\end{bmatrix}
\underbrace{\begin{bmatrix}
\lambda_2^i\\\vdots\\\lambda_N^i
\end{bmatrix}}_{=: \lambda^i}=(I\otimes Q)
\underbrace{\begin{bmatrix}
x_2^i\\\vdots\\x_N^i
\end{bmatrix}}_{=: x_{2:N}^i}.
\]
Solving this for the adjoint variables gives
\[
\lambda^i
=
\begin{bmatrix}
I &A^T &(A^T)^2 &\cdots &(A^T)^{N-2}\\
  &I &A^T&\cdots &(A^T)^{N-3}\\
  &  &\ddots &\ddots &\vdots\\
  &  &      &I&A^T\\
  &  &      & &I
\end{bmatrix}(I\otimes Q)
x_{2:N}^i,
\]
and substituting the latter into \eqref{eq:PMP_original_3} we obtain
\begin{align}
&-\underbrace{\begin{bmatrix}
u_1^i\\\vdots \\u_{N-1}^i
\end{bmatrix}}_{=: u^i}
=(I_{N-1} \otimes B^T)\lambda^i\nonumber\\
&= \! \underbrace{\begin{bmatrix}
B^T \! &B^TA^T \! &B^T(A^T)^2 \! &\cdots \! &B^T(A^T)^{N-2}\\
      &B^T \!     &B^TA^T \!     &\cdots \! &B^T(A^T)^{N-3}\\
      &           &\ddots \!     &\ddots \! &\vdots\\
      &           &              &B^T \!    &B^TA^T\\
      &           &              &          &B^T
\end{bmatrix}}_{= [\mathcal{S}_2\Gamma,\cdots,\mathcal{S}_N\Gamma]=\mathcal{S}(I_{N-1}\otimes\Gamma)} \! (I_{N-1} \otimes Q)
x_{2:N}^i\label{eq:PMP_u},
\end{align}
where $\mathcal{S} := [\mathcal{S}_2,\cdots,\mathcal{S}_N]$, $\mathcal{S}_N := I_{m(N-1)}$, $\mathcal{S}_{N-k} \in \mathbb{R}^{m(N-1) \times m(N-1)}$ is a block-matrix with identity matrices of size $n \times n$ on the $k$th upper block-diagonal, and
\[
\Gamma :=
\begin{bmatrix}
(A)^{N-2}B & (A)^{N-3}B & \cdots & B
\end{bmatrix}^T.
\]
Using \eqref{eq:PMP_u} and the fact that
\begin{align*}
x_t^i = \left[A^{t-2}B\: \cdots\:AB\: B\right] \left[u_1^{iT},\cdots,u_{t-1}^{iT}\right]^T+A^{t-1}x_1^i,
\end{align*}
we have
\begin{align*}
&x_{2:N}^i = \underbrace{\begin{bmatrix}
B&0&\cdots&0&0\\
AB&B&\cdots&0&0\\
\vdots&\vdots & & \vdots &\vdots\\
A^{N-1}B&A^{N-2}B&\cdots&AB&B
\end{bmatrix}}_{= (I_{N-1}\otimes \Gamma^T)\mathcal{S}^T}u^i+
\underbrace{\begin{bmatrix}
A\\A^2\\\vdots\\A^{N-1}
\end{bmatrix}}_{=: \tilde{A}}x_1^i,\nonumber\\
&=-\underbrace{(I_{N-1}\otimes \Gamma^T)\mathcal{S}^T\mathcal{S}(I_{N-1}\otimes \Gamma)(I_{N-1}\otimes Q)}_{=: \mathscr{F}(Q)}x_{2:N}^i+\tilde{A}x_1^i\\
&\implies(I_{(N-1)n}+\mathscr{F}(Q))x_{2:N}^i = \tilde{A}x_1^i
\end{align*}
Since \eqref{eq:forward_problem} has a unique solution, $I_{(N-1)n}+\mathscr{F}(Q)$ is intrinsically invertible. Therefore, $(I_{(N-1)n}+\mathscr{F}(Q))^{-1}\tilde{A}$ is another representation of the same model structure as $\mathcal{M}(Q, \{ \mathcal{P}_t \}_{t = 2}^N)$, if the former is also combined with the permutation matrices $\{\mathscr{P}_t\}_{t=2}^N$; for the sake of brevity we omit the details.

From above, it can be seen that the model structure is identifiable at $\bar{Q}$ if and only if
\begin{align}
\Upsilon \! := \! \left\{\img(\Delta Q)\mid \Delta Q\in\mathbb{S}^n,\bar{Q}+\Delta Q\in\mathbb{S}^n_+\right\}\cap \kernl(\Gamma) \! = \! \{0\}.
\label{eq:model_identifiability_condition}
\end{align}
Assuming that the system $(A,B)$ is controllable implies that the controllability matrix
\begin{align*}
\Gamma_n = \begin{bmatrix}
(A)^{n-1}B &\cdots &AB &B
\end{bmatrix}^T
\end{align*}
has full column-rank, and hence that $\kernl(\Gamma)=\{0\}$. The latter, in turn, means that \eqref{eq:model_identifiability_condition} is fulfilled. Nevertheless, in practice, if $\Gamma$ is ill-conditioned, the kernel of $\Gamma$ can be ``expanded" from a numerical perspective. In this case we can have $\Gamma \Delta Q \approx 0$ for some $\Delta Q$ that is not close to zero, and the set $\Upsilon$ might numerically not be the singleton $\{ 0 \}$. Thus, it is possible that
\[
(I_{(N-1)n}+\mathscr{F}(\bar{Q} + \Delta Q))^{-1}\tilde{A} \approx (I_{(N-1)n}+\mathscr{F}(\bar{Q}))^{-1}\tilde{A}
\]
for some feasible $\Delta Q$ that is not close to zero. Therefore, when the controllability matrix is ill-conditioned, we might have very similar models that corresponds to very different $Q$'s. In general, we expect that it will be numerically challenging to recover the ``true'' $\bar{Q}$ in these settings, regardless of which method is used.
However, although it might be numerically difficult to recover $\bar{Q}$ in these circumstances, the proof of Lemma~\ref{lem:true_Q_optimal} ensures that the control gains corresponding to $\bar{Q} + \Delta Q$ numerically coincides with the ``true" control gain, which is sufficient for predicting the systems' behaviors. Finally, a similar argument holds in the case of noisy observations.

\section{Numerical experiments and discussions}\label{sec:numerics}
In this section, we present a number of numerical experiments, performed on a number of different discrete-time systems, to illustrate the properties of the proposed algorithm.
In particular, the discrete-time systems are all generated by sampling continuous-time systems $\dot{x} = \hat{A}x + \hat{B}u$ via $A = e^{\hat{A}\Delta t}$ and $B =  \int_0^{\Delta t} e^{\hat{A}t} dt \hat{B}$, where the sampling period $\Delta t=0.05$.

All numerical examples are run on a Dell OptiPlex7080 with Intel(R) Core(TM) i7-10700 CPU and 32GB of RAM.
The solutions are obtained by implementing the optimization problems in Matlab using YALMIP \citep{lofberg2004yalmip} and solving them using MOSEK \citep{mosek}.

\subsection{Noiseless case}\label{subsec:num_noiseless}
In this experiment, the dimension of the system is set to $n = 3$, and $m = 1$. More specifically, we randomly generate system matrices $\hat{A} \in \mathbb{R}^{3 \times 3}$ and $\hat{B} \in \mathbb{R}^{3 \times 1}$ with entries drawn from a normal distribution with mean value zero and standard deviation one, i.e., with entries drawn from the distribution $\mathcal{N}(0,1)$. These are then sampled to generate discrete-time systems, as described above. Moreover,
the ``true "$\bar{Q}$ is randomly generated as $\bar{Q} = GG^T$, where $G \in \mathbb{R}^{3 \times 3}$ with entries drawn from $\mathcal{N}(0,1)$. We let $\bar{Q} \in \bar{\mathbb{S}}^3_+(5)$; any randomly generated $\bar{Q}$ would be discarded if it does not belong to $\bar{\mathbb{S}}^3_+(5)$ and another random $\bar{Q}$ would be generated.
In this way, 500 random triplets $(A, B, \bar{Q})$ are generated.
For each such random triplet, we set the time horizon to $N = 20$ and generate $M = 15$ random starting points $x_1^i$.
The latter are drawn from a uniform distribution on $[-10, 10] \times [-10, 10] \times [-10, 10]$.
The forward problem \eqref{eq:forward_problem} is solved for each starting point, and the noiseless data is then used to solve the optimization problem in \eqref{eq:opt_no_noise}, except that the cost function is scaled with $10^{-4}$ in order to give a better numerical scaling for the problem.
While this does not change any analytic properties, the obtained optimal solutions were observed to have a smaller relative error in general.
The results are presented in \figref{fig:noiseless-1} and \figref{fig:noiseless-2}.

The lower plot in \figref{fig:noiseless-1} illustrates the absolute value of the scaled objective function value at the theoretical optimal solution $(\bar{Q}, \{\bar{P}_t\})$, together with the absolute value of the difference between the objective function value at $(\bar{Q}, \{\bar{P}_t\})$ and at the solution $(Q_{est}, \{ P_{t, est} \})$ obtained with the solver.
As can be seen, the difference is in general several orders of magnitude smaller than the optimal value of the cost function, despite the fact that the obtained $Q_{est}$ is sometimes relatively far from $\bar{Q}$ (cf.~upper plot in \figref{fig:noiseless-1}).
This indicates that the cost function is ``flat" in a region around the optimal solution, which makes the ``true" $\bar{Q}$ hard to recover numerically with high accuracy. We believe that this is highly related to the fact that the controllable matrix is very ill-conditioned (the condition numbers of which varies from $2.899\times 10^2$ to $7.439\times 10^5$). Mitigation of this numerical difficulties is left for future work.

Nevertheless, as can be seen from \figref{fig:noiseless-2}, the corresponding control gain and closed loop system matrix are well-recovered, which serves the purpose of ``predicting the system's behaviour". In fact, the latter is in general recovered with better accuracy than the former.
This indicates that for certain systems, a larger mismatch in $Q$ can still give small mismatches in the control gains and the closed-loop system matrices. Hence it is harder to identify $Q$ numerically in these cases.
Since the control gain and the closed loop system are time-varying, the smallest and largest relative error over all time points are shown in \figref{fig:noiseless-2}.

The fact that the closed-loop system matrix in general seems to be better recovered than $\bar{Q}$ seems to indicate that the ``flatness'' of the cost function for certain problem instances is (at least partly) related to the discussion in Section~\ref{sec:ill-cond}. Namely, that for certain problem instances, substantially different $Q$'s can give rise to very similar closed-loop system matrices.

\begin{figure}[!htpb]
    \centering
    \includegraphics[width=\columnwidth]{noiseless-1-eps-converted-to.pdf}
    \caption{The upper plots shows a (normalized) histogram of the relative error in the estimate $Q_{est}$ obtained with noiseless data, as described in Section~\ref{subsec:num_noiseless}. The lower plots shows the absolute value of the (scaled) gobally optimal objective function value, as well as absolute value of the difference between the globally optimal objective function value and the objective function value in the obtained estimate.}
    \label{fig:noiseless-1}
\end{figure}

\begin{figure}[!htpb]
    \centering
    \includegraphics[width=\columnwidth]{noiseless-2-eps-converted-to.pdf}
    \caption{Histograms (normalized) of the relative error in the estimates of the closed-loop system matrix $A_{cl}(t, Q_{est})$ and the gain $K_t$, obtained from noiseless data as described in Section~\ref{subsec:num_noiseless}. Since the quantities are time-varying, the largest and smallest relative errors are shown.}
    \label{fig:noiseless-2}
\end{figure}

\subsection{Noisy case}\label{subsec:num_noisy}

Next, we illustrate the statistically consistency of the method. This is done on a dynamical system which does not have a numerically ill-conditioned controllability Gramian. More specifically, the continuous-time dynamics is given by
\[
\hat{A} = \begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix},
\qquad
\hat{B} = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix},
\]
which is the kinematic dynamics of a point mass that moves on a two-dimensional plane. In fact, the condition number of the controllability Gramian for this dynamical system is $1$. Moreover, for any $\bar{Q} \succ 0$, this is a model for the simplified setting of ``non-interacting'' individuals moving towards a common goal at the origin. We intend to extend this to interacting individuals in the future.

The `` true'' $\bar{Q}$ is generated as described in Section~\ref{subsec:num_noiseless}, and we set the time horizon to $N = 20$.
We then generate  $M = 4 +10 (k-1) $ random starting points $x_1^i$, drawn from a uniform distribution supported on $[-10, 10] \times [-10, 10] \times [-10, 10]$, for $k = 1, \ldots, 500$.
For each system, the forward problem \eqref{eq:forward_problem} is solved, and noise is added on the obtained optimal states (including the initial state).
The additive noise is drawn from a multi-variate zero-mean normal distribution with covariances matrix drawn from a Wishart distribution of degree $2$, i.e., with same number of degrees of freedom as the dimension of the state space.
The Wishart distribution has a random covariance generated as $0.02 GG^T$, where each element in $G \in \mathbb{R}^{2 \times 2}$ was drawn from $\mathcal{N}(0,1)$.
In total, this gives $500$ noisy data sets with a varying number of systems, for a fixed triplet $(A, B, \bar{Q})$.
Moreover, the signal-to-noise ratio (SNR) in the data sets varies between $27.8035$ and $32.1798$ dB.%
\footnote{The SNR in a data set is computed as the mean of the SNR for all trajectories in that data set, where the SNR of a trajectory is computed as the sum of the squared norm of all states divided by the sum of the squared norm of all noise realizations.}
For each data set, the problem \eqref{eq:opt_noise_empirical} is solved, using the corresponding cost function $H_S^{(\bm{Y}, M)}(Q,\{P_t\})|_{\bm{Y}=Y}$.
The results are shown in \figref{fig:noisy-1} and \figref{fig:noisy-2}.

\figref{fig:noisy-1} and the upper plot in \figref{fig:noisy-2} show the relative error of obtained estimates $Q_{est}$, $K_t$, and $A_{cl}(t, Q_{est})$. As in the previous example, since the control gain and the closed loop system are time-varying, the smallest and largest relative error over all time points are shown. Moreover, the thick dashed lines are averages calculated over a sliding window of length 31, centered around the corresponding point.%
\footnote{Observe that this means that for data sets $k = 1, \ldots, 15$, the average is computed over the index set $\{1, \ldots, k+15 \}$, and thus over a shorter window. An analogous observation holds for $k = 485, \ldots, 500$.}
From this ``trend lines'' it can be observed that the error in general decreases with increasing $M$, in line with the statistical consistency of the estimate as proved in Theorem~\ref{thm:statistical_consistency}.
Although the decreasing trend line seems to ``flatten out'' as $M$ increases, this is a visual effect that disappears if the same experiment is run with larger values of $M$.

Finally, from the lower plot in \figref{fig:noisy-2} we see that, as expected from Remark~\ref{rem:scaling_num_agents}, the time to solve the corresponding optimization problem does note scale with the number of systems $M$.%
\footnote{The solving times are returned by YALMIP.}

\begin{figure}[!htpb]
    \centering
    \includegraphics[trim=0.3cm 0.75cm 1cm 0.7cm, clip, width = \columnwidth]{noisy-1_point_mass_2-eps-converted-to.pdf}
    \caption{Relative error of the control gains and closed-loop system matrices as a function of the number of systems $M$. The estimates are obtained using noisy data, as described in Section~\ref{subsec:num_noisy}, and the dotted lines are averages over a sliding window of length 31.
    Since the quantities are time-varying, only the largest and smallest relative errors are shown.}
    \label{fig:noisy-1}
\end{figure}
\begin{figure}[!htpb]
    \centering
    \includegraphics[trim=0cm 0.75cm 1cm 0.7cm, clip, width = \columnwidth]{noisy-2_point_mass_2-eps-converted-to.pdf}
    \caption{The upper plot shows the relative error of $Q$, akin to the relative errors in \figref{fig:noisy-1}. Moreover, the lower plot shows the time it took (in seconds) to solve the corresponding optimization problem.}
    \label{fig:noisy-2}
\end{figure}

\section{Conclusions}\label{sec:conclusion}
In this work we have considered the linear-quadratic inverse optimal control problem in discrete time and with finite time horizon, but where the observed systems are indistinguishable.
In the case of exact measurements of the states, we show that the true parameter $\bar{Q}$ can be recovered as the unique globally optimal solution to a semidefinite programming problem.
Moreover, the size of this convex optimization problem is independent of the number of systems observed, and the formulation is thus suitable also for scenarios with a large number of systems.
Furthermore, in the case of noisy state observations the optimization problem is modified, and a statistically consistent estimator is obtained as the unique globally optimal solution to another semidefinite programming problem.
However, from numerical simulations it seems that that for certain problem instances the cost functions in both semidefinite programs are relatively flat around the globally optimal solutions, and hence accurate estimates of the parameters are difficult to recover.
Nevertheless, estimates of the time-varying control gains and closed-loop system matrices are obtained with higher accuracy.
An analysis of numerical ill-conditioning of the problem seem to suggest that this could be linked to the fact that for certain problem instances, different $Q$'s can give rise to similar close-loop system behaviour.
An important open question is therefore if and how the numerical conditioning of the estimators can be improved.

\section*{Acknowledgments}
The authors would like thank the reviewers for comments that helped improve the article.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


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In ArcGIS server I have published a dynamic map service, but in ArcGIS portal while accessing same map service using add data option "Add layer from web" getting error "Accessing Data failed" while loading service. In service pooling I have set service timeout upto 300 seconds(5 minutes).
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Publishing Geoprocessing Service with ArcGIS Server?
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text:
Fernando Díaz San Miguel (Salamanca, 1974) es un escritor español.

Biografía

Es socio fundador y director gerente de a.f. diseño y comunicación desde 1999. Es, además, profesor del Máster de escritura creativa en español de la Universidad de Salamanca.
Es Diplomado en Ciencias Empresariales por la Universidad de Salamanca, y Diplomado en Internacional Marketing en el Instituto Tecnológico de Sligo, y ha trabajado para Lloyds Bank en Oxford.

Fue fundador de la Tertulia Literaria Atril, que coordinó en el Ateneo de Salamanca, y director de la Revista Atril (1993-2013).
En 2018 publicó Momento, poesía reunida I, en donde el autor ordena y reescribe sus seis primeros libros: Poemas menores (Universidad Pontificia de Salamanca, 1996); Cartas en la manga, un libro de poemas con forma de baraja española ilustrado por cuatro autores (Ed. El gancho, 1998; segunda edición, 1999); Poemas mayores (Ed. El gancho, 1999); Poemas imperfectos, finalista del Premio Fray Luis de León de Poesía (Junta de Castilla y León, 2001); Poemas finales (Diputación de Salamanca, 2003), y Agosto (Amarú Ediciones, 2008).

A un segundo ciclo pertenecen Libro cero (Fundación Jorge Guillén, 2009), «Meiosis», Editado en forma de almanaque (2005), «Abandonos», en Revista Atlántica de Poesía (2010), y «Palimpsesto» en Revista de Occidente (2019).

Ha publicado relatos en revistas literarias de España y México, y ha participado en muy antologías como: A Celaya (1993), Las palabras de paso (Poesía en Salamanca, 1974-2001) (2001), Paisajes del infierno (2002), Los lugares del verbo (2005), o La sal de la palabra (2013).

Obras

Poesía
 Reunión en Cambridge (Poemas menores) (Universidad Pontificia de Salamanca, 1996)  ISBN: 978-84-7299-378-5
 Cartas en la manga, un libro de poemas con forma de baraja española ilustrado por cuatro autores (Ed. El gancho, 1998; segunda edición, 1999)  ISBN: 978-84-605-8316-5
 Poemas mayores (Ed. El gancho, 1999) ISBN: 978-84-605-8495-7
 Poemas imperfectos, finalista del Premio Fray Luis de León de Poesía (Junta de Castilla y León, 2001) ISBN: 978-84-9718-027-6
 Poemas finales (Diputación de Salamanca, 2003) ISBN: 978-84-7797-206-8
 Agosto (Amarú Ediciones, 2008)  ISBN: 978-84-8196-279-6
 Libro cero (Fundación Jorge Guillén, 2009)  ISBN: 978-84-89707-98-6
 Momento. Poesía reunida, I (Diputación de Salamanca, 2018) ISBN: 978-84-7797-552-6

Referencias

Enlaces externos
 Diario Salamanca al día, Entrevista de Charo Ruano
 Por la senda del arte. Entrevista con Marian Herrero Uña

Escritores de España

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text:
Hans Paasche (3 April 1881, in Rostock – 21 May 1920, in Waldfrieden, Neumark) was a German politician and pacifist. He was the son of the Reichstag vice president Hermann Paasche and Lisi Paasche, and was married to Gabriele (Ellen) Witting.

Hans (Johannes) Albert Ferdinand Paasche studied at the Joachimsthalsches Gymnasium in Berlin. He became a sea cadet in 1899. An Imperial Navy officer and combative pacifist, Hans Paasche was also a big game hunter and nature conservationist, explorer of Africa and life reformer, alcohol abstainer and vegetarian, author and revolutionary. His brief but active life was marked by attempts to change the Prussian Deutschland-über-alles military mindset.

His first experience with war was at the German African military campaign in the German East Africa colonies, where he learned first hand the horror and futility of war. In 1905, Paasche became commander in chief in the Rufiji-region, now southern Tanzania. His command was decisive, but he also took pains to reach a swift pacification. Refugees and defeated Africans found medical help and shelter at his headquarters at Mtanza. He was decorated with the Order of the Crown with Swords, but was removed from command because of his independent peace negotiations. His experiences during the uprising and his guilty feelings over his actions changed his life forever.

After his return to Germany, in 1908 he married Ellen Witting, daughter of Richard Witting, director of the Nationalbank für Deutschland and former Lord Mayor of Posen (Poznań). His honeymoon trip brought him back to Africa, where he undertook a major expedition with his wife Ellen, who became the first European woman to reach the source of the Nile and the first to ascend Kilimanjaro, as well as the recently erupted volcano Nyiragongo.

Paasche published his first book: Im Morgenlicht ("In the Morning Light"), which contained impressions of war and of hunting in Africa, but also valuable ethnographical materials. Paasche's fictional series of letters Lukanga Mukara offer a look at Germany through the eyes of an educated African and mock the decadence of the early pre-World War I Germany. His satirical parody of German lifestyles and attitudes gained immediate popularity. Hans Paasche became one of the most charismatic public figures of the German Empire, well known for openly advocating his provocative ideas.

Still in uniform but not on active duty, he undertook an assault on the many wrongs that he saw in the German society and its military obsessions. This brought him many avid listeners and readers, especially among the young. It also brought him vilification, including that of his own father, and the dangerous attention of military prosecutors.

A high treason prosecution was conducted against him by Abteilung IIIb, the intelligence service of the General Staff led by colonel Walter Nicolai. The trial ended in 1918 with a negotiated admission into a mental health institution. He was released at the end of 1918, and continued his work as a radical journalist. He served a brief term in the Berlin Workers' and Soldiers' Council and tried in vain to organize a national court. Eventually he was forced out by right-wing social democrats.

After the sudden death of his wife Ellen at twenty nine, he retreated to his estate at Waldfrieden to take care of his four young children, Jochen, Nils, Helga and Ivan. Nevertheless, he still took part in national and international campaigns for peace, understanding among nations, and a League of Nations. His estate Waldfrieden became a shelter for hunted leftist insurgents.

His assassination was ultimately decreed by the ultra-nationalist death squad Organisation Consul. In 1920, at the age of 39, he was shot by a commando of sixty soldiers from District Command III while supposedly trying to escape. The soldiers had mounted the operation under an anonymous call with the pretense of finding a hidden weapons cache, which was never found. Few were prosecuted and none were convicted.

In 1985, Helga Paasche removed the tombstone of her father from Waldfrieden to the archives of the German Youth Movement on Ludwigstein Castle, where it is part of a permanent Hans-Paasche-exhibition.

His life has been extensively documented by his daughter, Helga Paasche, the author Werner Lange and his grandson Dr. Gottfried Paasche (for many years a professor of sociology at Canada's York University), among others.  Hans Paasche's great-granddaughter, Sarah Paasche-Orlow, is a Conservative Rabbi.

Bibliography

 Im Morgenlicht. Kriegs-, Jagd- und Reise-Erlebnisse in Ostafrika 1907
 Die Forschungsreise des Afrikaners Lukanga Mukara ins innerste Deutschland 1912/1913
 Fremdenlegionär Kirsch - Eine abenteuerliche Fahrt von Kamerun in die deutschen Schützengräben in den Kriegsjahren 1914/15 1914
 Hans Paasche: Militant Pacifist in Imperial Germany, by Werner Lange, translated by David Koblick 2005
 MAGNUS SCHWANTJE: "Hans Paasche. Sein Leben und Wirken" (= Flugschriften des Bundes Neues Vaterland, Nr. 26/27) Berlin 1921
 OTTO WANDERER (d.i. Otto Buchinger): Paasche-Buch Hamburg 1921
 FRANZISKUS HÄHNEL: Erinnerungen an Hans Paasche. In: Junge Menschen, 3. Jg., Heft 11/12* Juni 1922
 HELMUT DONAT, WILFRIED KNAUER (Ê.): ""Auf der Flucht" erschossen. Schriften und Beiträge von und über Hans Paasche" (= Schriftenreihe Das Andere Deutschland, Nr. 1) Bremen/Zeven 1981
 REINHOLD LÜTGEMEIER-DAVIN: "Hans Paasche (1881–1920), Lebensreformer, Anti-Preuße, Revolutionär". In: Jahrbuch des Archivs der deutschen Jugendbewegung Bd. 13, Burg Ludwigstein 1981
 HELGA PAASCHE: "Ein Leben für unsere Zukunft. Hans Paasche zum 65. Todestag". In: Jahrbuch des Archivs der deutschen Jugendbewegung Bd. 15, Burg Ludwigstein 1984-85
 PETER MORRIS-KEITEL: Umwertung aller Werte. Hans Paasches
 HORST NAUMANN: "Hans Paasche. Pazifist - Revolutionär - Kommunist". In: Die Novemberrevolution und die Gründung der KPD. Protokoll der wissenschaftlichen Konferenz …, Teil 1 Berlin 1989
 HANS PAASCHE (E. HELMUT DONAT, HELGA PAASCHE): "Ändert Euren Sinn! Schriften eines Revolutionärs" (= Schriftenreihe Geschichte und Frieden, Bd. 2) Bremen 1992
 KARL H. SOLBACH: "Hans Paasche - Offizier, Reformer, Revolutionär". In: CORNELIUS NEUTSCH, KARL H. SOLBACH (E.), Reise in die Kaiserzeit. Ein deutsches Kaleidoskop Leipzig 1994
 ALAN NOTHNAGLE: "Metanoia! Hans Paasche - ein lebensreformerischer Visionär". In: Zeitschrift für Geschichtswissenschaft, 45. Jg., Nr. 9 Berlin 1997
 GOTTFRIED PAASCHE, JOAQUIN KUHN (E.): The Strange Story of the Shooting of Captain Hans Paasche. The Writings and Actions of a Peace Martyr Toronto 2001
 JERZY GIERGIELEWICZ: "Hans Paasche: fascynujaca postac Niemca, w Polsce prawie nie znana". In: Wedrowiec Zachodniopomorski, 10. Jg., Nr. 2 Szczecin 2003
 HANS PAASCHE, P. WERNER LANGE: "Die Legende von der Vertreibung der Kaiserin oder Potsdamer Beiträge zum deutsch-polnischen Jahr". In: Schriftzüge. Brandenburgische Blätter für Kunst und Literatur, 7. Jg., Nr.1 Potsdam 2005
 P. WERNER LANGE: "Die Toten im Maisfeld. Hans Paasches Erkenntnisse aus dem Maji-Maji-Krieg". In: FELICITAS BECKER, JIGAL BEEZ (E.), Der Maji-Maji-Krieg in Deutsch-Ostafrika, 1905-1907 Berlin 2005
 P. WERNER LANGE: ""Und ich zweifelte, ob ich ein Krieger sei …" Der Kolonialoffizier und Pazifist Hans Paasche". In: ULRICH VAN DER HEYDEN, JOACHIM ZELLER (E.), Macht und Anteil an der Weltherrschaft. Berlin und der deutsche Kolonialismus Münster 2005
 P. WERNER LANGE: "Die Treppe zum Himmel. Zur Eröffnung einer Gedenkstätte für Hans Paasche in der Wojewodschaft Wielkopolska". In: Inter Finitimos. Jahrbuch zur deutsch-polnischen Beziehungsgeschichte 3 Osnabrück 2006
ANDREAS SCHMID: "Deutschland im Spiegel Ostafrikas. Hans Paasches Lukanga Mukara (1912/13)". In: Zeitschrift für Kulturwissenschaften, 14. Jg., Nr. 2, 2020, 49–66.
Paasche, Karin Ilona. "Germany's Africa: A Literary and Historical Disconnect, " Procedia - Social and Behavioral Sciences 192 ( 2015 ) 398 – 407. https://www.sciencedirect.com/science/article/pii/S1877042815035272

References

External links

 Chronology of Hans Paasche's Life (by Werner Lange)

1881 births
1920 deaths
People from Rostock
People from the Grand Duchy of Mecklenburg-Schwerin
German politicians
German Youth Movement
Imperial German Navy personnel of World War I
Organisation Consul victims
Joachimsthalsches Gymnasium alumni


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More than ever, it will take shared expertise to come up with the kind of solutions we need for the state's environmental issues.
Approximately 1 million retirees live in Georgia, and they could help move the state's economy onto a higher growth path.
Ink & Elm brings a bit of grown-up fun and local fare to Decatur's Emory Village.
Groups in three Georgia cities have found different ways to bridge town and gown to create strong, diverse neighborhoods.


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Nectophrynoides frontierei är en groddjursart som beskrevs av Menegon, Salvidio och Simon Loader 2004. Nectophrynoides frontierei ingår i släktet Nectophrynoides och familjen paddor. IUCN kategoriserar arten globalt som otillräckligt studerad. Inga underarter finns listade i Catalogue of Life.

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TERN AusCover has a User Reference Group to provide advice on our activities, and ensure that we deliver products and services that are of value to the user community.
The membership of the User Reference Group is currently being refreshed, and will be listed here soon.
Our User Reference Group members are experienced remote sensing practitioners from a variety of departments and institutions, and working on a range of ecosystem science and management issues. None of the members are directly connected with TERN AusCover but, as with most disciplines, many know each other and those connected with TERN AusCover.


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"A great art and craft shop on the legendary Gloucester Road"
A review of Craftworks (Bristol) Ltd by Paul Bullivant written on Wednesday 14th of October 2020
Craftworks have an amazing range of art and craft products and materials and their framing service is excellent. Jenny and her team will always give useful and appropriate advice about frames and mounts and the end result is always exactly what I need and at a reasonable price - which is why I have used Craftworks for over 10 years!.
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Most of us have heard or had the chance to experience the unique southern combination of chicken and waffles. While it is a common find in the South, is not as easy to locate here on the west. It has almost become impossible to find in the Inland Empire, that was until March 2015 when Maple House Chicken and Waffles was established.


One week before the Super Bowl, Tom Brady shot down a storyline that would've dominated the airwaves right up until the moment the big game kicked off. On Sunday, Brady definitively ended the idea that he could retire after Super Bowl LIII, which will be played between the Rams and Patriots next Sunday.
Really, why would Brady retire?
At the age of 41, one year after he won MVP, he just bested the league's likely MVP on the road to reach his third-straight Super Bowl. Even in a so-called down season, Brady completed 65.8 percent of his passes for 4,355 yards, 29 touchdowns, 11 interceptions, and a 97.7 passer rating. He led the Patriots to playoff wins over two teams, the Chargers and Chiefs, that posted better regular-season win totals than them. Next weekend, he has a chance to capture his sixth Super Bowl and his fifth Super Bowl MVP by beating the franchise he defeated in his first-ever Super Bowl way back in February 2002.
Together, Brady and Bill Belichick have figured out the AFC to the point where going to the Super Bowl is a damn-near inevitability, which reminds me of something Brady said two years ago after capturing his fifth ring with an epic come-from-behind win over the Falcons.
"I have the answers to the test now," Brady told The Monday Morning Quarterback.
"You can't surprise me on defense. I've seen it all. I've processed 261 games, I've played them all. It's an incredibly hard sport, but because the processes are right and are in place, for anyone with experience in their job, it's not as hard as it used to be. There was a time when quarterbacking was really hard for me because you didn't know what to do. Now I really know what to do, I don't want to stop now. This is when it's really enjoyable to go out."
Why would Brady retire now?
He has his health. He's still one of the league's best quarterbacks. Come crunch time in the playoffs, he's still the league's best quarterback -- just ask a Chiefs defense that failed to stop him on three third-and-10s in overtime. He's back in the Super Bowl, and it wouldn't be at all surprising to see the Patriots in next season's Super Bowl, too.
That's why Brady isn't even entertaining the idea of retirement.