Control of partial differential equations

From Scholarpedia

Jean-Michel Coron (2009), Scholarpedia, 4(11):6451.
Curator and Contributors

1.00 - Jean-Michel Coron

0.33 -

Leo Trottier

0.33 -

Eugene M. Izhikevich

0.17 -

Milos Rancic

0.17 -

Benjamin Bronner

A control system is a dynamical system on which one can act by using suitable controls. In this article, the dynamical model is modeled by partial differential equations of the following type $\tag{1} \dot y=f(y,u).$

The variable $$y$$ is the state and belongs to some space $$\mathcal{Y}\ .$$ The variable $$u$$ is the control and belongs to some space $$\mathcal{U}\ .$$ In this article, the space $$\mathcal{Y}$$ is of infinite dimension and the differential equation (1) is a partial differential equation.

There are a lot of problems that appear when studying a control system. But the most common one is the controllability problem, which is, roughly speaking, the following one. Let us give two states. Is it possible to steer the control system from the first one to the second one? In the framework of (1), this means that, given the state $$a\in \mathcal{Y}$$ and the state $$b\in \mathcal{Y}\ ,$$ does there exit a map $$u:[0,T]\rightarrow \mathcal{U}$$ such that the solution of the Cauchy problem $$\dot y=f(y,u(t)), \, y(0)=a,$$ satisfies $$y(T)=b\ ?$$ If the answer is yes whatever the given states are, the control system is said to be controllable. If $$T>0$$ can be arbitrary small one speaks of small-time controllability. If the two given states and the control are restricted to be close to an equilibrium one speaks of local controllability at this equilibrium. (An equilibrium of the control system is a point $$(y_e,u_e)\in \mathcal{Y}\times \mathcal{U}$$ such that $$f(y_e,u_e)=0$$). If, moreover, the time $$T$$ is small, one speaks of small-time local controllability.

A general framework for control systems modeled by linear PDE's

The framework

For two normed linear spaces $$H_1$$ and $$H_2 \ ,$$ we denote by $$\mathcal{L}(H_1;H_2)$$ the set of continuous linear maps from $$H_1$$ into $$H_2$$ and denote by $$\|\cdot\|_{\mathcal{L}(H_1;H_2)}$$ the usual norm in this space.

Let $$H$$ and $$U$$ be two Hilbert spaces. Just to simplify the notations, these Hilbert spaces are assumed to be real Hilbert spaces (the case of complex Hilbert spaces follows directly from the case of real Hilbert spaces). The space $$H$$ is the state space and the space $$U$$is the control space. We denote by $$(\cdot, \cdot)_H$$ the scalar product in $$H\ ,$$ by $$(\cdot,\cdot)_U$$ the scalar product in $$U\ ,$$ by $$\|\cdot\|_H$$ the norm in $$H$$ and by$$\|\cdot\|_U$$ the norm in $$U\ .$$

Let $$S(t),\, t\in[0,+\infty)\ ,$$ be a strongly continuous semigroup of continuous linear operators on$$H\ .$$ Let $$A$$ be the infinitesimal generator of the semigroup $$S(t), \, t\in[0,+\infty)\ .$$ As usual, we denote by $$S(t)^*$$ the adjoint of $$S(t)\ .$$ Then $$S(t)^*, \ t\in[0,+\infty),$$ is a strongly continuous semigroup of continuous linear operators and the infinitesimal generator of this semigroup is the adjoint $$A^*$$ of $$A\ .$$ The domain $$D(A^*)$$ is equipped with the usual graph norm $$\|\cdot\|_{D(A^*)}$$ of the unbounded operator $$A^*\ :$$

$$\|z\|_{D(A^*)}:=\|z\|_{H}+\|A^*z\|_{H}, \, \forall z\in D(A^*).$$

This norm is associated to the scalar product in $$D(A^*)$$ defined by

$$(z_1,z_2)_{D(A^*)}:=(z_1,z_2)_{H}+(A^*z_1,A^*z_2)_H, \, \forall (z_1,z_2) \in D(A^*)^2.$$

With this scalar product, $$D(A^*)$$ is a Hilbert space. Let $$D(A^*)'$$ be the dual of $$D(A^*)$$ with respect to the pivot space $$H\ .$$ In particular,

$$D(A^*)\subset H\subset D(A^*)'.$$

Let

$\tag{2} B\in \mathcal{L}(U,D(A^*)').$

In other words, $$B$$ is a linear map from $$U$$ into the set of linear functions from $$D(A^*)$$ into $$\mathbb{R}$$ such that, for some $$C>0\ ,$$

$$|(Bu)z|\leqslant C \|u\|_{U}\|z\|_{D(A^*)},\, \forall u \in U, \, \forall z\in D(A^*).$$

We also assume the following regularity property (also called admissibility condition):

$\tag{3} \forall T>0, \exists C_T>0 \text{ such that } \int_0^T\|B^*S(t)^* z\|_{U}^2dt \leqslant C_T \|z\|^2_H, \, \forall z\in D(A^*).$

In (3) and in the following, $$B^*\in \mathcal{L}(D(A^*);U)$$ is the adjoint of $$B\ .$$ It follows from (3) that the operators

$$(z\in D(A^*)) \mapsto ((t\mapsto B^*S(t)^* z)\in C^0([0,T];U)),$$

$$(z\in D(A^*)) \mapsto ((t\mapsto B^*S(T-t)^* z)\in C^0([0,T];U))$$

can be extended in a unique way as continuous linear maps from $$H$$ into $$L^2((0,T);U)\ .$$ We use the same symbols to denote these extensions.

Note that, using the fact that $$S(t)^*\ ,$$ $$t\in[0,+\infty)\ ,$$ is a strongly continuous semigroup of continuous linear operators on$$H\ ,$$ it is not hard to check that (3) is equivalent to

$$\exists T>0, \exists C_T>0 \text{ such that } \int_0^T\|B^*S(t)^* z\|_{U}^2dt \leqslant C_T \|z\|^2_H, \, \forall z\in D(A^*).$$

The control system we consider here is

$\tag{4} \dot y =Ay +Bu, \, t\in(0,T),$

where, at time $$t\ ,$$ the control is $$u(t)\in U$$ and the state is$$y(t)\in H\ .$$

Let $$T>0\ ,$$ $$y^0\in H$$ and $$u\in L^2((0,T);U)\ .$$ We are interested in the Cauchy problem

$\tag{5} \dot y =Ay +Bu(t) , \, t\in (0,T),$

$\tag{6} y(0)=y^0.$

We first give the definition of a solution to (5)-(6). Let us first motivate our definition. Let $$\tau \in [0,T]$$ and $$\varphi :[0,\tau]\rightarrow H\ .$$ We take the scalar product in $$H$$ of (5) with $$\varphi$$ and integrate on$$[0,\tau]\ .$$ At least formally, we get, using an integration by parts together with (6),

$$(y(\tau),\varphi(\tau))_{H}-(y^0,\varphi(0))_{H}-\int_0^\tau (y(t),\dot \varphi (t) +A^*\varphi (t))_{H}dt=\int_0^\tau (u(t),B^*\varphi (t))_U dt.$$

Taking $$\varphi(t)=S(\tau-t)^*z^\tau\ ,$$ for every given $$z^\tau \in H\ ,$$ we have formally $$\dot \varphi (t) +A^*\varphi (t)=0\ ,$$ which leads to the following definition.

Definition (solution of the Cauchy problem)

Let $$T>0\ ,$$ $$y^0\in H$$ and $$u\in L^2((0,T);U)\ .$$ A solution of the Cauchy problem (5)-(6) is a function $$y\in C^0([0,T];H)$$ such that $\tag{7} (y(\tau),z^\tau)_H-(y^0,S(\tau)^*z^\tau)_H = \int_0^\tau (u(t),B^*S(\tau-t)^*z^\tau)_U dt, \, \forall \tau \in[0,T], \, \forall z^\tau \in H.$

Note that, by the regularity property (3), the right hand side of (7) is well defined.

With this definition one has the following theorem.

Theorem 1 (well posedness of the Cauchy problem)

Let $$T>0\ .$$ Then, for every $$y^0\in H$$ and for every $$u\in L^2((0,T);U)\ ,$$ the Cauchy problem (5)-(6) has a unique solution $$y\ .$$ Moreover, there exists $$C=C(T)>0\ ,$$ independent of $$y^0\in H$$ and $$u\in L^2((0,T);U)\ ,$$ such that

$\tag{8} \|y(\tau)\|_{H}\leqslant C (\|y^0\|_{H}+\|u\|_{L^2((0,T);U)}), \, \forall \tau \in [0,T].$

For a proof of this theorem, see, for example, (Jean-Michel Coron, 2007, pages 53-54).

Controllability of linear control systems

Different types of controllability

In this section we are interested in the controllability of the control system (4). In contrast to the case of linear finite-dimensional control systems, many types of controllability are possible and interesting. We define here three types of controllability.

Definition (exact controllability)

Let$$T>0\ .$$ The control system (4) is exactly controllable in time $$T$$ if, for every $$y^0\in H$$ and for every $$y^1\in H\ ,$$ there exists $$u \in L^2((0,T);U)$$ such that the solution $$y$$ of the Cauchy problem

$\tag{9} \dot y =Ay + Bu(t), \, y(0)=y^0,$

satisfies$$y(T)=y^1\ .$$

Definition (null controllability)

Let$$T>0\ .$$ The control system (4) is null controllable in time $$T$$ if, for every $$y^0\in H$$ and for every $$\tilde y^0\in H\ ,$$ there exists $$u \in L^2((0,T);U)$$ such that the solution of the Cauchy problem (8) satisfies $$y(T)=S(T)\tilde y^0\ .$$

Let us point out that, by linearity, we get an equivalent definition of "null controllable in time $$T$$" if, in the definition above, one assumes that $$\tilde y^0=0\ .$$ This explains the usual terminology "null controllability".

Definition (approximate controllability)

Let$$T>0\ .$$ The control system (4) is approximately controllable in time $$T$$ if, for every $$y^0\in H\ ,$$ for every $$y^1\in H\ ,$$ and for every $$\varepsilon>0\ ,$$ there exists $$u \in L^2((0,T);U)$$ such that the solution $$y$$ of the Cauchy problem (8) satisfies $$\|y(T)-y^1\|_H\leqslant \varepsilon\ .$$

Clearly

(exact controllability) $$\Rightarrow$$ (null controllability and approximate controllability).

The converse is false in general (see, for example, the heat control equation at this link). However, the converse holds if $$S$$ is a strongly continuous group of linear operators. More precisely, one has the following theorem.

Theorem 2 (null controllability/exact controllability)

Assume that $$S(t)\ ,$$ $$t\in \mathbb{R}\ ,$$ is a strongly continuous group of linear operators. Let $$T>0\ .$$ Assume that the control system (4) is null controllable in time $$T\ .$$ Then the control system (4) is exactly controllable in time $$T\ .$$

Proof of Theorem. Let $$y^0\in H$$ and $$y^1\in H\ .$$ From the null controllability assumption applied to the initial data $$y^0-S(-T)y^1\ ,$$ there exists $$u\in L^2((0,T);U)$$ such that the solution $$\tilde y$$ of the Cauchy problem

$$\dot {\tilde y}=A\tilde y +Bu(t), \,\tilde y (0)=y^0-S(-T)y^1,$$

satisfies

$\tag{10} \tilde y(T)=0.$

One easily sees that the solution $$y$$ of the Cauchy problem

$$\dot y=A y +Bu(t), \, y (0)=y^0,$$

is given by

$\tag{11} y(t)=\tilde y(t)+S(t-T)y^1, \, \forall t \in [0,T].$

In particular, from (10) and (11),

$$y(T)=y^1.$$

This concludes the proof of the theorem.

Methods to study controllability

Rouhgly speaking there are essentially two types of methods to study the controllability of linear PDE, namely direct methods and duality methods.

Direct methods

Among these methods, let us mention in particular

• The extension method. See, for example, (David L. Russell, 1974), the proof of Theorem 5.3, pages 688-690, in (David L. Russell, 1978), (Walter Littman, 1978) and Section 2.1.2.2, pages 30-34, in (Jean-Michel Coron, 2007).
• Moment theory. See, for example, (Werner Krabs, 1992), (Vilmos Komornik and Paola Loreti, 2005), (Sergei Avdonin and Sergei Ivanov, 1995), and Section 2.6, pages 95-99, in (Jean-Michel Coron, 2007). For an example of an application of the moment theory for a Schrödinger equation, see at this link.
• Flatness. This approach has been initiated in the framework of control theory in finite dimension in (Michel Fliess, Jean Lévine, Philippe Martin and Pierre Rouchon, 1995). For applications of this method to the control of linear PDE, see, in particular, (Hugues Mounier, Joachim Rudolph, Michel Fliess and Pierre Rouchon, 1998), (Béatrice Laroche, Philippe Martin and Pierre Rouchon, 2000), (Nicolas Petit and Pierre Rouchon, 2001), as well as this article by Pierre Rouchon in Scholarpedia.

Duality methods

Let us now introduce some "optimal control maps". Let us first deal with the case where the control system (4) is exactly controllable in time $$T\ .$$ Then, for every $$y^1\ ,$$ the set $$U^T(y^1)$$ of $$u\in L^2((0,T);U)$$ such that

$$(\dot y=Ay+Bu(t), \, y(0)=0)\Rightarrow (y(T)=y^1)$$

is nonempty. Clearly the set $$U^T(y^1)$$ is a closed affine subspace of $$L^2((0,T);U)\ .$$ Let us denote by $$\mathcal{U}^T(y^1)$$ the projection of $$0$$ on this closed affine subspace, i.e., the element of $$U^T(y^1)$$ of the smallest $$L^2((0,T);U)$$-norm. Then it is not hard to see that the map

$$\begin{array}{rrcl} \mathcal{U}^T:&H&\rightarrow&L^2((0,T);U) \\ &y^1&\mapsto&\mathcal{U}^T(y^1) \end{array}$$

is a linear map. Moreover, using the closed graph theorem (see, for example, Theorem 2.15 on page 50 in (Rudin, 1973)) one readily checks that this linear map is continuous.

Let us now deal with the case where the control system (4) is null controllable in time $$T\ .$$ Then, for every $$y^0\ ,$$ the set $$U_T(y^0)$$ of $$u\in L^2((0,T);U)$$ such that

$$(\dot y=Ay+Bu(t), \, y(0)=y^0)\Rightarrow (y(T)=0)$$

is nonempty. Clearly the set $$U_T(y^0)$$ is a closed affine subspace of $$L^2((0,T);U)\ .$$ Let us denote by $$\mathcal{U}_T(y^0)$$ the projection of $$0$$ on this closed affine subspace, i.e., the element of $$U_T(y^0)$$ of the smallest $$L^2((0,T);U)$$-norm. Then, again, it is not hard to see that the map

$$\begin{array}{rrcl} \mathcal{U}_T:&H&\rightarrow&L^2((0,T);U) \\ &y^0&\mapsto&\mathcal{U}_T(y^0) \end{array}$$

is a continuous linear map.

The main results of this section are the following ones.

Theorem 3 (exact controllability)

Let $$T>0\ .$$ The control system (4) is exactly controllable in time $$T$$ if and only if there exists $$c>0$$ such that

$\tag{12} \int_0^T\|B^*S(t)^* z\|_U^2dt \geqslant c \|z\|_{H}^2, \, \forall z \in D(A^*).$

Moreover, if such a $$c>0$$ exists and if $$c^T$$ is the maximum of the set of $$c>0$$ such that (12) holds, one has

$\tag{13} \left\|\mathcal{U}^T\right\|_{\mathcal{L}(H;L^2((0,T);U))}=\frac{1}{\displaystyle \sqrt{c^T}}.$

Theorem 4 (approximate controllability)

The control system (4) is approximately controllable in time $$T$$ if and only if, for every $$z\in H\ ,$$

$\tag{14} (B^*S(\cdot)^*z =0 \text{ in }L^2((0,T);U))\Rightarrow (z=0).$

Theorem 5 (null controllability)

Let$$T>0\ .$$ The control system (4) is null controllable in time $$T$$ if and only if there exists $$c>0$$ such that $\tag{15} \int_0^T\|B^*S(t)^* z\|_U^2dt \geqslant c \|S(T)^*z\|^2_{H}, \, \forall z \in D(A^*).$

Moreover, if such a $$c>0$$ exists and if $$c_T$$ is the maximum of the set of $$c>0$$ such that (15) holds, then $\tag{16} \left\|\mathcal{U}_T\right\|_{\mathcal{L}(H;L^2((0,T);U))}=\frac{1}{\displaystyle \sqrt{c_T}}.$

Theorem 6 (null controllability/approximate controllability)

Assume that, for every $$T>0\ ,$$ the control system (4) is null controllable in time $$T\ .$$ Then, for every $$T>0\ ,$$ the control system (4) is approximately controllable in time $$T\ .$$

For a proof of these theorems, see, for example Section 2.3.2 in (Jean-Michel Coron, 2007). Inequalities (12) and (15) are usually called observability inequalities for the abstract linear control system $$\dot y =Ay +Bu\ .$$ The difficulty is to prove them! For this purpose, there are many methods available (but still many open problems). Among these methods, let us mention in particular

• Multiplier methods. See in particular, (Jacques-Louis Lions, 1988), (Vilmos Komornik, 1994) and (Enrique Zuazua, 2006). For a simple example where this method is used, see at this link.
• Microlocal analysis. See in particular (Claude Bardos, Gilles Lebeau and Jeffrey Rauch, 1992) and the appendix 2 of the book (Jacques-Louis Lions, 1988).

Remark. In contrast with Theorem 6, note that, for a given$$T>0\ ,$$ the null controllability in time $$T$$ does not imply the approximate controllability in time $$T\ .$$ For example, let $$L>0$$ and let us take $$H:=L^2(0,L)$$ and $$U:=\{0\}\ .$$ We consider the linear control system

$\tag{17} y_t+y_x=0,\, t\in (0,T),\, x\in (0,L),$

$\tag{18} y(t,0)=u(t)=0,\, t\in (0,T).$

Through examples in the next section, we shall see how to put this control system in the abstract framework $$\dot y =Ay +Bu\ .$$ As one can see in the section at this link, whatever $$y^0\in L^2(0,L)$$ is, the solution to the Cauchy problem

$$y_t+y_x=0,\, t\in (0,T),\, x\in (0,L),$$

$$y(t,0)=u(t)=0,\, t\in (0,T),$$

$$y(0,x)=y^0(x),\, x\in (0,L),$$

satisfies

$$y(T,\cdot)=0, \text{ if } T\geqslant L.$$

In particular, if $$T\geqslant L\ ,$$ the linear control system (17)-(18) is null controllable but is not approximately controllable.

Again, there are two possibilities to study numerically the controllability of a linear control system: direct methods, duality methods. The most popular ones use duality methods and in particular the Hilbert Uniqueness Method (HUM) introduced in (Jacques-Louis Lions, 1988). For the numerical approximation, one uses often discretization by finite difference methods. However a new problem appear: the control for the discretized model does not necessarily lead to a good approximation to the control for the original continuous problem. In particular, the classical convergence requirements, namely stability and consistency, of the numerical scheme used does not suffice to guarantee good approximations to the controls that one wants to compute. Observability/controllability may be lost under numerical discretization as the mesh size tends to zero. To overcome this problem, several remedies have been used, in particular, filtering, Tychonoff regularization, multigrid methods, and mixed finite element methods. For precise informations and references, we refer to the survey papers (Enrique Zuazua, 2005; 2006).

Complements

On the controllability of linear PDE, we have already given references to books and papers. But there are of course many other references which must also be mentioned. If one restricts to books or surveys we would like to add in particular (but this is a very incomplete list):

• The survey (Fatiha Alabau-Boussouira and Piermarco Cannarsa, 2008) which deals, in particular, with abstract evolution equations, wave equations, heat equations and quadratic optimal control for linear PDE.
• The book (Alain Bensoussan, 1992) on stochastic control.
• The books (Alain Bensoussan, Giuseppe Da Prato, Michel Delfour and Sanjoy Mitter, 1992; 1993) which deal, in particular, with differential control systems with delays and partial differential control systems with specific emphasis on controllability, stabilizability and the Riccati equations.
• The book (Ruth Curtain and Hans Zwart, 1995) which deals with general infinite-dimensional linear control systems theory. It includes the usual classical topics in linear control theory such as controllability, observability, stabilizability, and the linear-quadratic optimal problem. For a more advanced level on this general approach, one can look at the book (Olof Staffans, 2005).
• The book (René Dáger and Enrique Zuazua, 2006) on partial differential equations on planar graphs modeling networked flexible mechanical structures (with extensions to the heat, beam and Schrödinger equations on planar graphs).
• The book (Abdelhaq El Jaï and Anthony Pritchard, 1988) on the input-output map and the importance of the location of the actuators/sensors for a better controllability/observability.
• The books (Hector Fattorini, 1999; 2005) on optimal control for infinite-dimensional control problems (linear or nonlinear, including partial differential equations).
• The book (Andrei Fursikov, 2000) on the study of optimal control problems for infinite-dimensional control systems with many examples coming from physical systems governed by partial differential equations (including the Navier-Stokes equations).
• The book (Vilmos Komornik and Paola Loreti, 2005) on harmonic (and nonharmonic) analysis methods with many applications to the controllability of various time-reversible systems.
• The book (John Lagnese and Günter Leugering, 2004) on optimal control on networked domains for elliptic and hyperbolic equations, with a special emphasis on domain decomposition methods.
• The books (Irena Lasiecka and Roberto Triggiani, 2000a; 2000b) which deal with finite horizon quadratic regulator problems and related differential Riccati equations for general parabolic and hyperbolic equations with numerous important specific examples.
• The survey (David Russell, 1978) which deals with the hyperbolic and parabolic equations, quadratic optimal control for linear PDE, moments and duality methods, controllability and stabilizability.
• The book (Marius Tucsnak and George Weiss, 2006) on passive and conservative linear systems, with a detailed chapter on the controllability of these systems.
• The survey (Enrique Zuazua, 2006) on recent results on the controllability of linear partial differential equations. It includes the study of the controllability of wave equations, heat equations, in particular with low regularity coefficients, which is important to treat semi-linear equations, fluid-structure interaction models.

There are many important problems which are not discussed in this paper. Perhaps the more fundamental ones are optimal control theory and the stabilization problem. For the optimal control theory, see references already mentioned above. The stabilization problem is the following one. We have an equilibrium which is unstable (or not enough stable) without the use of the control. Let us give a concrete example. One has a stick that is placed vertically on one of his fingers. In principle, if the stick is exactly vertical with a speed exactly equal to 0, it should remain vertical. But, due to various small errors (the stick is not exactly vertical, for example), in practice, the stick falls down. In order to avoid this, one moves the finger in a suitable way, depending on the position and speed of the stick; one uses a feedback law (or closed-loop control) which stabilizes the equilibrium. The problem of the stabilization is the existence and construction of such stabilizing feedback laws for a given control system. More precisely, let us consider the control system (1) and let us assume that $$f(0,0)=0\ .$$ The stabilization problem is to find a feedback law $$y\rightarrow u(y)$$ such that 0 is asymptotically stable for the closed loop system $$\dot y = f(y,u(y))\ .$$

Again, as for the controllability, the first step to study the stabilization problem is to look at the linearized control system at the equilibrium. Roughly speaking one expects that a linear feedback which stabilizes (exponentially) the linearized control system stabilizes (locally) the nonlinear control system. This is indeed the case in many important situations. For example, for the Navier control system mentioned in the section Controllability of nonlinear control systems, see in particular (Viorel Barbu, 2003), (Viorel Barbu and Roberto Triggiani, 2004), (Viorel Barbu, Irena Lasiecka and Roberto Triggiani, 2006), (Andrei Fursikov, 2004), (Jean-Pierre Raymond, 2006; 2007) and (Rafael Vázquez, Emmanuel Trélat and Jean-Michel Coron, 2008).

When the linearized control system cannot be stabilized it still may happen that the nonlinearity helps. This is for example the case for the Euler control system described in the section at this link. See (Jean-Michel Coron, 1999) and (Olivier Glass, 2005). (See also at this link for the controllability.)

The most popular approach to construct stabilizing feedbacks relies on Lyapunov functions. See Chapter 12 in (Jean-Michel Coron, 2007) for various methods to design Lyapunov functions.

References

• Andrei A. Agrachev. Newton diagrams and tangent cones to attainable sets. In Analysis of controlled dynamical systems (Lyon, 1990), volume 8 of Progr. Systems Control Theory, pages 11–20. Birkhäuser Boston, Boston, MA, 1991.
• Andrei A. Agrachev and Andrei V. Sarychev. Navier-Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech., 7(1):108–152, 2005.
• Andrei A. Agrachev and Andrei V. Sarychev. Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing. Comm. Math. Phys., 265(3):673–697, 2006.
• Fatiha Alabau-Boussouira and Piermarco Cannarsa. Control of partial differential equations, 2009.
• Claudio Baiocchi, Vilmos Komornik, and Paola Loreti. Ingham-Beurling type theorems with weakened gap conditions. Acta Math. Hungar., 97(1-2):55–95, 2002.
• Viorel Barbu. Feedback stabilization of Navier-Stokes equations. ESAIM Control Optim. Calc. Var., 9:197–206 (electronic), 2003.
• Viorel Barbu, Irena Lasiecka, and Roberto Triggiani. Tangential boundary stabilization of Navier-Stokes equations. Mem. Amer. Math. Soc., 181(852):x+128, 2006.
• Viorel Barbu and Roberto Triggiani. Internal stabilization of Navier-Stokes equations with finite-dimensional controllers. Indiana Univ. Math. J., 53(5):1443–1494, 2004.
• Claude Bardos, Gilles Lebeau, and Jeffrey Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim., 30(5):1024–1065, 1992.
• Karine Beauchard. Local controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. (9), 84(7):851–956, 2005.
• Karine Beauchard. Controllability of a quantum particle in a 1D variable domain. ESAIM Control Optim. Calc. Var., 14(1):105–147, 2008.
• Karine Beauchard. Local controllability of a one-dimensional beam equation. SIAM J. Control Optim., 47(3):1219–1273 (electronic), 2008.
• Karine Beauchard and Jean-Michel Coron. Controllability of a quantum particle in a moving potential well. J. Funct. Anal., 232(2):328–389, 2006.
• Rosa Maria Bianchini. High order necessary optimality conditions. Rend. Sem. Mat. Univ. Politec. Torino, 56(4):41–51, 1998.
• Rosa Maria Bianchini and Gianna Stefani. Sufficient conditions for local controllability. Proc. 25th IEEE Conf. Decision and Control, Athens, pages 967–970, 1986.
• Rosa Maria Bianchini and Gianna Stefani. Controllability along a trajectory: a variational approach. SIAM J. Control Optim., 31(4):900–927, 1993.
• Jerry Bona and Ragnar Winther. The Korteweg-de Vries equation, posed in a quarter-plane. SIAM J. Math. Anal., 14(6):1056–1106, 1983.
• Nicolas Burq. Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mém. Soc. Math. France (N.S.), (55):126, 1993.
• Eduardo Cerpa. Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim., 46(3):877–899 (electronic), 2007.
• Eduardo Cerpa and Emmanuelle Crépeau. Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Inst. H. Poincaré Anal. Non Linéaire, in press, 2008.
• Marianne Chapouly. Global controllability of nonviscous Burgers type equations. C. R. Math. Acad. Sci. Paris, 344(4):241–246, 2007.
• Wei-Liang Chow. Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann., 117:98–105, 1939.
• Marco Cirinà. Boundary controllability of nonlinear hyperbolic systems. SIAM J. Control, 7:198–212, 1969.
• Jean-Michel Coron. Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems, 5(3):295–312, 1992.
• Jean-Michel Coron. Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris Sér. I Math., 317(3):271–276, 1993.
• Jean-Michel Coron. On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions. ESAIM Control Optim. Calc. Var., 1:35–75 (electronic), 1995/96.
• Jean-Michel Coron. On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. (9), 75(2):155–188, 1996.
• Jean-Michel Coron. On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain. SIAM J. Control Optim., 37(6):1874–1896, 1999.
• Jean-Michel Coron. Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations. ESAIM Control Optim. Calc. Var., 8:513–554, 2002. A tribute to J. L. Lions.
• Jean-Michel Coron. On the small-time local controllability of a quantum particle in a moving one-dimensional infinite square potential well. C. R. Math. Acad. Sci. Paris, 342(2):103–108, 2006.
• Jean-Michel Coron. Control and nonlinearity, volume 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. ISBN 0-8218-3668-4; 978-08218-3668-2.
• Jean-Michel Coron and Emmanuelle Crépeau. Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. (JEMS), 6(3):367–398, 2004.
• Jean-Michel Coron and Andrei V. Fursikov. Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys., 4(4):429–448, 1996.
• Jean-Michel Coron and Emmanuel Trélat. Global steady-state controllability of one-dimensional semilinear heat equations. SIAM J. Control Optim., 43(2):549–569 (electronic), 2004.
• Jean-Michel Coron and Emmanuel Trélat. Global steady-state stabilization and controllability of 1D semilinear wave equations. Commun. Contemp. Math., 8(4):535–567, 2006.
• Lokenath Debnath. Nonlinear water waves. Academic Press Inc., Boston, MA, 1994.
• François Dubois, Nicolas Petit, and Pierre Rouchon. Motion planning and nonlinear simulations for a tank containing a fluid. In European Control Conference (Karlruhe, Germany, September 1999), 1999.
• Caroline Fabre. Résultats de contrôlabilité exacte interne pour l’équation de Schrödinger et leurs limites asymptotiques: application à certaines équations de plaques vibrantes. Asymptotic Anal., 5(4):343–379, 1992.
• Caroline Fabre. Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems. ESAIM Contrôle Optim. Calc. Var., 1:267–302 (electronic), 1995/96.
• Caroline Fabre, Jean-Pierre Puel, and Enrique Zuazua. Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A, 125(1):31–61, 1995.
• Hector O. Fattorini. Infinite-dimensional optimization and control theory, volume 62 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999. ISBN 0-521-45125-6.
• Hector O. Fattorini and David L. Russell. Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal., 43:272–292, 1971.
• Enrique Fernández-Cara, Sergio Guerrero, Oleg Yu. Imanuvilov, and Jean-Pierre Puel. Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. (9), 83(12):1501–1542, 2004.
• Enrique Fernández-Cara and Enrique Zuazua. Null and approximate controllability for weakly blowing up semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 17(5):583–616, 2000.
• Michel Fliess, Jean Lévine, Philippe Martin, and Pierre Rouchon. Flatness and defect of non-linear systems: introductory theory and examples. Internat. J. Control, 61(6):1327–1361, 1995.
• Andrei V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete Contin. Dyn. Syst. Series A, 10(1-2):289–314, 2004. Partial differential equations and applications.
• Andrei V. Fursikov and Oleg Yu. Imanuvilov. Controllability of evolution equations, volume 34 of Lecture Notes Series. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.
• Andrei V. Fursikov and Oleg Yu. Imanuvilov. Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys, 54:565–618, 1999.
• Olivier Glass. Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles en dimension 3. C. R. Acad. Sci. Paris Sér. I Math., 325(9):987–992, 1997.
• Olivier Glass. Exact boundary controllability of 3-D Euler equation. ESAIM Control Optim. Calc. Var., 5:1–44 (electronic), 2000.
• Olivier Glass. On the controllability of the Vlasov-Poisson system. J. Differential Equations, 195(2):332–379, 2003.
• Olivier Glass. Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case. SIAM J. Control Optim., 44(3):1105–1147 (electronic), 2005.
• Olivier Glass. On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc. (JEMS), 195(2):332-379, 2006.
• Pierre Grisvard. Singularities in boundary value problems, volume 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1992. ISBN 2-225-82770-2.
• Richard S. Hamilton. The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.), 7(1):65–222, 1982.
• Henry Hermes. Controlled stability. Ann. Mat. Pura Appl. (4), 114:103–119, 1977.
• Henry Hermes. Control systems which generate decomposable Lie algebras. J. Differential Equations, 44(2):166–187, 1982. Special issue dedicated to J. P. LaSalle.
• Lars Hörmander. On the Nash-Moser implicit function theorem. Ann. Acad. Sci. Fenn. Ser. A I Math., 10:255–259, 1985.
• Thierry Horsin. On the controllability of the Burgers equation. ESAIM Control Optim. Calc. Var., 3:83–95 (electronic), 1998.
• Oleg Yu. Imanuvilov. Boundary controllability of parabolic equations. Uspekhi Mat. Nauk, 48(3(291)):211–212, 1993.
• Oleg Yu. Imanuvilov. Controllability of parabolic equations. Mat. Sb., 186(6):109–132, 1995.
• Oleg Yu. Imanuvilov. On exact controllability for the Navier-Stokes equations. ESAIM Control Optim. Calc. Var., 3:97–131 (electronic), 1998.
• Oleg Yu. Imanuvilov. Remarks on exact controllability for the Navier-Stokes equations. ESAIM Control Optim. Calc. Var., 6:39–72 (electronic), 2001.
• Albert Edward Ingham. Some trigonometrical inequalities with applications to the theory of series. Math. Z., 41:367–369, 1936.
• Stéphane Jaffard. Contrôle interne exact des vibrations d’une plaque carrée. C. R. Acad. Sci. Paris Sér. I Math., 307(14):759–762, 1988.
• Stéphane Jaffard and Sorin Micu. Estimates of the constants in generalized Ingham’s inequality and applications to the control of the wave equation. Asymptot. Anal., 28(3-4):181–214, 2001.
• Stéphane Jaffard, Marius Tucsnak, and Enrique Zuazua. Singular internal stabilization of the wave equation. J. Differential Equations, 145(1):184–215, 1998.
• Jean-Pierre Kahane. Pseudo-périodicité et séries de Fourier lacunaires. Ann. Sci. École Norm. Sup. (3), 79:93–150, 1962.
• Tosio Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. ISBN 3-540-58661-X.
• Matthias Kawski. High-order small-time local controllability. In H.J. Sussmann, editor, Nonlinear controllability and optimal control, volume 133 of Monogr. Textbooks Pure Appl. Math., pages 431–467. Dekker, New York, 1990.
• Vilmos Komornik and Paola Loreti. A further note on a theorem of Ingham and simultaneous observability in critical time. Inverse Problems, 20(5):1649–1661, 2004.
• Béatrice Laroche, Philippe Martin, and Pierre Rouchon. Motion planning for the heat equation. Internat. J. Robust Nonlinear Control, 10(8):629–643, 2000.
• Irena Lasiecka and Roberto Triggiani. Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems. Appl. Math. Optim., 23(2):109–154, 1991.
• Irena Lasiecka and Roberto Triggiani. Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet control. Differential Integral Equations, 5(3):521–535, 1992.
• Claude Le Bris. Control theory applied to quantum chemistry: some tracks. In Contrôle des systèmes gouvernées par des équations aux dérivées partielles (Nancy, 1999), volume 8 of ESAIM Proc., pages 77–94 (electronic). Soc. Math. Appl. Indust., Paris, 2000.
• Gilles Lebeau. Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. (9), 71(3):267–291, 1992.
• Gilles Lebeau and Luc Robbiano. Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations, 20(1-2):335–356, 1995.
• Ta Tsien Li. Controllability and Observability for Quasilinear Hyperbolic Systems, volume 2 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2008.
• Ta Tsien Li and Bo-Peng Rao. Exact boundary controllability for quasi-linear hyperbolic systems. SIAM J. Control Optim., 41(6):1748–1755 (electronic), 2003.
• Ta Tsien Li and Wen Ci Yu. Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series, V. Duke University Mathematics Department, Durham, NC, 1985.
• Ta Tsien Li and Bing-Yu Zhang. Global exact controllability of a class of quasilinear hyperbolic systems. J. Math. Anal. Appl., 225(1):289–311, 1998.
• Jacques-Louis Lions. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics]. Masson, Paris, 1988. Contrôlabilité exacte. [Exact controllability], With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch.
• Walter Littman. Boundary control theory for hyperbolic and parabolic partial differential equations with constant coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5(3):567–580, 1978.
• Elaine Machtyngier. Exact controllability for the Schrödinger equation. SIAM J. Control Optim., 32(1):24–34, 1994.
• Elaine Machtyngier and Enrique Zuazua. Stabilization of the Schrödinger equation. Portugal. Math., 51(2):243–256, 1994.
• Hugues Mounier, Joachim Rudolph, Michel Fliess, and Pierre Rouchon. Tracking control of a vibrating string with an interior mass viewed as delay system. ESAIM Control Optim. Calc. Var., 3:315–321 (electronic), 1998.
• Nicolas Petit and Pierre Rouchon. Flatness of heavy chain systems. SIAM J. Control Optim., 40(2):475–495 (electronic), 2001.
• Kim-Dang Phung. Observability and control of Schrödinger equations. SIAM J. Control Optim., 40(1):211–230 (electronic), 2001.
• Petr K. Rashevski. About connecting two points of complete nonholonomic space by admissible curve. Uch Zapiski Ped. Inst. Libknexta, 2:83–94, 1938.
• Jean-Pierre Raymond. Feedback boundary stabilization of the two-dimensional Navier-Stokes equations. SIAM J. Control Optim., 45(3):790–828 (electronic), 2006.
• Jean-Pierre Raymond. Feedback boundary stabilization of the three-dimensional incompressible Navier-Stokes equations. J. Math. Pures Appl. (9), 87(6):627–669, 2007.
• Ray M. Redheffer. Remarks on incompleteness of {$$e^{i\lambda nx}$$}, nonaveraging sets, and entire functions. Proc. Amer. Math. Soc., 2:365–369, 1951.
• Lionel Rosier. Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var., 2:33–55 (electronic), 1997.
• Pierre Rouchon. Control of a quantum particle in a moving potential well. In Lagrangian and Hamiltonian methods for nonlinear control 2003, pages 287–290. IFAC, Laxenburg, 2003.
• Walter Rudin. Functional analysis. McGraw-Hill Book Co., New York, 1973. McGraw-Hill Series in Higher Mathematics.
• David L. Russell. Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl., 18:542–560, 1967.
• David L. Russell. Exact boundary value controllability theorems for wave and heat processes in star-complemented regions. In Differential games and control theory (Proc. NSF—CBMS Regional Res. Conf., Univ. Rhode Island, Kingston, R.I., 1973), pages 291–319. Lecture Notes in Pure Appl. Math., Vol. 10. Dekker, New York, 1974.
• David L. Russell. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev., 20(4):639–739, 1978.
• Adhémar Jean Claude Barré de Saint-Venant. Théorie du mouvement non permanent des eaux, avec applications aux crues des riviéres et à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris Sér. I Math., 53:147–154, 1871.
• Laurent Schwartz. Étude des sommes d’exponentielles. 2ième éd. Publications de l’Institut de Mathématique de l’Université de Strasbourg, V. Actualités Sci. Ind. Hermann, Paris, 1959.
• Armen Shirikyan. Approximate controllability of three-dimensional Navier-Stokes equations. Comm. Math. Phys., 266(1):123–151, 2006.
• Héctor J. Sussmann. Lie brackets and local controllability: a sufficient condition for scalarinput systems. SIAM J. Control Optim., 21(5):686–713, 1983.
• Héctor J. Sussmann. A general theorem on local controllability. SIAM J. Control Optim., 25(1):158–194, 1987.
• Alexander I. Tret'yak. Necessary conditions for optimality of odd order in a time-optimality problem for systems that are linear with respect to control. Mat. Sb., 181(5):625–641, 1990.
• Marius Tucsnak and George Weiss. Passive and conservative linear systems. Preliminary version. Université de Nancy, 2006.
• Rafael Vázquez, Emmanuel Trélat, and Jean-Michel Coron. Control for fast and stable laminar-to-high-Reynolds-number transfer in a 2D Navier-Stokes channel flow. Discrete Contin. Dyn. Syst. Series B, 10(4):925–956, 2008.
• Enrique Zuazua. Exact boundary controllability for the semilinear wave equation. In Haïm Brezis and Jacques-Louis Lions, editors, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. X (Paris, 1987–1988), volume 220 of Pitman Res. Notes Math. Ser., pages 357–391. Longman Sci. Tech., Harlow, 1991.
• Enrique Zuazua. Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(1):109–129, 1993.
• Enrique Zuazua. Remarks on the controllability of the Schrödinger equation. In Quantum control: mathematical and numerical challenges, volume 33 of CRM Proc. Lecture Notes, pages 193–211. Amer. Math. Soc., Providence, RI, 2003.
• Enrique Zuazua. Control and numerical approximation of the wave and heat equations. In Marta Sanz-Solé, Javier Soria, Varona Juan Luis, and Joan Verdera, editors, Proceedings of the International Congress of Mathematicians, Vol. I, II, III (Madrid, 2006), volume III, pages 1389–1417. European Mathematical Society, 2006.
• Enrique Zuazua. Controllability and observability of partial differential equations: Some results and open problems. In C. M. Dafermos and E. Feireisl, editors, Handbook of differential equations: evolutionary differential equations, volume 3, pages 527–621. Elsevier/North-Holland, Amsterdam, 2006.

Internal references

• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Victor M. Becerra (2008) Optimal control. Scholarpedia, 3(1):5354.
• Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.

Sponsored by: Jean-Jacques Slotine, Nonlinear Systems Lab, MIT, Cambridge, MA
Reviewed by: Anonymous
Accepted on: 2009-11-04 15:22:43 GMT
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 193, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.7500427961349487, "perplexity_flag": "middle"}