Welcome to the Simoneirani.com website ('the website'). We take great pride in our website and aim to provide our customers with an effortless online shopping experience.
Please take a few minutes to read the information below about the legal Terms & Conditions that apply to your use of this website and any services or products that you order from us.
By accessing the website you signify you agree to these Terms & Conditions which form a legally binding contract between you and Simone Irani Pte Ltd ('Simone Irani'/'we'/'our'/'us') and its successors and assignees.
The website is provided solely for your personal use for the sale of clothing, accessories and other items ('items').
Please read the Terms carefully and make sure that you understand them before making any purchases from our website.
Your purchase from the website indicates that you have read, accepted and will comply with the Terms. You also represent that you are at least eighteen (18) years old, or you are at least the minimum legal age to enter into a contract in the jurisdiction in which you are using the site and you are not purchasing, nor will you use the goods in the course of a business.
In addition, when you use any current or future version of the web site, you also will be subject to the Terms of this contract. Please print a copy of this contract for your records. We may, from time to time amend the Terms at our sole discretion. By using the website after any such change(s), you agree to comply with, and be bound by, the most recent version of the Terms.
Whilst every effort is made to ensure that all details, prices, product images and descriptions displayed are correct at the time, there may however, from time to time, be technical inaccuracies, typographical errors or omissions relating to a product description, product image, price, promotion, special offer or availability. There may be slight variations , from time to time, in style or color reproduction, or in text descriptions.
While we have attempted to show textures, appearances and colors of our products as accurately as possible, the images of the items on the website are for illustrative purposes only and what you see will vary to the delivered product depending on your computer monitor. We are unable to promise or guarantee that your monitor's display of any color will actually reflect the color of the product delivered to you.
If you have registered your email address for notification of the arrival of a specific product featured on the website, we will attempt to notify you by email within 48 hours of the product becoming available on the website. Please be aware that on occasion certain products that are in particularly high demand will sell out during this period.
Simoneirani.com will store a record of your transactions for a minimum of one year.
Our shopping pages will guide you through the steps you need to take to place an order with us. Our order process allows you to check and amend any errors before submitting your order to us. Please take the time to read and check your order at each page of the order process. You place your order for items from the website by clicking on the Complete Order button at the end of the online checkout process.
Once you have placed your order, we will send you an email acknowledging the exact details of your order, providing you with an order number and the value of your order that will be debited from your credit/debit card. This is not an order confirmation or order acceptance from Simoneirani.com.
Unless we have notified you that we do not accept your order or you have cancelled your order in accordance with the Exchanges & Returns policy set out below, our acceptance of your order and the completion of the contract between you and us will take place when you receive an email from us notifying you that we have dispatched the Items ordered by you.
Our acceptance of your order will be deemed complete and received by you at the time and date of sending of that email, which time and date is specified on the email. The deemed time and date of receipt by you shall apply regardless of whether, for reasons outside our control, you have not received that email.
We may not accept your order if :(i) an item you have ordered is out of stock, (ii) your card is due to expire within 28 days of your purchase date (iii) we are unable to obtain authorisation for your payment (iv) we identify a product or pricing error (vi) shipping restrictions apply or (v) it is intended for commercial use such as resale or distribution for commercial purposes.
The resale of any products purchased on the website without the express written agreement of Simone Irani is not permitted.
If we are unable to fulfill your order following our order acknowledgement, we will contact you by email advising you of this and re-credit your account as soon as possible and in any event within 30 days of your order being placed.
All items are subject to availability. We will use our best endeavours to ensure that products ordered are available for delivery, and in most cases will notify you when a product is unavailable soon after you have completed your order.
In some cases, immediate notification is not possible and we may need to reject an order where we later establish that it is unavailable for delivery. If your order is delayed awaiting supply of stock or for any other reason, we will make every effort to keep you informed and discuss revised delivery timing.
Payment can be made securely online by all major credit cards and PayPal (Visa, AmEx, MC). The credit/debit card that is being used must be yours. All credit/debit card holders are subject to validation checks and authorization by the card issuer.
By submitting an order to us through our website you represent and warrant that the payment details provided on your order are valid and correct and that when your order is accepted and processed by us, payment will be made in full. Failure to do so will mean that we will not process your order and your items will not be dispatched. Please note that payment is not deemed to have been made unless or until payment is showing as cleared funds in our account.
If the issuer of your payment card refuses to authorize payment, or subsequently refuses to, or does not, for any reason, authorize payment to us, we will not be liable for any delay or non-delivery of your order.
In the event that your card authorization and validation is declined, we reserve the right to cancel your order. For first time online customers, we may request that shipping be to the credit card billing address only.
In the case of exchanges/returns, risk in the returned items will only pass to us when the items are delivered to us.
We aim to deliver your chosen Items in perfect condition and in the shortest possible time. Our delivery charges are dependent on your shipping destination.
We require a signature for any goods delivered, at which point responsibility for your purchased goods passes to you. If you have specified a recipient who is not you for delivery purposes then you accept that evidence of a signature by them (or at that delivery address) is evidence of delivery and fulfillment by Simone Irani, and transfer of responsibility in the same way. The goods are your responsibility from the time we deliver them to the address you gave us.
You will own the goods on and from the later of the point in time when: (a) we receive payment in full, or (b) we deliver the goods to you. If your payment is declined or reversed for any reason we reserve the right to reclaim the items from your possession, custody or control even if they have been delivered to you or moved from the delivery address. We reserve the right to keep or sell the Items.
In estimating your delivery time, please note that delivery times may vary from product to product. Most orders will be dispatched within one to three business days of placing an order.
Delivery times appearing on the website are estimates and are to be used as a guide only and commence from the date of dispatch. We attempt to deliver the Items within the estimated time frame; however, WE WILL NOT ACCEPT ANY RESPONSIBILITY OR LIABILITY FOR ANY LATE SHIPMENTS. For further detail, please refer to the DISCLAIMERS; LIMITATION OF LIABILITY section of these Terms.
Please order from Simoneirani.com with enough lead time to prevent any loss or disappointment resulting from the delivery time as we cannot be responsible for this.
Shipping fees will be charged on all orders in accordance with our shipping policy stated in our FAQs which may be updated at our discretion from time to time.
Once your order has been dispatched, you will receive an email with tracking details, which will allow you to track your order every step of the way. You can also check the status of your order by logging into your account. We use SingPost for local shipments and UPS for international shipments. Please ensure that an authorized person is available at your shipping address as a signature will be requested upon receipt of shipment. If an authorized person is not able to sign for your shipment, the driver will leave a card and the shipment will be returned to the nearest collection centre for you to collect. Please ensure your shipping details are correct as we are unable to re-direct shipments once your order has been dispatched.
A free shipping calculator at checkout will notify you how much more you need to spend to qualify for the free shipping.
From time to time, we may offer free shipping promotions. Please note that free shipping promotions will be applied to your order on the payment page and may be subject to terms and conditions (such as minimum spend or specific dates).
If you are buying from outside Singapore, in addition to our shipping costs, you may incur import duties and taxes levied by the country to which you have specified delivery of your order. Please note that we have no control over these charges and you will be responsible for payment of any such import duties and taxes.
Simone Irani reserves the right to deduct any return costs incurred from the total return amount for orders that are refused at customs should duties and taxes be unpaid.
During sale periods, you may experience a delay in receiving your order, we apologize in advance for any inconvenience. Please allow at least 2 to 3 business days for us to dispatch your order during these busy periods.
To be notified when we have new arrivals, events, sales, promotions simply sign up to receive all the latest Simone Irani news by entering your email and joining the Inner Circle at the bottom of any page.
Gift cards are non-transferable and may not be returned or redeemed for cash or used to purchase items in our stores.
If your order total is less than the value of the gift card any remaining balance will be retained for use against future purchases.
If your order exceeds the value of the gift card the balance must be paid using store credit from your account or by credit or debit card.
If you return products you have purchased using a gift card, the remaining balance will be credited to your account as credit and will be redeemed against subsequent orders. You may check your store credit balance online via your account page.
Gift cards will be e-mailed to the recipient once the full order has been processed and payment has been taken.
Simone Irani is not liable for delivery of a gift card to an incorrect or non-existent email address. This is the sole responsibility of the purchaser.
Simone Irani reserves the right to cancel a gift card if we deem such action necessary.
Shipping is applicable on any products purchased with a gift card.
You must request your exchange or return and send your unwanted items back to us within 14 days. We do not accept exchanges or returns for sale/marked-down items or on earrings. For full details on how to return your purchase see our Exchanges & Returns policy.
Please note you can only exchange items for a different size, based on stock availability. If you wish to exchange your item for an alternative product, we suggest that you return it for online store credit and purchase the new item separately.
Items should be returned unused and with all garment tags still attached. Returns that are damaged or soiled will not be accepted and will be sent back to the customer and exchange refused.
Please email [email protected] if any of your purchases have been delivered without Simone Irani tags.
Original shipping charges are non-refundable and charges for exchanged/returned items are your responsibility. Further shipping charges apply on all replacement Items.
We take no responsibility, and will not be liable, for loss of returned items prior to delivery to Simone Irani. We encourage you to return your Item(s) via registered or traceable postal services as return parcels remain your responsibility until received by Simone Irani. Returned items must be sent by a prepaid method of shipping.
Returns are processed in accordance with our processing procedures. If you are to receive a credit for Simoneirani.com, the credits are applied to your Simoneirani.com account once your return has been processed. Simoneirani.com credits are valid for 6 months from date of issue and can be used only for purchases via the website.
You can check your Simoneirani.com balance by logging on to your account. To use your Simoneirani.com credits you will have the option in the checkout on the payment step to use Simoneirani.com credits.
During our sale clearance, products marked as sale/marked-down cannot be returned or exchanged.
We offer a flexible returns policy to make your online shopping experience even easier. We do monitor the number of returns made by customers, and continued returns will be flagged and may, at our discretion, lead to the closure of your Simoneirani.com account or future orders being refused.
We may offer unique promotional codes for a specific promotional activity from time to time. These codes may be redeemed for product discounts for specific time periods. Code must be entered and applied at the checkout to redeem.
Offers are valid exclusively online in-between the dates advertised and cannot be transferred to another location, extended or combined with any other offer. This offer is available to customers who receive the code directly from Simoneirani.com and cannot be forwarded or shared with other customers. Normal exchanges/returns policy applies. Each code is unique to the recipient, is not transferable, has no monetary value and under no circumstances can it be converted into currency. Each code has an expiry date, which will be stated at the time of promotion or on accompanying promotional materials which cannot be modified. Some discount codes may be redeemed both in store and online, or when specified, for online purchases only. Again, this will be stipulated at the time of promotion and on accompanying promotional materials.
When an order is placed using a promotional code and contains a number of items, the value of the discount is spread over each item according to the pro-rated value of that item. Any return of an item purchased in such an order will be reduced by the value of discount attached to it. The 'discount' will not be refunded.
Under no circumstances can the value of a discount code be deducted from an order if that code is defective or out of date: no replacement codes will be issued. Only one discount voucher can be used per order.
Free shipping offers apply to all deliveries within Singapore and on all orders valued at US $300 or more. In all cases, only the delivery is free. Any taxes, duties or customs fees are chargeable.
We may run competitions and promotions throughout the year. Competitions and promotions may have their own separate terms and conditions which will apply in addition to these Terms. Please carefully read those terms and conditions before entering or participating in a competition or promotion. If a competition or a promotion does not have its own specific terms, these general Terms will apply.
Your use of the website and its contents, including but not limited to the text, graphics, logos, button icons, images, audio clip, videos, data compilations and software (collectively 'the content') and the trademarks, service marks and logos contained therein ('marks') , grants no rights to you in relation to any copyright, designs, trademarks and all other intellectual property and material rights relating to the content and marks.
All such content and marks are owned by or licensed to Simone Irani and are protected by national intellectual property and other laws and international treaty provisions.
The content and the marks on the website are provided to you as is for your information and personal, non-commercial use only. For your personal use, you may view, copy, and print pages of the website. Otherwise, the website may not be used, copied, modified, downloaded, reproduced, duplicated, distributed, published, performed, streamed, transmitted, broadcast, displayed, sold, resold, licensed, or otherwise exploited, in full or in part, for any other purposes whatsoever without our prior written consent.
Any unauthorized use of any materials on this website may violate copyright, trademark, and other laws.
You agree not to use, copy, download, or distribute any of the content other than as expressly permitted herein, including any use, copying, downloading, making derivative works from, or use of account information for the benefit of any third party, or distribution of user submissions of third parties obtained through the website for any commercial purposes.
You shall not, and shall not permit others to, license, sublicense, sell, resell, transfer, assign, distribute, or otherwise commercially exploit or make available to any third party, all or any part of the rights granted to you under these terms, the website, or any content or materials accessible through the website, in whole or part.
You may not create, recreate, advertise or distribute any portion of the content unless authorized by us. You may not incorporate the content into, or stream or transmit the content via any hardware or software application. You may not frame, link to, or utilise any framing or linking techniques to enclose the website or any portion thereof without our prior written consent which may be withheld in our absolute discretion.
You may not build a business using the content, whether or not for profit. If you copy or print pages of the content for personal use, you must retain all copyright and other proprietary notices contained therein. You agree not to circumvent, disable or otherwise interfere with security related features of the website or features that prevent or restrict use or copying of any content or enforce limitations on use of the website or the content therein. Simone Irani reserves the right to refuse service to any individual at its sole discretion and without limitation if it believes that a user's conduct violates applicable laws or is otherwise harmful to Simone Irani.
We grant you a limited, revocable, non-exclusive, non-transferable license to access and make personal and non-commercial use of the website.
In no event shall Simone Irani, its directors, members, employees or agents be liable for any direct, special, indirect or consequential damages or any other damages of any kind, including, but not limited to, loss of use, loss of profits or loss of data, whether in an action in contract, tort (including, but not limited to, negligence) or otherwise, arising out of or in any way connected to the use of, or inability to use, the complete website, including without limitation any damages caused by or resulting from reliance by the user on any information obtained from Simone Irani, or that result from mistakes, omissions, interruptions, deletion of files or email, errors, defects, viruses, delays in operation or transmission or failure of performance, whether or not resulting from acts of god, communications failure, theft, destruction or unauthorized access to Simone Irani's records, programs or services.
In no event shall the aggregate liability of Simone Irani, whether in contract, warranty, tort (including negligence, whether active, passive or imputed), product liability, strict liability or other theory, arising out of or relating to the use of or inability to use the site exceed any compensation you pay, if any, to Simone Irani for access to or use of the complete site. Some jurisdictions do not allow the limitation of liability in contracts with consumers, so some or all of these limitations of liability may not apply to you.
You agree to indemnify, defend and hold harmless Simone Irani Pte Ltd, its subsidiaries, affiliates, officers, agents, partners, members, employees, independent contractors, service providers and consultants, and their respective directors, employees and agents from and against any claims, losses, obligations, demands, liabilities, expenses, damages, costs, debt, actions, suits, and legal fees resulting from (1) your use of, access to, or misuse of the website or Simone Irani products or services; (2) your violation of any of these terms; (3) your violation of any third party right, including without limitation any copyright, property, or privacy right; (4) any claim that one of your user submissions caused damage to a Simone Irani party or a third party; (5) any activity related to your Simone Irani account, related to your internet account, or related to your email address and password (including negligent or wrongful conduct) by you or any other person accessing the website using your internet account or your email address; or (6) any transaction you conduct as a result of the contact facilitated by the website. You agree to cooperate with us (at your own expense) in the handling of disputes, complaints, investigations or litigation that arise as a result of your use of our products including but not limited to disputes, complaints, investigations or litigation that arises out of or relates to incorrect information you have given us.
The obligations under this clause will survive termination of these terms. If Simone Irani does take any legal action against you as a result of your violation of these terms, Simone Irani will be entitled to recover from you, and you agree to pay, all reasonable legal fees and costs of such action, in addition to any other relief granted to Simone Irani.
ALL PRODUCTS ARE SOLD ON AN "AS IS" BASIS WITHOUT ANY WARRANTIES AT ALL. YOU EXPRESSLY AGREE THAT YOUR USE OF THE WEBSITE SHALL BE AT YOUR SOLE RISK AND THAT THE WEBSITE IS PROVIDED "AS IS" AND "AS AVAILABLE." SIMONE IRANI MAKES NO REPRESENTATIONS OR WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, AS TO THE OPERATION OF THE WEBSITE, OR THE INFORMATION, CONTENT, MATERIALS, PRODUCTS (INCLUDING SOFTWARE) OR OTHER SERVICES INCLUDED ON OR OTHERWISE MADE AVAILABLE TO YOU THROUGH THE WEBSITE, UNLESS OTHERWISE SPECIFIED IN WRITING. TO THE FULL EXTENT PERMISSIBLE BY APPLICABLE LAW, SIMONE IRANI DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE, SATISFACTORY QUALITY, SECURITY, ACCURACY, THE USE OF REASONABLE CARE AND SKILL AND NON-INFRINGEMENT AND WARRANTIES ARISING FROM A COURSE OF DEALING , COURSE OF PERFORMANCE OR TRADE USAGE . WITHOUT LIMITING THE FOREGOING, SIMONE IRANI DOES NOT REPRESENT OR WARRANT THAT THE INFORMATION ON THE WEBSITE IS ACCURATE, COMPLETE OR CURRENT; THAT THE WEBSITE OR ANY SERVICE WILL OPERATE WITHOUT INTERRUPTION OR ERROR; OR THAT THE QUALITY, SAFETY ,OR LEGALITY OF ANY CONTENT, PRODUCTS, SERVICES, INFORMATION OR OTHER MATERIAL PURCHASED OR OBTAINED BY YOU THROUGH THE WEBSITE WILL MEET YOUR EXPECTATIONS.
THIS SITE MAY INCLUDE INACCURACIES, MISTAKES OR TYPOGRAPHICAL ERRORS. THE SIMONE IRANI PARTIES ASSUME NO LIABILITY OR RESPONSIBILITY FOR ANY ERRORS, MISTAKES, OR INACCURACIES OF CONTENT, PERSONAL INJURY OR PROPERTY DAMAGE, OF ANY NATURE WHATSOEVER (INCLUDING, WITHOUT LIMITATION, DATA LOSS OR CORRUPTION), RESULTING FROM YOUR ACCESS TO AND USE OF THE WEBSITE, UNAUTHORIZED ACCESS TO OR USE OF OUR SERVERS AND/OR ANY AND ALL PERSONAL INFORMATION AND/OR INFORMATION STORED THEREIN, INTERRSATION OF TRANSMISSION TO OR FROM THE WEBSITE, BUGS, VIRUSES, TROJAN HORSES, DESTRUCTIVE COMPUTER CODES, OR THE LIKE WHICH MAY BE TRANSMITTED TO OR THROUGH THE WEBSITE BY ANY THIRD PARTY, AND/OR LOSS OR DAMAGE OF ANY KIND INCURRED AS A RESULT OF THE USE OF ANY CONTENT POSTED, EMAILED, TRANSMITTED, OR OTHERWISE MADE AVAILABLE VIA THE WEBSITE.
TO THE FULL EXTENT PERMISSIBLE BY LAW, SIMONE IRANI WILL NOT BE LIABLE FOR ANY DAMAGES OF ANY KIND THROUGH THE USE OF THE WEBSITE OR FROM ANY INFORMATION, CONTENT, MATERIALS, PRODUCTS (INCLUDING SOFTWARE) OR OTHER SERVICES INCLUDED ON OR OTHERWISE MADE AVAILABLE TO YOU THROUGH THE WEBSITE, INCLUDING, BUT NOT LIMITED TO DIRECT, INDIRECT, INCIDENTAL, SPECIAL, PUNITIVE, EXEMPLARY, CONSEQUENTIAL OR OTHER SIMILAR DAMAGES, (INCLUDING LOSS OF PROFITS, REVENUE, PRODUCTION, OPPORTUNITY, GOODWILL, REPUTATION, USE OR ANY INDIRECT, REMOTE, ABNORMAL OR UNFORESEEABLE LOSS, OR ANY LOSS OR DAMAGE RELATING TO BUSINESS INTERRUPTION, LOSS OF PROGRAMS OR OTHER DATA ON YOUR INFORMATION SYSTEMS OR COSTS OF REPLACEMENT GOODS) SUFFERED BY YOU OR CLAIMS MADE AGAINST YOU, ARISING OUT OF OR IN CONNECTION WITH THE WEBSITE, INABILITY TO ACCESS OR USE THE WEBSITE, THE PRODUCTS, THE SERVICES, THE LATE SUPPLY OF PRODUCTS, THE TERMS, ERRORS, MISTAKES, OMISSIONS, OR INACCURACIES OF CONTENT, PERSONAL INJURY OR PROPERTY DAMAGE, OF ANY NATURE WHATSOEVER, RESULTING FROM YOUR ACCESS TO AND USE OF THE WEBSITE, UNAUTHORIZED ACCESS TO OR USE OF OUR SERVERS AND/OR ANY AND ALL INFORMATION STORED THEREIN, INTERRUPTION OR CESSATION OF TRANSMISSION TO OR FROM THE WEBSITE, BUGS, VIRUSES, OR THE LIKE, WHICH MAY BE TRANSMITTED TO OR THROUGH THE WEBSITE BY ANY THIRD PARTY, USER SUBMISSIONS, THIRD PARTY WEBSITES; OR ANY LOSS OR DAMAGE OF ANY KIND INCURRED AS A RESULT OF YOUR USE OF ANY CONTENT, TO THE FULL EXTENT PERMISSIBLE BY APPLICABLE LAW, THE SIMONE IRANI PARTIES DISCLAIM ALL OTHER WARRANTIES, EXPRESS OR IMPLIED (OR IF DISCLAIMING IMPLIED WARRANTIES IS NOT PERMITTED BY APPLICABLE LAW) THE SIMONE IRANI PARTIES EXPRESSLY LIMIT THE DURATION OF ALL OTHER WARRANTIES TO THIRTY (30) DAYS FROM RECEIPT OF THE MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, SATISFACTORY QUALITY, SECURITY, ACCURACY, THE USE OF REASONABLE CARE AND SKILL, AND NON-INFRINGEMENT AND WARRANTIES ARISING FROM A COURSE OF DEALING, COURSE OF PERFORMANCE, OR TRADE USAGE. WITHOUT LIMITING THE FOREGOING, SIMONE RANI DOES NOT REPRESENT OR WARRANT THAT THE INFORMATION ON THE WEBSITE OR ON ANY WEBSITES OR ONLINE SERVICES LINKED TO THE WEBSITE IS ACCURATE, COMPLETE OR CURRENT; THAT THE WEBSITE OR ANY SERVICE WILL OPERATE WITHOUT INTERRUPTION OR ERROR; OR THAT THE QUALITY, SAFETY OR LEGALITY OF ANY CONTENT, PRODUCTS, SERVICES, INFORMATION OR OTHER MATERIAL PURCHASED OR OBTAINED BY YOU THROUGH THE WEBSITE.
TO THE MAXIMUM EXTENT PERMITTED UNDER LAW, WHERE OUR LIABILITY CANNOT BE EXCLUDED, OUR TOTAL LIABILITY ARISING OUT OF OR IN CONNECTION WITH THE PRODUCTS, THE SERVICES OR THE TERMS AND THIS WEBSITE AND ITS CONTENT, HOWEVER ARISING, INCLUDING UNDER CONTRACT, TORT (INCLUDING NEGLIGENCE), IN EQUITY, UNDER STATUTE OR OTHERWISE, WILL BE NO MORE THAN THE AMOUNT YOU PAID FOR THE PRODUCT THAT IS THE SUBJECT OF A CLAIM.
ANY CLAIMS RELATING TO USE OF THE WEBSITE MUST BE BROUGHT WITHIN ONE(1) YEAR FROM THE DATE THE CAUSE OF ACTION AROSE. CLAIMS BROUGHT AFTER SUCH PERIOD ARE VOID.
We may include hyperlinks on this website to other websites or resources owned and operated by parties other than Simone Irani. We are not responsible for the content or operation of any other websites that are not under our control nor are we responsible for the availability of such external websites or resources. You access those websites and/or use the website's products and services solely at your own risk. The links are provided solely for your convenience and do not indicate, expressly or impliedly, an endorsement by us of the websites or the products or services provided on the website.
By using the website, you expressly release Simone Irani, its affiliates, and their respective officers, directors, employees, agents, shareholders, licensors, licensees, and suppliers (collectively 'the parties') from any and all liability arising from your use of any third-party website or online service and from any loss or damage of any sort you may incur from dealing with any third party. Any concerns regarding an external link, should be directed to its responsible website administrator.
You shall be solely responsible for your own user submissions and the consequences of posting or publishing them. However, by submitting the user submissions to Simone Irani or using the branded hashtags #SimoneIrani, #SimoneIraniOnline or #SimoneIraniResortWear on any social media or on any of Simone Irani's social media networks, you hereby grant Simone Irani and its successors, assigns, and affiliates a worldwide, non-exclusive, royalty-free, perpetual, irrevocable, transferable, license to use, reproduce, distribute, prepare derivative works of, display, publish, broadcast, perform, make, use, import, offer to sell and otherwise transfer and exploit all copyrights, inventions, and other intellectual property rights in the user submissions in connection with the website and Simone Irani's (and its successors', assigns', and affiliates') business, including without limitation for promoting and redistributing part or all of the website (and derivative works thereof) in any form and media formats and through any media channels.
You also hereby grant each user of the website a worldwide, non-exclusive, royalty-free license to access your user submissions through the website, and to use, reproduce, distribute, prepare derivative works of, display and perform such user submissions as permitted through the functionality of the website and under these terms.
In connection with user submissions, you represent and warrant that you will not: (i) submit or post material that is copyrighted, protected by trade secret or otherwise subject to third party proprietary rights, including privacy and publicity rights, unless you are the owner of such rights or have permission from their rightful owner to post the material and to grant Simone Irani all of the license rights granted herein; (ii) publish falsehoods or misrepresentations that could damage Simone Irani or any third party; (iii) submit or post material that involves gambling, gaming, sweepstakes, and/or any other activity with an entry fee and a prize, or (iv) submit or post material that is otherwise inappropriate. In addition, you agree to pay for all royalties, fees, and other payments owed to any party by reason of your posting or submitting user submissions.
Simone Irani does not endorse any user submission or any opinion, recommendation, or advice expressed therein, and the Simone Irani parties expressly disclaim any and all liability in connection with user submissions. Simone Irani does not permit copyright infringing activities, infringement of rights of publicity, or infringement of other intellectual property rights on its website, and Simone Irani will remove all content (including user submissions) if properly notified that such content or user submission infringes on another's rights. Simone Irani will not necessarily monitor user submissions. However, Simone Irani reserves the right to monitor, alter, edit, or remove content (including user submissions) and block users without prior notice and for any reason. Simone Irani also reserves sole discretion to decide whether any content (including a user submission) is appropriate and complies with these terms.
You understand that when using the website, you will be exposed to user submissions from a variety of sources, and that Simone Irani is not responsible for the accuracy, usefulness, safety, or intellectual property rights of or relating to such user submissions. You further understand and acknowledge that you may be exposed to user submissions that are inaccurate, offensive, indecent, or objectionable, and you agree to waive, and hereby do waive, any legal or equitable rights or remedies you have or may have against the Simone Irani parties with respect thereto, and agree to indemnify and hold the Simone Irani parties harmless to the fullest extent allowed by law regarding all matters related to your use of the website.
All notices given by you to us must be given to 501 Bukit Timah Road, 02-17 Cluny Court, Singapore 259760 or sent to [email protected].
We may give notice to you at either the email or postal address you provide to us when placing an order, or in any of the ways specified in the para above. Notice will be deemed received and properly served immediately when posted on our website, 24 hours after an e-mail is sent, or 5 days after the date of posting of any letter. In proving the service of any notice, it will be sufficient to prove, in the case of a letter, that such letter was properly addressed, stamped and placed in the post and, in the case of an email, that such e-mail was sent to the specified e-mail address of the addressee.
We reserve the right to amend any typographical errors, inaccuracies or omissions and to correct or update content or cancel orders at any time without prior notice.
Please note that this also applies to orders that have been submitted. If a product is quoted at an incorrect price due to typographical error, inaccuracy or oversight we will let you know as soon as possible and not accept your order, further: (a) we reserve the right to refuse or cancel any orders placed for the product quoted at the incorrect price, even if we have confirmed the receipt of your order and charged your credit or debit card and the order is in transit; and (b) if we have charged your credit or debit card but subsequently cancelled your order, we will promptly issue a refund to your credit or debit card account for the amount charged.
To the extent permitted by law Simone Irani Pte Ltd shall not be liable for any claims either direct or in terms of consequential loss relating to the accuracy of the information contained in any of the sections of this website, whether this arises from breach of duty, breach of contract, negligence or any other way.
If you believe that there is a typographical error, inaccuracy or oversights on this website please contact us by email at [email protected].
We will not be liable for any delay or failure to perform our obligations under the Terms if such delay is due to any circumstance beyond our reasonable control, including, but not limited to, acts of God, war, riot, civil commotion, fire, flood, storm, strike, breakdown of plant or machinery, default of suppliers or sub-contractors or inability to procure materials required for performance of the contract, failure of a utility service or transport network, explosion, accident, malicious damage, compliance with any law or government order, rule, regulation or direction, or unforeseen circumstances ("Force Majeure Event").
Our performance under the Terms is deemed to be suspended for the period that the Force Majeure Event continues, and we will have an extension of time for performance for the duration of that period. We will use our reasonable endeavours to bring the Force Majeure Event to a close or to find a solution by which our obligations under the Terms may be performed despite the Force Majeure Event. If the Force Majeure Event continues for a continuous period of more than 2 months, we may terminate our contract with you by giving you notice in writing.
Any waiver of any of the provisions of these Terms by Simone Irani shall not be deemed a waiver of any subsequent breach or default and shall in no way affect the other provisions of these Terms.
These Terms, and any rights and licenses granted hereunder, may not be transferred or assigned by you without Simone Irani's prior written consent, but may be assigned by Simone Irani without restriction and without notice to you.
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text:
\section{Introduction}\label{sec:intro}

\IEEEPARstart{T}{his} paper
 considers the feedback control of linear systems subject to
 structural constraints on the controller. We are interested in
 characterizing when the set of achievable closed-loop maps is
 convex. Convexity is important because in many cases it makes the
 problem of synthesizing structurally constrained controllers that
 optimize some performance objective a tractable one.

Suppose $w\in \mathcal{W}$ is the exogenous disturbance,
$u\in\mathcal{U}$ is the control input, $z\in\mathcal{Z}$ is the
regulated output, and $y\in\mathcal{Y}$ is the measurement.  We assume
that the plant is a linear and continuous map
$P:(\mathcal{W,U})\to(\mathcal{Z,Y})$. The controller is a map
$K:\mathcal{Y}\to\mathcal{U}$ that is connected to the plant in
feedback. We partition the four blocks of $P$ as $P_{ij}$ and by
convention we let $G = P_{22}$. The closed-loop interconnection
mapping $w\mapsto z$ is depicted in Figure~\ref{fig:PK_diagram}.

\begin{figure}[ht]
\centering
\begin{tikzpicture}[thick,node distance=1.4cm]
\node [block](P){$\begin{matrix}P_{11}\rule[-1.3ex]{-1.3ex}{0pt}& P_{12}\\
P_{21}\rule{-1.3ex}{0pt} & G\end{matrix}$};
\node [block,below of=P](K){$K$};
\draw [<-] (P.east)+(0,-0.3) -- +(0.6,-0.3) |- node[pos=0.25,anchor=west]{$u$} (K);
\draw [<-] (P.east)+(0,0.3) -- +(1,0.3) node [anchor=west]{$w$};
\draw [->] (P.west)+(0,-0.3) -- +(-0.6,-0.3) |- node[pos=0.25,anchor=east]{$y$} (K);
\draw [->] (P.west)+(0,0.3) -- +(-1,0.3) node [anchor=east]{$z$};
\end{tikzpicture}
\caption{\label{fig:PK_diagram}Plant $P$ connected to a controller $K$.}
\end{figure}

The set of \emph{achievable} closed-loop
maps is
\[
C = \set{ P_{11} + P_{12}K(I-GK)^{-1}P_{21} }{ K \in S }
\]
Here the set $S$ is the set of $K$ satisfying the structural constraints
imposed in the problem. Examples of such structural constraints
include sparsity requirements, where for example the controller
responsible for choosing~$u_j$ cannot measure~$y_i$. Other
possibilities include constraints arising as a consequence of
measurement delays.  To make notation more compact, we define the
function
\begin{equation}\label{eq:h}
\hg(K) = -K(I-GK)^{-1}
\end{equation}
and then we may write $C = P_{11}-P_{12}\hg(S)P_{21}$ where $\hg(S)$
is the image of $S$ under the map $\hg$.  Note that $\hg$ implicitly
depends on the map $G$.  In this paper we refer to $\hg$ and $G$
together with the understanding that $\hg$ is defined as a function of
$G$.  A constrained synthesis problem may take the form
\begin{equation}\label{opt:synth}
	\begin{aligned}
	\text{minimize}&\qquad \norm{X} \\
	\text{subject to}&\qquad X \in C
	\end{aligned}
\end{equation}
where $\norm\cdot$ is some convex measure of performance, such as a
norm. It is desirable to know when $C$ is a convex set, because then
if~\eqref{opt:synth} is a convex optimization problem.

A set $S$ is called \emph{quadratically invariant} under $G$ if
$KGK\in S$ for all $K\in S$.  It is shown
in~\cite{rotkowitz02,rotkowitz06} in that, roughly speaking,
$\hg(S)=S$ if and only if $S$ is quadratically invariant under
$G$. This provides a sufficient condition under which $C$ is convex.

This work builds on the notion of quadratic invariance developed
in~\cite{rotkowitz02,rotkowitz06}.  This earlier work showed that
quadratic invariance is sufficient for $\hg(S)$ and $C$ to be convex.
In this paper, our main contributions are as follows.

\begin{enumerate}[(i)]

\item We show that quadratic invariance is a \emph{necessary} condition for convexity of $\hg(S)$.

\item We show that, subject to some additional assumptions, quadratic invariance is also a necessary
condition for convexity of~$C$.

\item We give examples that illustrate both the generality of the results
 as well as the limitation encountered when the additional assumptions
 in (ii) are violated.

\end{enumerate}

The paper is organized as follows. We begin by reviewing related works
in the literature and covering necessary mathematical preliminaries,
including a review of quadratic invariance. In Section~\ref{sec:main}
we prove our main results and in Section~\ref{sec:examples} we show
some illustrative examples. Finally, we conclude in
Section~\ref{sec:conclusion}.

\subsection{Prior work}

Optimal controller synthesis subject to information constraints is
known to be hard in general~\cite{blondel}. Even when the plant is
linear, the noise is Gaussian, and the cost function is quadratic, the
optimal controller may not be linear~\cite{witsenhausen}. Furthermore,
there is currently no known efficient algorithm for finding the
optimal linear controller in general.

Despite these difficulties, the pervasiveness of decentralized
information in large-scale systems has driven researchers to seek
subclasses of problems that are tractable. Early work by
Radner~\cite{radner} showed that some static team decision problems
admit optimal controllers that are linear. This was extended to
dynamic teams by Ho and Chu~\cite{hochu} under the assumption that the
information structure is~\emph{partially nested}.

More recently, many works have addressed the broad
area of decentralized control synthesis, including~\cite{lin_2011,nayyar_2013,sabau_2011,tanaka_2011}.
Specific  efforts have focused on characterizing when the set of
achievable closed-loop maps as detailed in Section~\ref{sec:intro} is
convex. For example,~\cite{qimurti04,nonclassical} shows several
classes of information constraints that lead to a convex~$C$.
In~\cite{rotkowitz06} it is shown that the simple algebraic condition
of quadratic invariance encompasses a wide class of problems for
which~$C$ is convex.

However, quadratic invariance is only a sufficient condition for
convexity of~$C$. Recent
work~\cite{lessard2009reduction,lessard2010iqi} introduces the concept
of~\emph{internal quadratic invariance}, which gives a more complete
characterization of~$C$. This condition highlights cases where
quadratic invariance does not hold, but a suitable transformation can
produce a new problem for which~$C$ is unchanged and quadratic
invariance now holds.

One may also characterize~$C$ more directly. Using tools from
algebraic geometry, specifically elimination theory,
the paper~\cite{shin2012decentralized} gives a method for
computing a representation of the smallest algebraic variety
containing~$C$. This does not necessarily
make the problem of verifying convexity easier, but it does open the
door to other tools for identifying convexity such as sum-of-squares
relaxations. See~\cite{sos-convexity} and references therein.

This paper gives conditions under which quadratic invariance is both
necessary and sufficient for convexity. Preliminary versions of some
of these results appeared
in~\cite{lessard_thesis,lessard2011qi_necc_suff} for the Banach space
case only. In this paper, we include new results that cover the
extended spaces~$\ell_{2e}$ and~$L_{2e}$, and we present new examples
in Section~\ref{sec:examples} that illustrate both the generality and
limitations of our results.

\section{Preliminaries}\label{sec:prelim}

If $\mathcal{X}$ and $\mathcal{Y}$ are topological vector spaces
(TVS), we let $\L(\mathcal{X},\mathcal{Y})$ denote the set of all maps
$T:\mathcal{X}\to\mathcal{Y}$ such that $T$ is linear and
continuous. This paper concerns properties of linear operators. What
we are able to prove depends on the underlying structure of the vector
spaces involved, so we distinguish two classes of topological vector
spaces: Banach spaces, and the extended spaces $\ell_{2e}$ and
$L_{2e}$.

\subsubsection*{Banach spaces}
A Banach space is a TVS whose topology is induced by a norm, and the
space is complete. Any linear and continuous map from one Banach space
to another is bounded. Common examples of Banach spaces include
$\ell_p$ and $L_p$; the set of functions $f:\Z_+\to \R$ and
Lebesgue-measurable functions $f:\R_+\to\R$ respectively, for which
the $p$-norm is finite. Simpler examples include the Hilbert spaces
$\R^n$, $\ell_2$, or $L_2$. Suppose $\mathcal{U}$ and $\mathcal{Y}$
are Banach spaces, and $G \in \L(\mathcal{U},\mathcal{Y})$. Define the
following set.
\begin{equation}\label{def:M}
\mg = \set{ K \in \L(\mathcal{Y},\mathcal{U}) }{ I-GK \text{ is invertible} }
\end{equation}
Note that the set $\mg$ is precisely the domain of $\hg$. For any
$A \in \L(\mathcal{Y},\mathcal{Y})$, the \eemph{resolvent set} is
defined as $
\rho(A) = \set{ \lambda \in \C }{ (\lambda I - A) \text{ is invertible}}
$.  It is a fact that $\rho(A)$ is always an open set, and contains
all $\lambda \in \C$ for which $|\lambda| > \norm{A}$. Therefore, one
may define $\rho_\text{uc}(A)$, the unbounded connected component of
$\rho(A)$. Now define the subset $\ng\subset \mg$ as follows.
\begin{equation}\label{def:N}
\ng = \set{K \in \L(\mathcal{Y},\mathcal{U}) }{ 1 \in \rho_\text{uc}(GK) }
\end{equation}

\subsubsection*{Extended spaces}

Banach spaces are well-suited for representing a wide variety of
systems, but when the time horizon is infinite, such as with $\ell_2$
and $L_2$, only bounded maps are permitted. In order to represent
unbounded maps as well, such as unstable systems, we use the notion
of \emph{extended spaces}. First, define the truncation and shift
operators $P_T$ and $D_T$ which operate on functions $f:\Z_+\to\R$
or $f:\R_+\to\R$, as follows. For $T >0$, define
\begin{align*}
P_T f = \begin{cases}
	f(t)	& \text{if } t \le T \\
	0		& \text{otherwise}
\end{cases}
\end{align*}
and
\begin{align*}
D_T f = \begin{cases}
	0			& \text{if } t < T \\
	f(t-T)	& \text{otherwise}
\end{cases}
\end{align*}
The extended spaces $\ell_{pe}$ and $L_{pe}$ are defined as follows.
\begin{align*}
\ell_{pe} &= \sett{f:\Z_+\to\R}{P_Tf\in\ell_p\text{ for all }T\in\Z_+} \\
L_{pe} &= \sett{f:\R_+\to\R}{P_Tf\in L_p\text{ for all }T\in\R_+}
\end{align*}
Note that $\ell_{pe}$ is the same for every $p$, and is the set of
real sequences, which we abbreviate as $\ell_e$. However, $L_{pe}$ is
the set of functions that are $L_p$ on finite intervals, and so is
different for different~$p$.

A map $A\in\L(L_{2e}^m,L_{2e}^n)$ or $A\in\L(\ell_{2e}^m,\ell_{2e}^n)$ is called causal if,
for all $ T \geq 0 $, we have $P_T A P_T = P_T A $.
A causal system $A$  is called time-invariant if  $D_T A = A D_T $ for all $T >0$.
We denote the set of linear, causal, time-invariant maps  as $\LTI$.

The extended spaces are not Banach spaces, but we may endow them with
a suitable topology and recover notions of convergence and
continuity. Let the topology on $L_{2e}$ be generated by all the open
$\norm{\cdot}_T$-balls for every $T\in\R_+$, where we define
$\norm{f}_T = \norm{P_T f}_{L_2}$. Also, we let the topology on
$\L(L_{2e}^m,L_{2e}^n)$ be generated by all the open
$\norm{\cdot}_T$-balls for every $T\in\R_+$, where $\norm{A}_T
= \norm{P_TA}_{L_2^m\to L_2^n}$. It can be shown that these topologies
are Hausdorff, and thus $L_{2e}$ and $\L(L_{2e}^m,L_{2e}^n)$ are
topological vector spaces (TVS). Furthermore, convergence in this
topology is equivalent to convergence in every $\norm{\cdot}_T$-norm,
and continuity of a linear operator in this topology is equivalent to
continuity in every $\norm{\cdot}_T$-norm. A similar topology is
defined for $\ell_{2e}$. For a thorough treatment of these concepts,
see for example~\cite{zimmer}. We also require the concept of
an \emph{inert} subspace, which we give below.

\begin{defn}\label{def:inert}
The set $S \subset \LTI(L_{2e}^{n_u},L_{2e}^{n_y})$ is \eemph{inert}
with respect to $G \in \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$ if for all
$K\in S$, $(gk)_{ij} \in L_{\infty e}$ for all $i,j=1,\dots,n_y$ where
$(gk)$ is the impulse response matrix of $GK$. We overload our
notation and also define
$S\subset \LTI(\ell_{2e}^{n_u},\ell_{2e}^{n_y})$ to be inert with
respect to $G \in \LTI(\ell_{2e}^{n_u},\ell_{2e}^{n_y})$ if for all
$K\in S$, $(gk)_{ij}\in\ell_e$ for all $i,j=1,\dots,n_y$ and
$r((gk)(0)) < 1$ where $(gk)$ is the discrete impulse response matrix
of $GK$ and $r(\cdot)$ denotes the spectral radius.
\end{defn}

Note in particular that if $S$ is inert with respect to $G$, then
$I-GK$ is invertible for all $K\in S$. A proof of this result is
in~\cite{rotkowitz06}.

\subsubsection*{Quadratic invariance}

We now summarize the definitions and main results regarding quadratic
invariance. The following definition may be found
in~\cite{rotkowitz02,rotkowitz06}.

\begin{defn}\label{def:qi}
Suppose $\mathcal{U}$ and $\mathcal{Y}$ are TVS. Suppose $S \subset \L(\mathcal{Y},\mathcal{U})$ and $G\in\L(\mathcal{U},\mathcal{Y})$.
The set $S$ is \eemph{quadratically invariant} under $G$ if $KGK \in S$ for all $K \in S$.
\end{defn}

\noindent The first quadratic invariance result we present applies to Banach spaces.

\begin{thm}[from \cite{rotkowitz02}] \label{thm:qi_banach} Suppose $\mathcal{U}$ and $\mathcal{Y}$ are Banach spaces. Suppose $S \subset \L(\mathcal{Y},\mathcal{U})$ is a closed subspace and $G \in \L(\mathcal{U},\mathcal{Y})$. Define $\mg$ and $\ng$ as in~\eqref{def:M}--\eqref{def:N}, and suppose $S \cap \ng = S \cap \mg$. Then
$S$ is quadratically invariant under $G$ if and only if
$ \hg(S\cap \mg) = S \cap \mg$.
\end{thm}

\noindent
The second quadratic invariance result applies to the topological
vector spaces~$\ell_{2e}$ and~$L_{2e}$.

\begin{thm}[from \cite{rotkowitz06}] \label{thm:qi_extended} Suppose $S\subset\LTI(L_{2e}^{n_u},L_{2e}^{n_y})$ is an inert closed subspace and $G \in \LTI(L_{2e}^{n_u},L_{2e}^{n_y})$, or suppose $S\subset\LTI(\ell_{2e}^{n_u},\ell_{2e}^{n_y})$ is an inert closed subspace and $G \in \LTI(\ell_{2e}^{n_u},\ell_{2e}^{n_y})$. Then
$S$ is quadratically invariant under $G$ if and only if $\hg(S) = S$.
\end{thm}

Roughly speaking, $S$ being quadratically invariant under $G$ means
that the set $S$ is invariant under the application of
$\hg$. Technical conditions arise only to ensure that we avoid $K$ for
which $\hg(K)$ is not well-defined. In the case of
Theorem~\ref{thm:qi_banach}, this amounts to intersecting $S$ with
$\mg$, the domain of $\hg$. In Theorem~\ref{thm:qi_extended}, we
make the technical assumption that $S$ is an inert subspace, and this
ensures that $\hg(K)$ is always well-defined.

\section{Main results}\label{sec:main}

We begin by defining two types of sets that will be useful in our main
results. These definitions apply to any vector space over $\R$, so in
particular they apply to both Banach spaces and the topological vector
spaces $\ell_{2e}$ and $L_{2e}$.

\begin{defn}
Suppose $\mathcal{V}$ is a vector space over~$\R$, and
$S \subset \mathcal{V}$. We call $S$ \eemph{homogeneous} if for all
$v \in S$ and $\alpha \in \R$, we have $\alpha v \in S$.
\end{defn}

\noindent
Homogeneous sets are collections of lines that pass through the
origin. If a point belongs to a homogeneous set, so does the line that
passes through that point and the origin. Note that every subspace is
homogeneous, but not all homogeneous sets are subspaces.

\begin{defn}
Suppose $\mathcal{V}$ is a vector space over~$\R$, and
$T \subset \mathcal{V}$. We call $T$ \eemph{star-shaped} if for all
$v \in T$ and $\alpha \in [0,1]$, we have $\alpha v \in T$.
\end{defn}

\noindent
If a point belongs to a homogeneous set, so does the line segment
connecting it to the origin. Therefore, every homogeneous set is
star-shaped, but not vice-versa. Furthermore, every convex set containing
the origin is star-shaped, but not vice-versa. For examples of these
sets in $\R^2$, see Figure~\ref{fig:linestar_sets}.

\begin{figure}[ht]
\centering
\begin{minipage}[b]{.45\textwidth}
	\centering
	\begin{tikzpicture}[bline,twopic]
	\def\w{.3}  \def\x{.7}  \def\y{.8}  \def\z{.3}
	\draw[white,fill=blue!20] (-\x,-1) -- (\x,1) -- (1,\x) -- (-1,-\x);
	\draw[<->] (-\x,-1) -- (\x,1);
	\draw[<->] (-1,-\x) -- (1,\x);
	\draw[white,fill=blue!20] (-\y,1) -- (\y,-1) -- (1,-\z) -- (-1,\z);
	\draw[<->] (-\y,1) -- (\y,-1);
	\draw[<->] (1,-\z) -- (-1,\z);
	\draw[<->] (1,\w) -- (-1,-\w);
	\draw[axes] (-1.3,0) -- (1.3,0);
	\draw[axes] (0,-1.1) -- (0,1.1);
	\end{tikzpicture}
	\subcaption{homogeneous set
\end{minipage}
\begin{minipage}[b]{.45\textwidth}
	\centering
	\begin{tikzpicture}[bline,twopic]
	\def\w{.5}  \def\x{.5}
	\draw[white,fill=blue!20] (0,0) -- (-\x,-1) -- (-1,-\x);
	\draw[->] (0,0) -- (-\x,-1);
	\draw[->] (0,0) -- (-1,-\x);
	\draw[->] (0,0) -- (\w,-1);
	\draw (0,0) -- (-.7,.4);
	\draw (0,0) -- (-.7,.9);
	\draw (0,0) -- (-.25,1);
	\draw[white,fill=blue!20] (0,0) -- (.3,.8) -- (.45,.35) -- (1,.2) -- (0,0);
	\draw[bline] (0,0) -- (.3,.8) -- (.45,.35) -- (1,.2) -- (0,0);
	\draw[axes] (-1.3,0) -- (1.3,0);
	\draw[axes] (0,-1.1) -- (0,1.1);
	\end{tikzpicture}
	\subcaption{star-shaped set
\end{minipage}
\caption{\label{fig:linestar_sets}
Example of a homogeneous set and a star-shaped set in $\R^2$. Lines
with arrows indicate that they extend to infinity. }
\end{figure}

\subsection{Banach space results}\label{sec:main_banach}

Our first main result states that if $\hg$ maps a homogeneous subset
of its domain to a star-shaped set, it must in fact map that
homogeneous set to itself. We will see that this result has
implications concerning convexity.

\begin{thm}\label{thm:main_banach}
Suppose $\mathcal{U}$ and $\mathcal{Y}$ are Banach spaces. Suppose
$S \subset \L(\mathcal{Y},\mathcal{U})$ is closed and
homogeneous and $G \in \L(\mathcal{U},\mathcal{Y})$.
 Define $\mg$ and $\ng$ as
in~\eqref{def:M}--\eqref{def:N}, and suppose $S \cap \ng = S \cap
\mg$. If $\hg(S \cap \mg) = T \cap \mg$ for some star-shaped set $T$, then $T\cap \mg = S\cap \mg$.
\end{thm}

\begin{proof}
Suppose $\hg(S\cap \mg) = T\cap \mg$, where $T$ is a star-shaped
set. Fix some $K\in S\cap \mg$. Therefore, $I-GK$ is invertible, and
$1\in \rho(GK)$. The resolvent set of a bounded linear operator is an
open set, so there exists a sufficiently small $\eps \in (0,1]$ such
that $1-\alpha \in \rho(GK)$ for all $\alpha \in [0,\eps]$. For any
such $\alpha$, it follows that $I-(1-\alpha)GK$ is
invertible. Therefore, $\left(I-(1-\alpha)GK\right)(I-GK)^{-1}$ is
invertible as well. Expanding this expression, we find that it equals
$I-\alpha G\hg(K)$. Thus $\alpha \hg(K) \in \mg$.

Also, we have $\hg(K) \in T$ by assumption, and so $\alpha \hg(K) \in
T$ whenever $\alpha\in[0,1]$, because $T$ is a star-shaped set. It
follows that for $\alpha~\in [0,\eps]$, $\alpha \hg(K) \in
T\cap \mg$. Applying $\hg$ to both sides, we conclude that
$\hg(\alpha \hg(K)) \in \hg(T\cap \mg) = S\cap \mg$, where we made use
of the involutive property of $\hg$. Expanding $\hg(\alpha \hg(K))$,
we find that it equals $\alpha K(I-(1-\alpha)GK)^{-1}$. Since $S$
is a homogeneous set, we may multiply this expression by $-1/\alpha$
when $\alpha\ne 0$, and the result will still lie in $S$. Thus,
$-K(I-(1-\alpha)GK)^{-1} \in S$.  \\Now define the function
$g:[0,\eps] \rightarrow \mathcal{L}(\mathcal{Y},\mathcal{U})$ by
\begin{equation}\label{def:g}
g(\alpha) = -K(I-(1-\alpha)GK)^{-1}
\end{equation}
Notice that $g$ is continuous at $0$, since $I-(1-\alpha)GK$ is
invertible for sufficiently small $\alpha \geq 0$ as above, and the inversion
map is continuous on its domain.  Since $S$ is closed
and $g(\alpha)\in S$ for $\alpha \in (0,\eps]$, we
have
\[
\lim_{\alpha \rightarrow 0^+} g(\alpha) \in S
\]
We may take the limit $\alpha\to 0^+$ by simply evaluating $g$ at
$\alpha=0$. Thus, we conclude that $\hg(K) \in S$. Now $\hg$ is a
bijection from $\mg$ to $\mg$, and so we actually have $\hg(K) \in
S\cap \mg$. Since $K$ was an arbitrary element of $S\cap \mg$, it
follows that $\hg(S\cap \mg) \subset S\cap \mg$. Using the involutive
property of $\hg$ once more, $\hg(S\cap \mg) = S\cap \mg$, as
required.
\end{proof}

Theorem~\ref{thm:main_banach} may be specialized to the case where $S$
is a subspace, and combined with Theorem~\ref{thm:qi_banach} to yield
a necessary condition for convexity of $\hg$.

\begin{cor}\label{cor:main_banach}
Suppose $\mathcal{U}$ and $\mathcal{Y}$ are Banach spaces. Suppose
$S \subset \L(\mathcal{Y},\mathcal{U})$ is a closed subspace and
$G \in \L(\mathcal{U},\mathcal{Y})$. Define $\mg$ and $\ng$ as
in~\eqref{def:M}--\eqref{def:N}, and suppose $S \cap \ng =
S \cap \mg$. Then the following statements are equivalent.
\begin{enumerate}[(i)]
\item $S$ is quadratically invariant under $G$
\item $\hg(S\cap \mg)=S\cap \mg$
\item $\hg(S\cap \mg)= \Gamma \cap \mg$ for some convex set  $\Gamma$
\end{enumerate}
\end{cor}

\begin{proof}
{\it(i)$\iff$(ii)} is precisely Theorem~\ref{thm:qi_banach}.  The case
{\it(ii)$\implies$(iii)} is immediate.  Finally, to show
{\it(ii)$\impliedby$(iii)}, notice that if \textit{(iii)} holds then
$\Gamma$ must contain the origin, and since it is convex it must
therefore be star-shaped. Then, since every subspace is homogeneous,
\textit{(ii)} follows from Theorem~\ref{thm:main_banach}.
\end{proof}

Corollary~\ref{cor:main_banach} shows that quadratic invariance is
necessary and sufficient for convexity of $\hg$. We now show that
under additional invertibility assumptions, this result also applies
to the set of achievable closed-loop maps described in
Section~\ref{sec:intro}.

\begin{cor}\label{cor:clmap_banach}
Suppose the conditions of Corollary~\ref{cor:main_banach}
hold. Additionally, suppose $\mathcal{W}$ and $\mathcal{Z}$ are Banach
spaces, and $P_{11}\in\L(\mathcal{W},\mathcal{Z})$,
$P_{12}\in\L(\mathcal{U},\mathcal{Z})$, and
$P_{21}\in\L(\mathcal{W},\mathcal{Y})$. Finally, suppose $S \cap \mg =
S$.  If $P_{12}$ is left-invertible and $P_{21}$ is right-invertible,
then the following statements are equivalent.
\begin{enumerate}[(i)]
\item $S$ is quadratically invariant under $G$
\item $P_{11}-P_{12}\hg(S)P_{21} = \Gamma$, where $\Gamma$ is a convex set.
\end{enumerate}
\end{cor}

\begin{proof}
The proof of {\it(i)$\implies$(ii)} follows directly from
Theorem~\ref{thm:main_banach}. Conversely, suppose that {\it(ii)}
holds. Let $P_{12}^\dagger$ be a left-inverse of $P_{12}$ and let
$P_{21}^\dagger$ be a right-inverse of $P_{21}$. Then we have
\begin{align*}
\hg(S) &= P_{12}^\dagger P_{11} P_{21}^\dagger - P_{12}^\dagger \Gamma P_{21}^\dagger
\end{align*}
where the right-hand side is a convex set, because it is an affine
transformation of the convex set $\Gamma$. Applying
Corollary~\ref{cor:main_banach}, we conclude that $S$ is quadratically
invariant under $G$, as required.
\end{proof}

In Section~\ref{sec:examples}, we will show some examples that
illustrate why the invertibility requirements are necessary in this
result.

\subsection{Extended space results}\label{sec:main_extended}

We now present results analogous to those of
Section~\ref{sec:main_banach}, but now for the extended
spaces~$\ell_{2e}$ and~$L_{2e}$.

\begin{thm}\label{thm:main_extended}
Suppose $S \subset \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$ is inert, closed,
and homogeneous and $G \in \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$. If
$\hg(S)=T$ where $T$ is a star-shaped set, then $T = S$. This result
also holds when $L_{2e}$ is replaced by $\ell_{2e}$.
\end{thm}

\begin{proof}
This proof is similar to that of Theorem~\ref{thm:main_banach}, except
we do not need to worry about the invertibility of $I-GK$, since it is
guaranteed by the inertness property of $S$. Suppose $\hg(S) = T$,
where $T$ is a star-shaped set. Applying $\hg$ to both sides, we
conclude that $\hg(T)=S$. Together with the star-shaped property of
$T$, it follows that $\hg(\alpha \hg(S)) \subset S$ for any $\alpha \in
[0,1]$. As in the proof of Theorem~\ref{thm:main_banach}, we conclude
that $g(\alpha) \in S$ for all $\alpha\in(0,1]$, where $g$ is defined
in~\eqref{def:g}. The rest of the proof follows as in the proof of
Theorem~\ref{thm:main_banach}. The only difference is that we must
verify continuity of $g$ at $0$ using the topology defined in
Section~\ref{sec:prelim}.
\end{proof}

As in Section~\ref{sec:main_banach}, Theorem~\ref{thm:main_extended}
may be specialized to the case where $S$ is a subspace, and combined
with Theorem~\ref{thm:qi_extended} to yield a necessary condition for
convexity of $\hg$.

\begin{cor}\label{cor:main_extended}
Suppose $S \subset \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$ is an inert and
closed subspace and $G \in \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$. Then the
following statements are equivalent
\begin{enumerate}[(i)]
\item $S$ is quadratically invariant under $G$
\item $\hg(S)=S$
\item $\hg(S)$ is convex
\end{enumerate}
This result also holds when $L_{2e}$ is replaced by $\ell_{2e}$.
\end{cor}

\begin{proof}
See the proof of Corollary~\ref{cor:main_banach}.
\end{proof}

Corollary~\ref{cor:main_extended} shows that quadratic invariance is
necessary and sufficient for convexity of $\hg$. As in
Section~\ref{sec:main_banach}, this result also applies to the set of
achievable closed-loop maps when we make additional invertibility
assumptions.

\begin{cor}\label{cor:clmap_extended}
Suppose the conditions of Corollary~\ref{cor:main_extended}
hold, and  $P\in\LTI(L_{2e},L_{2e})$.  If $P_{12}$ is
left-invertible and $P_{21}$ is right-invertible, then the following
statements are equivalent
\begin{enumerate}[(i)]
\item $S$ is quadratically invariant under $G$
\item $P_{11}-P_{12}\hg(S)P_{21}$ is convex
\end{enumerate}
This result also holds when $L_{2e}$ is replaced by $\ell_{2e}$.
\end{cor}

\begin{proof}
See the proof of Corollary~\ref{cor:clmap_banach}.
\end{proof}

\section{Examples}\label{sec:examples}

\subsubsection*{LQG with sparsity}

Consider the extended space of signals $L_{2e}$, and suppose we have
the LTI map
\[
\bmat{P_{11} & P_{12} \\ P_{21} & G}:
	\bmat{L_{2e}^{n_w}\\L_{2e}^{n_u}} \to
	\bmat{L_{2e}^{n_z}\\L_{2e}^{n_y}}
\]
given by the following minimal state-space realization.
\[
\bmat{P_{11} & P_{12} \\ P_{21} & G} =
	\left[ \begin{array}{c|cc}
	A & B_1 & B_2 \\ \hlinet
	C_1 & 0 & D_{12} \\
	C_2 & D_{21} & 0
	\end{array}\right]
\]
We now make the classical assumptions typically made in $\Htwo$
synthesis. These assumptions ensure that the controller that achieves
a closed-loop map with minimum norm exists, is unique, and is
rational~\cite{zdg}. They are as follows.
\begin{enumerate}[(i)]\itemsep 0.2\baselineskip
\item $(C_2,A,B_2)$ is stabilizable and detectable

\item
For all $\omega\in\R$ the matrices $\bmat{ A-j\omega I & B_2 \\ C_1 &
D_{12}}$ and $D_{12}$ have full column rank

\item
For all $\omega\in\R$ the matrices $\bmat{A-j\omega I & B_1 \\ C_2 &
D_{21}}$ and $D_{21}$ have full row rank

\end{enumerate}
Now suppose we seek a controller
$K \in \LTI(L_{2e}^{n_y},L_{2e}^{n_u})$ where $K$ has some prescribed
sparsity structure. So $K\in S$ for some closed subspace
$S$. Assumptions (ii) and (iii) ensure that $P_{12}$ is
left-invertible and $P_{21}$ is right-invertible,
respectively. Therefore, we may apply
Corollary~\ref{cor:clmap_extended} and conclude that quadratic
invariance is necessary and sufficient for convexity of the set of
achievable closed-loop maps. Thus for this class of problems,
quadratic invariance may be tested to determine convexity. In
particular, if $S$ is defined as the set of transfer functions with
desired sparsity or delays, then quadratic invariance may be
computationally tested using the methods in~\cite{rotkowitz06}.

It is worth noting that under the above assumptions there are only two
possibilities. Either $S$ is quadratically invariant, in which case
the set of closed-loop maps $C$ must be given by $C =
P_{11}-P_{12}SP_{21}$, or $S$ is not quadratically invariant in which
case $C$ is not convex. Additionally, one can conclude structural
properties. For example, if $S$ is a subspace, then for any plant $P$
satisfying these assumptions the only possible form of $C$ when it is
convex is that $C$ is affine; it cannot, for example, be a ball.

\subsubsection*{Non-affine example}
In the results of Section~\ref{sec:main_banach}, as well as in the
first example of this section, the set of achievable closed loop maps
is affine whenever it is convex.  In this example, we show that
more complicated sets are achievable as well. Consider the Banach
space of real matrices under the standard induced 2-norm. Define the
matrices {\advance\arraycolsep-1pt
\begin{align*}
P_{11} &= \bmat{0\\0}
&
G &= \bmat{
0 & 1  & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & -1 & 0}
&
K &= \bmat{t&0&0&0\\0&t&0&0\\0&0&s&0\\0&0&0&s}
\end{align*}
Here, the subspace of admissible controllers $S$ is the set of all $K$
above parameterized by $(t,s)\in\R^2$. Now consider two possible pairs
of values for $P_{12}$ and $P_{21}$, and their corresponding sets of
achievable closed-loop maps $C = P_{11}-P_{12}\hg(K)P_{21}$.
\begin{enumerate}[(a)]
\item
$P_{12} = \bmat{0 & 2 & 0 & 0 \\ 0 & 0 & 0 & -2}$
and
$P_{21} = \bmat{0\\1\\0\\-1}$,
\if\MODE3{}\else{\hfill\break}\fi
which leads to
$C = \bmat{ \frac{2s}{1+s^2+t^2} \\[1.5mm]
\frac{2t}{1+s^2+t^2}}$.

\item $P_{12} = \bmat{1 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0}$
and
$P_{21} = \bmat{0\\2\\0\\-1}$,
\if\MODE3{}\else{\hfill\break}\fi
which leads to
$C = \bmat{ \frac{(s^2+2)t^2}{1+s^2+t^2} \\[1.5mm]
\frac{s^2(1-t^2)}{1+s^2+t^2}}$.
\end{enumerate}
As we vary $(s,t)\in\R^2$, one may check that $C$ is the unit disc in
Case~(a), and a more complicated nonconvex shape in
Case~(b). Figure~\ref{fig:ex_shapes} shows the sets $C$ in each case.

\begin{figure}[ht]
\centering\begin{minipage}[b]{.45\textwidth}
	\centering
	\begin{tikzpicture}[twopic]
	\def\th{0.05}
	\draw[blfill] (0,0) circle (1);
	\draw[axes] (-1.5,0) -- (1.5,0);
	\draw[axes] (0,-1.4) -- (0,1.4);
	\draw[axes] (0,-1.4) -- (0,1.4);
	\draw[ticks] (1,-\th) -- (1,\th);
	\draw[ticks] (-1,-\th) -- (-1,\th);
	\draw[ticks] (-\th,1) -- (\th,1);
	\draw[ticks] (-\th,-1) -- (\th,-1);
	\node[anchor=south east] at (0,1) {$1$};
	\node[anchor=north east] at (0,-1) {$-1$};
	\node[anchor=north west] at (1,0) {$1$};
	\node[anchor=north east] at (-1,0) {$-1$};
	\end{tikzpicture}
	\subcaption{first case, $C$ is convex.}
\end{minipage}
\begin{minipage}[b]{.45\textwidth}
	\centering
	\begin{tikzpicture}[twopic]
	\def\xoff{.35}
	\def\poff{0.0}
	\def\th{0.05}
	\path[bfill] (0,0) -- (0,1) -- (2+\xoff,-1-\xoff) -- (2+\xoff,-\xoff) -- (2,0) -- (0,0);
	\draw[bline] (0,1) -- (0,0) -- (2,0);
	\draw[bdash,->] (0,1) -- (2+\xoff,-1-\xoff);
	\draw[bdash,->] (2,0) -- (2+\xoff,-\xoff);
	\draw[axes] (-.5-\poff,0) -- (2.5-\poff,0);
	\draw[axes] (0,-1.4) -- (0,1.4);
	\draw[ticks] (1,-\th) -- (1,\th);
	\draw[ticks] (2,-\th) -- (2,\th);
	\draw[ticks] (-\th,1) -- (\th,1);
	\draw[ticks] (-\th,-1) -- (\th,-1);
	\node[anchor=north east] at (1,0) {$1$};
	\node[anchor=south east] at (2,0) {$2$};
	\node[anchor=east] at (0,1) {$1$};
	\node[anchor=east] at (0,-1) {$-1$};
	\end{tikzpicture}
	\subcaption{second case, $C$ is nonconvex.}
\end{minipage} \\
\caption{\label{fig:ex_shapes}The set of achievable closed-loop
 maps described by both cases above. Solid lines are included in the
 set, while dashed lines are not.}
\end{figure}

Both cases above have the same $G$ and $S$. It is straightforward to
check that $S$ is not quadratically invariant under $G$. Therefore,
Corollary~\ref{cor:main_banach} implies that $\hg(S)$ is not a convex
set. Unfortunately, we cannot apply Corollary~\ref{cor:clmap_banach}
to deduce anything about the convexity of $C$ because in both cases,
$P_{12}$ is not left-invertible and $P_{21}$ is not
right-invertible. Indeed, we have shown that both a convex set~(a) and
a nonconvex set~(b) are achievable depending on the choice of $P_{12}$
and $P_{21}$.

This example shows that in general, the convexity of $C$ does not
depend on $S$ and $G$ alone, but also on $P_{12}$ and $P_{21}$. This
idea is further discussed in~\cite{lessard2010iqi}, where $P_{12}$ and
$P_{21}$ are used as part of a sufficient condition for convexity that
is more general than quadratic invariance.

\newcommand{\B}{\rule[-1.2ex]{0pt}{0pt}}

\subsubsection*{Affine example}
We now show an example where the set of closed-loop maps is a convex set, and in particular is affine, even though $S$ is not quadratically invariant under $G$. This example is a variant on an example shown in~\cite{lessard2009reduction}. Suppose
\begin{align*}
  P &= \left[\addtolength{\arraycolsep}{-1pt}\begin{array}{c;{1pt/2pt}ccc} \B a & b_1 & b_2 & b_2\\ \hdashline[1pt/2pt] \T
       c_1 & g_1 & 0 & 0 \\ \T
       c_1 &  g_1 & 0 & 0 \\ \T
        c_2 & g_2 & g_3 & g_3
       \end{array}\right] &
  S &= \left.\left\{ \addtolength{\arraycolsep}{-1pt}\bmat{\T k_1 & 0 & 0\\\T 0 & k_2 & 0\\\T 0 & 0 & k_3} \,\right\vert\, k_i \in \R \right\}
\end{align*}
Here $a$, $b_i$, $c_i$ and $g_i$ are real numbers for simplicity, although it is straightforward to construct similar examples over the rational transfer matrices. The matrix $P$ is partitioned into its four blocks as shown with $P_{22}=G$; the dashed lines do not denote a state-space representation. The information constraint $S$ is the subspace of controllers with a diagonal structure. Note that $KGK$ is not diagonal for all  diagonal $K$, so $S$ is not quadratically invariant under $G$. We will show that nonetheless that the set of achievable closed-loop maps $C$ is affine. To see why, define
\[
\t P = \left[\begin{array}{c;{1pt/2pt}cc} \B a & b_1 & b_2\\
    \hdashline[1pt/2pt] \T
    c_1 & g_1 & 0 \\ \T
    \T\B c_2 & g_2 & g_3
  \end{array}\right] \quad
\t S = \left.\left\{
    \bmat{k_1 & 0\\k_2 & k_3} \,\right\vert\, k_i \in \R \right\}
\]
Here $\t S$ is clearly quadratically invariant under $\t G$. Therefore the set of achievable closed-loop maps $\t C$ is affine. It is straightforward to check that $C =\t C$. In fact, the parameterizations are exactly equal provided the $k_i$ are the same. So $C$ is affine despite $S$ not being quadratically invariant under $G$. Such transformations $(P,S)\mapsto(\t P,\t S)$ are explored in~\cite{lessard2009reduction}.

\section{Conclusion}\label{sec:conclusion}

It was previously known that when the set of decentralized controllers
is a subspace~$S$, quadratic invariance is a necessary and sufficient
condition under which $\hg(S) = S$. In this paper, we showed that
quadratic invariance is in fact necessary and sufficient for $\hg(S)$
to be convex. Furthermore, we showed that under certain invertibility
conditions, quadratic invariance is also necessary and sufficient for
the convexity of~$C$, the set of achievable closed-loop maps.

This work therefore strengthens the utility of quadratic invariance as
an indicator for tractability of decentralized control synthesis
problems. However, there remains a nontrivial case, that when $S$ is
not quadratically invariant and the aforementioned invertibility
conditions do not hold. In that case one cannot draw any conclusions
about the convexity of~$C$. This was illustrated in
Section~\ref{sec:examples}, where we gave an example of a problem that
is not quadratically invariant, but~$C$ can be either convex or
nonconvex, depending on the choice of the system parameters.

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


text:
A Conversation with Amb. Raouf Adly Saad
Wed, 8/14/2013 12:00 pm to 1:00 pm The Middle East Institute is pleased to welcome H.E. Ambassador Raouf Adly Saad, former special envoy to Africa of the interim president of Egypt, for a conversation about Egypt's political transition and future. In light of Egypt's many challenges, Ambassador Saad will discuss the transitional government's goals and priorities and offer his insights about how to get Egypt on a path toward economic stability and democracy.Bios:H.E. Ambassador Raouf Adly Saad served as the special envoy to Africa for Interim Egyptian President Adly Mansour. Prior to that he served as ambassador to both the Russian Federation and the European Union. Amb. Saad has held a number of other prominent positions within the Egyptian government, including as senior assistant to the minister of foreign affairs, assistant foreign minister for International Economic Relations, and member of parliament and chairman of the Foreign Relations Committee in the People's Assembly. He has led several Egyptian delegations to international events, including the 7th Forum on Democracy, Development, and Free Trade in Doha, and the Meeting of the Economic and Social Council of the League of Arab States. He obtained his B.A. in economics and political science from Cairo University.Kate Seelye is senior vice president of the Middle East Institute, where she oversees MEI's programs and communications. Prior to joining MEI, Seelye worked as a radio and television journalist covering the Arab world from 2000-2009 from her base in Beirut, Lebanon. She reported on the region for NPR, BBC's The World, PBS' Frontline/World and the renowned Channel Four British investigative news series, Unreported World. Prior to that she worked as a producer for the Newshour with Jim Lehrer on PBS.
Interconnected: Trade, food security, and stability in the GCC and MENA

redpajama_set_name: RedPajamaCommonCrawl


text:
'My home was bombed and I lost my best friend': A Syrian refugee's story
Mohamed fled Aleppo. Now he's working with Cardiff students taking action to help refugees
"In Aleppo there are no more doctors. They need more doctors," begins Mohamed.
As the fighting in Aleppo has intensified recently, it has rarely been out of the news. By some estimates, 4.8 million people have left Syria. Mohamed is one of them.
"Aleppo is a big city, the second biggest in Syria. In comparison to Cardiff it is way bigger. It was sad to hear that many weeks ago the last hospital stopped functioning. When I was a child my dream was to be a doctor and help the people. My late mother always wanted me to be a doctor. I worked hard to achieve this dream and I successfully got into Aleppo medical school. Once I passed the first two years I had to leave Syria because of the war.
"The Syrian army asked why we didn't want to fight with them. I told them that we don't want to fight. I don't want to be a part of killing anyone. In my last month in Aleppo I was living in an area that was close to the fighting. My home was bombed and I lost my best friend. That was really hard for me. We decided to leave."
Mohamed's story is being told as part of Framing Identity, a project by the Cardiff arm of the student-led charity STAR (Student Action for Refugees). The brainchild of Steven Curry, it aims to provide a platform through which refugees, volunteers, and asylum seekers can tell their story.
Whilst Mohamed's experiences are hard to genuinely comprehend, and his origins geographically distant, there is, as the late MP Jo Cox put it, more in common than things that divide us. In Aleppo, Mohamed was a student, studying to improve his future. After that came a big choice, he says:
"When I finished I suggested to my family that I would leave Syria – we didn't have any safety. My family, they accepted my decision."
It's easy to express concern about the situation in Syria, and easier still to condemn that concern as a lack of actual action. Framing Identity forms part of the wider work that students across the country are taking to help Mohamed and many others like him as they adjust to life in the UK. One of 38 STAR organisations nationwide, Cardiff STAR run a weekly English conversation club. These confront one of the biggest obstacles facing those who have left their home country – the language barrier.
Their event 'Refugee Rhythms' drew over 500 people to dance together with refugees and asylum seekers. It won the Cardiff Students' Union's award for 'Best Large Event'.
This work helps people who have been through things like those Mohamed has to be able to look towards the future. Speaking of his hopes, Mohamed says:
"I want to travel and find about more places and countries. Hopefully after I finish my studies. I like Germany. There's a lot of Syrians settling in Germany. I have a lot of friends who I went to medical school with there and they decided to stay. I decided to carry on to the U.K. I'm in contact with them now. They started their study again from where they left off. We left Syria together.
"Sometimes when I think about my situation, I was in medical school for two years. In four years I would have been a doctor. Now I'm trying to get back into medical school again."
Despite pursuing a Medicine degree for two years, Mohamed isn't able to study in the UK. Cardiff STAR are working to change this through their Equal Access campaign, says President Michael Hatcher. He explains that it aims for "equal access and fees to higher education for asylum seekers and refugees."
Working with groups such as Hope not Hate, the NUS, and the Welsh Refugee Council, the campaign hopes "to make Wales the first nation in the UK in which all universities are Equal Access accredited."
For all that Mohamed has been through, he remains optimistic. "Sometimes when I think that I'll eventually be a student in British University it's a really good feeling, I just need time."
To get involved with Cardiff STAR, join them on Facebook. Elsewhere, join their national network or search for your university's group.
More information on the Equal Access campaign and how you can help at your university, can be found here.
To follow Framing Identity, go to Cardiff STAR's Instagram.
For more information on what you can do to help the people in Aleppo – and you can – read this.
Photo credit: Steven Curry