The next talk of the webinar Control in Times of Crisis http://ctcseminar.mat.utfsm.cl/ will be given by Mariano Mateos (Oviedo, Spain) on Thursday, December 9, see title and abstract below.

The talk will be given at
    10 AM in Mexico / 1: 00 PM in Chile  / 5:00 PM in European Central Time (Berlin, Roma, Madrid, Paris) on Thursday December 9
on the Zoom videoconference platform (ID and access code are given below).

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Speaker : Mariano Mateos (Oviedo, Spain)

Title: Optimal control problems with non-smooth integral constraints in space.

Abstract: Sparsity is often a desirable property for the solution of least squares problems that may appear, for instance, in image processing or optimal control problems. Two possible techniques to obtain sparse solutions are the following: we can either penalize the functional with an $L^1$-type term or impose an upper bound on the norm of the control variable in $L^1$. There are several examples in the literature about optimal control problems where the usual Tikhonov-regularized tracking type cost functional is penalized, by instance, with the norm of the control variable in $L^1(\Omega)$ --for problems governed by elliptic equations-- or in $L^2(0,T;L^1(\Omega))$ --in the case of problems governed by parabolic equations--. In this talk, we discuss the second technique: we impose an upper bound on the norm of the control variable in $L^1(Omega)$ --for problems governed by elliptic equations-- or in $L^\infty(0,T;L^1(\Omega))$ --in the case of problems governed by parabolic equations--. Notice that this is a quite natural condition: the ``amount of control'' that you can feed into the system at each instant of time is usually limited by technological or economic reasons. On the other hand, the lack of differentiability of the constraint leads to special difficulties in the analysis of the problem that require new techniques of proof. We first prove existence of solution, and obtain necessary first order optimality conditions, from which we deduce the sparsity properties of local solutions. Second order necessary and sufficient optimality conditions with a minimal gap are also obtained. In the second part of the talk, we show the correct way to discretize the problem, so that the discrete solutions still satisfy the sparsity properties, prove convergence and obtain error estimates. Finally, we discuss optimization methods and present several examples. In order to avoid some of the most technical details, we will focus on a problem governed by a semilinear elliptic equation, and will comment on the differences and extra difficulties that appear when the governing equation is parabolic. This presentation is based on a collaboration with E. Casas (U. de Cantabria, Spain) and K. Kunisch (U. Graz, Austria).

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Meeting Zoom:
https://us02web.zoom.us/j/81379950776?pwd=c013ck1LSzYwUFkyK0tidnQ1bFpkQT09
ID metting: 813 7995 0776
Access code: CTC2021

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All talks will be available online at the webpage http://ctcseminar.mat.utfsm.cl/.
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