Averaged control

We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter. First we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action. We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system

Switching control

We analyze the problem of switching controls for control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to another in a systematic way so that, at each instant of time, only one actuator is active. We first address a finite-dimensional model and show that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. To do this we introduce a new variational principle building a new functional based on the adjoint system whose minimizers yield the switching controls. When the above rank condition fails, the same variational strategy applies but the controls obtained this way fail to be of switching form since they may be, for some instants of time, convex combinations of both controllers. We then address the same issue for the 1-d heat equation endowed with two pointwise controls. We show that, due to the time analyticity of solutions, under suitable conditions on the location of the controllers, switching control strategies exist. We also show that the controls we obtain are optimal in the sense that, for instance, for two scalar valued controls, they are of minimal L2 (0, T; ℝ2)-norm, the space ℝ2 being endowed with the l1-norm. We also discuss some possible extensions to multi-dimensional heat equations which require a preliminary analysis of generic properties of the spectrum that, as far as we know, are not yet well understood

Spike controls for elliptic and parabolic PDEs

We analyze the use of measures of the minimal norm to control elliptic and parabolic equations. We prove the sparsity of the optimal control. In the parabolic case, we prove that the solution of the optimization problem is a Borel measure supported in a set of Lebesgue measure zero. In both cases, the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations. We also analyze the corresponding dual problem

Large Time Asymptotics for Partially Dissipative Hyperbolic Systems

This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t → ∞. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover

High frequency wave packets for the Schrödinger equation and its numerical approximations [Paquets d'ondes à haute fréquence pour l'équation de Schrödinger et ses approximations numériques]

We build Gaussian wave packets for the linear Schrödinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat and Zuazua (2009) [6]. It is by now well known that bigrid algorithms provide filtering mechanisms allowing to recover the uniformity of the dispersive properties as the mesh size goes to zero. We analyze and illustrate numerically how these high frequency wave packets split and propagate under these bigrid filtering mechanisms, depending on how the fine grid/coarse grid filtering is implemented

The Hardy inequality and the heat equation in twisted tubes

We show that a twist of a three-dimensional tube of uniform cross-section yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of self-similar variables and weighted Sobolev spaces for the heat equation. © 2010 Elsevier Masson SAS

Recovery of an initial temperature from discrete sampling

The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class

Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces

We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua, constituted by two n-dimensional wave equations coupled with a (n - 1)- dimensional one along a flexible interface. That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other. Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces. In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too. These results are illustrated by several numerical experiments