A spillover phenomenon in the optimal location of actuators

In this paper, we are interested in finding the optimal location and shape of the actuators in a stabilization problem. Namely, we consider the one-dimensional wave equation damped by an internal feedback supported on a subdomain $\omega$ of given length. The criterion we want to optimize represents the rate of decay of the total energy of the system. It theoretically involves all the eigenmodes of the operator. From an engineering point of view, it seems more realistic to consider only a finite number of modes, say the $N$ first ones. In that context, we are able to prove existence and uniqueness of an optimal domain $\omega_N^*$: it is the better possible location for the actuators. We characterize this optimal domain and we point out the following strange phenomenon (at least for small lengths): the optimal domain $\omega_N^*$ which is the better one for the $N$ first modes is actually the worse one for the $N+1$-th mode. This looks like to the well-known spillover phenomenon in Control Theory. At last, we will give some possible extension and open problems in higher dimension

A deterministic optimal design problem for the heat equation

For the heat equation on a bounded subdomain $\Omega$ of $\mathbb{R}^d$, we investigate the optimal shape and location of the observation domain in observability inequalites. A new decomposition of $L^2(\mathbb{R}^d)$ into heat packets allows us to remove the randomisation procedure and assumptions on the geometry of $\Omega$ in previous works. The explicit nature of the heat packets gives new information about the observability constant in the inverse problem.Comment: 22 page

Optimal design of boundary observers for the wave equation

In this article, we consider the wave equation on a domain of $\mathbb{R}^n$ with Lipschitz boundary. For every observable subset $\Gamma$ of the boundary $\partial\Omega$ the observability constant provides an account for the quality of the reconstruction in some inverse problem. Our objective is here to determine what is, in some appropriate sense, the best observation domain. After having defined a \textit{randomized observability constant}, more relevant tan the usual one in applications, we determine the optimal value of this constant over all possible subsets $\Gamma$ of prescribed measure $L|\partial\Omega|$, with $L\in(0,1)$, under appropriate spectral assumptions on $\Omega$. We compute the maximizers of a relaxed version of the problem, and then study the existence of an optimal set of particular domains $\Omega$. We then define and study an approximation of the problem with a finite number of modes, showing existence and uniqueness of an optimal set, and provide some numerical simulations

Optimal design of sensors for a damped wave equation

International audienceIn this paper we model and solve the problem of shaping and placing in an optimal way sensors for a wave equation with constant damping in a bounded open connected subset Ω of IR n. Sensors are modeled by subdomains of Ω of a given measure L|Ω|, with 0 < L < 1. We prove that, if L is close enough to 1, then the optimal design problem has a unique solution, which is characterized by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

We consider the homogeneous wave equation on a bounded open connected subset Ω of IRn. Some initial data being specified, we consider the problem of determining a measurable subset ω of Ω maximizing the L2-norm of the restriction of the corresponding solution to ω over a time interval [0, T], over all possible subsets of Ω having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components

Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions

We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset $\Omega$ of $\R^n$. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a $L^1$ constraint on densities, the so-called {\it Rellich functions} maximize this functional.Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the $L^\infty$-norm of {\it Rellich functions} may be large, depending on the shape of $\Omega$, we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of {\it bang-bang} functions and {\it Rellich densities} for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation.Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations

Optimal observation of the one-dimensional wave equation

International audienceIn this paper, we consider the homogeneous one-dimensional wave equation on $[0,\pi]$ with Dirichlet boundary conditions, and observe its solutions on a subset $\omega$ of $[0,\pi]$. Let $L\in(0,1)$. We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets $\omega$ of $[0,\pi]$ of Lebesgue measure $L\pi$. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for $L = 1/2$. When $L \neq 1/2$ we prove the existence of solutions of a convexified minimization problem, proving a no gap result. We then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem

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