Controllability of Evolution Equations with Memory

Felipe W. Chaves-Silva, Xu Zhang, and Enrique Zuazua Controllability of Evolution Equations with Memory
SIAM J. Control Optim., 55(4), 2437–2459. (23 pages) DOI: 10.1137/151004239

Abstract: This article is devoted to studying the null controllability of evolution equations with memory terms. The problem is challenging not only because the state equation contains memory terms but also because the classical controllability requirement at the final time has to be reinforced, involving the contribution of the memory term, to ensure that the solution reaches the equilibrium. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the adjoint system. We first consider finite-dimensional dynamical systems involving memory terms and derive rank conditions for controllability. Then the null controllability property is established for some parabolic equations with memory terms, by means of Carleman estimates.

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Null controllability for wave equations with memory

Lu Q., Zhang X., Zuazua E. Null controllability for wave equations with memory
Journal de Mathématiques Pures et Appliquées DOI: 10.1016/j.matpur.2017.05.001

Abstract: We study the memory-type null controllability property for wave equations involving memory terms. The goal is not only to drive the displacement and the velocity (of the considered wave) to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. This memory-type null controllability problem can be reduced to the classical null controllability property for a coupled PDE–ODE system. The latter can be viewed as a degenerate system of wave equations, in which the velocity of propagation for the ODE component vanishes. This fact requires the support of the control to move to ensure the memory-type null controllability to hold, under the so-called Moving Geometric Control Condition. The control result is proved by duality by means of an observability inequality which employs measurements done on a moving observation open subset of the domain where the waves propagate.

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Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation

Gosse L., Zuazua E Filtered gradient algorithms for inverse design problems of one-dimensional Burgers equation
Innovative Algorithms and Analysis,Gosse, Laurent, Natalini, Roberto (Eds.), Springer INdAM Series 16, 2017, 197-228. DOI: 10.1007/978-3-319-49262-9_7

Abstract: Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a N-wave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presented.

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Averaged controllability of parameter dependent conservative semigroups

Lohéac J., Zuazua E. Averaged controllability of parameter dependent conservative semigroups
Journal of Differential Equations, 262 (3) (2017), 1540–1574 DOI: 10.1016/j.jde.2016.10.017

Abstract: We consider the problem of averaged controllability for parameter depending (either in a discrete or continuous fashion) control systems, the aim being to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do it in the context of conservative models, both in an abstract setting and also analysing the specific examples of the wave and Schrödinger equations.
Our first result is of perturbative nature. Assuming the averaging probability measure to be a small parameter-dependent perturbation (in a sense that we make precise) of an atomic measure given by a Dirac mass corresponding to a specific realisation of the system, we show that the averaged controllability property is achieved whenever the system corresponding to the support of the Dirac is controllable.
Similar tools can be employed to obtain averaged versions of the so-called Ingham inequalities.
Particular attention is devoted to the 1d wave equation in which the time-periodicity of solutions can be exploited to obtain more precise results, provided the parameters involved satisfy Diophantine conditions ensuring the lack of resonances.

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Minimal controllability time for the heat equation under unilateral state or control constraints

Lohéac J., Trélat E., Zuazua E. Minimal controllability time for the heat equation under unilateral state or control constraints
Mathematical Models and Methods in Applied Sciences, Volume 27, Issue 09, August 2017 DOI: 10.1142/S0218202517500270

Abstract: The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve non-negativity. Besides, due to infinite velocity of propagation, the heat equation is null controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady state to a positive final one, requiring that the state remains non-negative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity of propagation, realizing controllability under the unilateral non-negativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in the multi-dimensional case. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.

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Local elliptic regularity for the Dirichlet fractional Laplacian

Biccari U., Warma M., Zuazua E. Local elliptic regularity for the Dirichlet fractional Laplacian
Advanced Nonlinear Studies 17 (2017), 387-409. DOI: 10.1515/ans-2017-0014

Abstract: We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian (-\Delta)^s on an arbitrary bounded open set \Omega\subset\mathbb{R}^N. For 1<p<2, we obtain regularity in the Besov space B^{2s}_{p,2,\textrm{loc}}(\Omega), while for 2\leq p<\infty we show that the solutions belong to W^{2s,p}_{\textrm{loc}}(\Omega). The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

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Large time control and turnpike properties for wave equations

Zuazua E. Large time control and turnpike properties for wave equations
Annual Reviews in Control DOI: 10.1016/j.arcontrol.2017.04.002

Abstract: In the last decades mathematical control theory has been extensively developed to handle various models, including Ordinary and Partial Differential Equations (ODE and PDE), both of deterministic and stochastic nature, discrete and hybrid systems.
However, little attention has been paid to the length of the time horizon of control, which is necessarily long in many applications, and to how it affects the nature of controls and controlled trajectories. The turnpike property refers precisely to those aspects and stresses the fact that, often, optimal controls and trajectories, in long time intervals, undergo some relevant asymptotic simplification property ensuring that, during most of the time-horizon of control, optimal pairs remain close to the steady-state optimal one.
Due to the intrinsic finite velocity of propagation and the oscillatory nature of solutions of the free wave equation, optimal controls for waves are typically of oscillatory nature. But, despite this, as we shall see, under suitable coercivity conditions on the cost functional to be minimised and when controllability holds, the turnpike property is also fulfilled for the wave equation.
When this occurs, the approximation of the time-depending control problem by the steady-state one is justified, a fact that is often employed in applications to reduce the computational cost.
We present some recent results of this nature for the wave equation and other closely related conservative systems, and discuss some other related issues and a number of relevant open problems that arise in this field.

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Actuator design for parabolic distributed parameter systems with the moment method

Privat Y., Trélat E., Zuazua E. Actuator design for parabolic distributed parameter systems with the moment method
SIAM Journal on Control and Optimization, 55(2), 1128–1152. DOI: 10.1137/16M1058418

Abstract: In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset Ω of IR n. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in Ω, over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L 2-energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional parabolic operators, but we also provide several examples in higher dimensions. We consider two types of controllers: either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results.

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Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Privat Y., Trélat E., Zuazua E. Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Journal of the European Mathematical Society, Volume 18, Issue 5, 2016, pp. 1043–1111 DOI: 10.4171/JEMS/608

Abstract: We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [0,T] with T>0. It is well known that, if the pair (ω,T) satisfies the Geometric Control Condition (ω being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in ω×(0,T).
We address the problem of the optimal location of the observation subset ω among all possible subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset ω has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain Ω).
We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures.
Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.

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