Section: New Results

Observability, controllability and stabilization in the time domain

Participants : Fatiha Alabau, Xavier Antoine, Thomas Chambrion, Antoine Henrot, Karim Ramdani, Lionel Rosier, Mario Sigalotti, Takéo Takahashi, Marius Tucsnak, Jean-Claude Vivalda, Ghislain Haine, Roberto Guglielmi.

Observability

The PhD of Ghislain HAINE is devoted to the analysis of observers based techniques for solving inverse problems. In [34] , we provide a convergence analysis of the iterative reconstruction algorithm proposed by Ramdani et al. in [81] . More precisely, we propose a complete numerical analysis for semi-discrete (in space) and fully discrete approximations of the iterative algorithm using finite elements in space and an implicit Euler method in time. In order to disseminate our reconstruction method in the community of Automatic and control engineering, we wrote an engineer's oriented note [33] presenting the main ideas of our algorithm.

Control

In [48] , we develop a model that describes the impact of the amount of soot in the filter on the Diesel engine performance. This model is used to determine the optimal amount of soot on which the regeneration of the particulate filter shall start.

In [49] , we give sufficient conditions for the simultaneous approximate controllability of a bilinear Schrödinger equation driven by a single scalar control in the case where every energy level is non-degenerate and the control potential couples each pair of energy levels.

In [25] , we give sufficient conditions for the simultaneous approximate controllability of a bilinear Schrödinger equation driven by a single scalar control under a generic condition of coupling of all energy levels via a chain of non-degenerate transitions. The result applies for systems with degenerate energy levels or when the coupling operator does not couple directly each pair of energy levels.

In [16] we prove exact controllability for symmetric coupled wave equations by a single control in the case of coupling and control regions which do not intersect. For this, we use and extend the two-level energy method introduced by Alabau-Boussouira (2001, 2003). Using transmutation, we derive null controllability results for coupled parabolic and Schrödinger equations. This is the first positive quantitative result, in a multi-dimensional framework with control and coupling regions with empty intersection. Such questions have been considered using Carleman estimates but no positive quantitative results could be derived in the case of control and coupling regions which do not intersect.

In [30] we propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator. The algorithm combines Russell's “stabilizability implies controllability” principle with the Galerkin method. The main new feature of this work consists of giving precise error estimates.

Stabilization

In [44] we consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain. We prove exponential stability of the solution, by introducing suitable energies and Lyapounov functionals. Such analysis is also extended to a nonlinear case.

In [52] we present a course on stabilization of hyperbolic equations given at a CIME session on Control of PDE's in Italy in july 2010, including well-known results, together with recent ones including nonlinear stabilization, memory-damping and stabilization of coupled systems by a reduced number of controls. In particular, we present the optimal-weight convexity method (Alabau-Boussouira 2005, 2010) in both the finite dimensional and infinite dimensional framework and give applications to semi-discretization of hyperbolic PDE's.

In [14] , we consider stabilization of coupled systems of hyperbolic PDE's with hybrid boundary conditions, by a reduced number of closed loop globally distributed controls. We establish polynomial stabilization for such systems under a new compatibility condition. We also derive decay rates for explicit initial data using interpolation theory.

In [15] , we consider stabilization of coupled systems of wave-type, with localized couplings and either localized internal closed loop controls or boundary control. We establish polynomial decay rates for coupling and damping regions which do not intersect in the one-dimensional case. We also derive results in the multi-dimensional case, under multiplier type conditions for both the coupling and damping regions. The novelty and difficulty is to consider localized couplings.

In [13] we give a constructive proof of Gibson's stability theorem, some extension and further positive and negative applications of this result.

Very few lower energy estimates are available in the literature. The main one has been proved in the one-dimensional case for a locally distributed power-like damping for the wave equation in 1995 by Haraux. This approach does not generalize to multi-dimensional cases and for systems of equations. In [11] , we prove strong energy and weak velocity lower estimates for the nonlinearly damped Timoshenko beams (coupled system), and for Petrowsky equations in two space dimensions.

In [12] , we show that if a linear system is observable through a locally distributed (resp. boundary) observation, then any dissipative nonlinear feedback locally distributed (resp. active only on a part of the boundary) stabilize the system and we give quasi-optimal energy decay rates, under the optimal condition of geometric optics of Bardos-Lebau-Rauch (1992). The approach is based on the optimal-weight convexity method (Alabau-Boussouira 2005, 2010). Our results generalize previous results by Haraux (1989) and Ammari and Tucsnak (2001) for linear feedbacks.

In [17] , we study the stabilization of Bresse system, which models vibrations of a beam through three coupled wave equations. We establish polynomial stabilization of the full system by a single feedback control.

In [23] , Badra (University of Pau) and Takahashi consider the stabilization of the system $y^{\text{'}}=Ay+Bu$ where $A:𝒟\left(A\right)\to 𝒳$ be the generator of an analytic semigroup and $B:𝒰\to {\left[𝒟\left(A^{*}\right)\right]}^{\text{'}}$ a quasi-bounded operator. They consider controls $u$ which are the linear combination of a finite family $(v_1,...,v_K)$. They show that if $(A^{*},B^{*})$ satisfies a unique continuation property and if $K$ is greater or equal to the maximum of the geometric multiplicities of the the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,...,v_K)$. With the same functional framework, they also prove the stabilizability of a class of nonlinear system when using feedback or dynamical controllers. They apply these results to stabilize the Navier–Stokes equations in 2D and in 3D by using boundary control with an optimal number of controllers.

In [32] we tackle an unsolved difficulty in the control of vibrating systems, consisting in the fact that a small delay in the application of a feedback control may destroy the stabilizing effect of the control. We consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end and we show that certain delays (large, in general) in the boundary feedback preserve the exponential stability of the system.

In [18] we consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a chain beginning with a string. We study the strong and polynomial stabilities of this system on this network and the spectrum of the corresponding conservative system.

In [45] we study the asymptotic behavior of the solution of the non-homogeneous elastic systems with voids and a thermal effect. Our main results concern strong and polynomial stabilities (since this system suffers of exponential stability).

In [24] we are interested in an inverse problem for the wave equation with potential on a star-shaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate.

In [35] we consider switched systems on Banach and Hilbert spaces governed by strongly continuous one-parameter semigroups of linear evolution operators. We provide necessary and sufficient conditions for their global exponential stability, uniform with respect to the switching signal, in terms of the existence of a Lyapunov function common to all modes.

In [47] , we investigate sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We apply our result to the synthesis of an observer for the three-cell converter.

Other problems

In [37] , we study a spectral problem related to a reaction-diffusion model where the preys and the predators do not live on the same area. We are interested in the optimal zone where the control should take place. First we prove existence of an optimal domain in a natural class. Then, it seems plausible that the optimal domain is localized in the intersection of the living areas of the two species. We prove this fact in one dimension for small size of domains.

In [41] , we explain why Donnelly's proof of the gap conjecture is not correct.

In [22] , we study the set of points, in the plane, defined by $\{(x,y)=({\lambda }_1\left(\Omega \right),{\lambda }_2\left(\Omega \right)),|\Omega |=1\},$ where $({\lambda }_1\left(\Omega \right),{\lambda }_2\left(\Omega \right))$ are either the two first eigenvalues of the Dirichlet-Laplacian, or the two first non trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

In [28] , we look for the minimizers of the functional $J_{\lambda }\left(\Omega \right)=\lambda \left|\Omega \right|-P\left(\Omega \right)$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\lambda$, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.