## Null-controllability properties of the wave equation with a second order memory term

Abstract: We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. We assume that the control is acting on an open subset $\omega(t)\subset\mathbb{T}$, which is moving with a constant velocity $c\in\mathbb{R}\setminus\{-1,0,1\}$. The main result of the paper shows that the equation is null controllable in a sufficiently large time $T$ and for initial data belonging to suitable Sobolev spaces. Its proof follows from a careful analysis of the spectrum associated to our problem and from the application of the classical moment method.

## Phase portrait control for 1D monostable and bistable reaction-diffusion equations

Pouchol C., Trélat E., Zuazua E. , DOI: 10.1088/1361-6544/aaf07e

Abstract: We consider the problem of controlling parabolic semilinear equations arising in population dynamics, either in finite time or infinite time. These are the monostable and bistable equations on (0,L) for a density of individuals 0≤y(t,x)≤1, with Dirichlet controls taking their values in [0,1]. We prove that the system can never be steered to extinction (steady state 0) or invasion (steady state 1) in finite time, but is asymptotically controllable to 1 independently of the size L, and to 0 if the length L of the interval domain is less than some threshold value L⋆, which can be computed from transcendental integrals. In the bistable case, controlling to the other homogeneous steady state 0<θ<1 is much more intricate. We rely on a staircase control strategy to prove that θ can be reached in finite time if and only if L < L⋆. The phase plane analysis of those equations is instrumental in the whole process. It allows us to read obstacles to controllability, compute the threshold value for domain size as well as design the path of steady states for the control strategy.

## Decay rates for elastic-thermoelastic star-shaped networks

Han Z., Zuazua E. Decay rates for elastic-thermoelastic star-shaped networks
Network and Heterogeneous Media September 2017, 12(3): 461-488. DOI: 10.3934/nhm.2017020

Abstract:
This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths.

First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.

## Controllability under positivity constraints of multi-d wave equations

Pighin, D., Zuazua, E. Controllability under positivity constraints of multi-d wave equations , DOI:

Abstract: We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems.

## Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces

Trélat E., Zhang C., and Zuazua E.
SIAM J. Control Optim. (2018), no. 2, 1222-1252. DOI: 10.1137/16M1097638

Abstract: In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.

## Optimal shape design for 2D heat equations in large time

Trélat E., Zhang C., and Zuazua E. Pure and Applied Functional Analysis DOI:

Abstract: In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.

## Minimal controllability time for finite-dimensional control systems under state constraints

Lohéac J., Trélat E., Zuazua, E.

DOI: 10.1016/j.automatica.2018.07.010

Abstract: We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.

## Corrigendum and addendum to “Hierarchic control for a coupled parabolic system”

Hernández-Santamaría V., de Teresa L., Poznyak A. , DOI: 10.4171/PM/1998

Abstract: In [2] we used three controls for a system of two coupled parabolic equations. We defined three functionals to be minimized and a hierarchy on the controls obtaining from the optimality condition a system of six coupled equations. In order to prove the null controllability, by means of the leader control acting only on the first equation, we give a proof of a Carleman inequality (Proposition 6.4) that in fact is incorrect. In this corrigendum we slightly modify the followers functionals given by (3) page 118 [2] in such a way that for the corresponding hierarchic system a correct Carleman inequality can be proved. This modification allows to introduce a coefficient a12≠0 (a12 was zero in [2]).

## Insensitizing controls for a semilinear parabolic equation: a numerical approach

Hernández-Santamaría V., de Teresa L., Boyer, F. , DOI:

Abstract: In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semidiscrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.