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## Section: New Results

### Controllability: new results

Let us list here our new results on controllability beyond the quantum control framework.

• In [13], we study approximate and exact controllability of linear difference equations using as a basic tool a representation formula for its solution in terms of the initial condition, the control, and some suitable matrix coefficients. When the delays are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. The paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in ${L}^{2}$. The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.

• In [14] we consider the 2D incompressible Navier-Stokes equation in a rectangle with the usual no-slip boundary condition prescribed on the upper and lower boundaries. We prove that for any positive time, for any finite energy initial data, there exist controls on the left and right boundaries and a distributed force, which can be chosen arbitrarily small in any Sobolev norm in space, such that the corresponding solution is at rest at the given final time. Our work improves earlier results where the distributed force is small only in a negative Sobolev space. It is a further step towards an answer to Jacques-Louis Lions' question about the small-time global exact boundary controllability of the Navier-Stokes equation with the no-slip boundary condition, for which no distributed force is allowed. Our analysis relies on the well-prepared dissipation method already used for Burgers and for Navier-Stokes in the case of the Navier slip-with-friction boundary condition. In order to handle the larger boundary layers associated with the no-slip boundary condition, we perform a preliminary regularization into analytic functions with arbitrarily large analytic radius and prove a long-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontal analyticity.

• We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $\omega$ along a time interval $\left[0,T\right]$ with $T>0$. It is well known that, if $\omega$ is open and if the pair $\left(\omega ,T\right)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $\omega ×\left(0,T\right)$. The observability constant ${C}_{T}\left(\omega \right)$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In [17], we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a measurable subset $\omega$. We also establish that, as $T\to +\infty$, the ratio ${C}_{T}\left(\omega \right)/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions, the second one is of a geometric nature and involves the average time spent in $\omega$ by Riemannian geodesics.

• In [22] 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 $\left(0,L\right)$ for a density of individuals $0\le y\left(t,x\right)\le 1$, with Dirichlet controls taking their values in $\left[0,1\right]$. 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 $0<\theta <1$ is much more intricate. We rely on a staircase control strategy to prove that $\theta$ 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.

• The paper [27] deals with the controllability problem of a linearized Korteweg-de Vries equation on bounded interval. The system has a homogeneous Dirichlet boundary condition and a homogeneous Neumann boundary condition at the right end-points of the interval, a non homogeneous Dirichlet boundary condition at the left end-point which is the control. We prove the null controllability by using a backstepping approach, a method usually used to handle stabilization problems.

• The paper [44] is devoted to the controllability of a general linear hyperbolic system in one space dimension using boundary controls on one side. Under precise and generic assumptions on the boundary conditions on the other side, we previously established the optimal time for the null and the exact controllability for this system for a generic source term. In this work, we prove the null-controllability for any time greater than the optimal time and for any source term. Similar results for the exact controllability are also discussed.

• Given any measurable subset $\omega$ of a closed Riemannian manifold and given any $T>0$, we study in [49] the smallest average time over $\left[0,T\right]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation subset.

• Our goal is to study controllability and observability properties of the 1D heat equation with internal control (or observation) set an interval of size $ϵ\to 0$. For any $ϵ$ fixed, the heat equation is controllable in any time $T>0$. It is known that depending on arithmetic properties of the center of the interval, there may exist a minimal time of pointwise control of the heat equation. We relate these two phenomena in [54].

• Our goal in [55] is to relate the observation (or control) of the wave equation on observation domains which evolve in time with some dynamical properties of the geodesic flow. In comparison to the case of static domains of observation, we show that the observability of the wave equation in any dimension of space can be improved by allowing the domain of observation to move.

• In [57] we consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.

Let us also mention t the book chapter [31], which has been published this year.