Workshop on Control Problems

Welcome, introductory words by organizers, technical information, and soundcheck for first talk.

We study the null controllability of linear parabolic equations posed on the whole space $\mathbb{R}^n$ by means of a source term locally distributed on a subset of $\mathbb{R}^n$. We want to know to which extend the results known for the heat equation still hold in this degenerate setting. We want to identify their geometric control condition. First, we recall known results for the heat equation. Then, we present a variante of the Lebeau-Robbiano's method, less greedy in spectral analysis. It relies on projections for which a spectral inequality holds, and appropriate smoothing properties to compensate the lack of commutation between the projection and the evolution. Finally, we provide interesting sufficient conditions for the null controllability of Ornstein-Uhlenbeck equations and quadratic equations with zero singular space.

In this lecture we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning and Universal Approximation. We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control. We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies. This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics. This lecture is inspired in joint work, among others, with Borjan Geshkovski (MIT), Carlos Esteve (Cambridge), Domènec Ruiz-Balet (IC, London) and Dario Pighin (Sherpa.ai).

Consider the Baouendi-Grushin equation $(\partial_t - \partial_x^2 - x^2\partial_y^2)g(t,x,y) = 0$ with Dirchlet boundary conditions on $\mathbb R \times [0,\pi]$. The functions $\exp(-nt + iny - nx^2\!/2) \sin(ny)$ are solutions of this equation. Thus, functions of the form $P(\exp(-t+iy-x^2\!/2)$ where $P$ is a polynomial, are also solutions. That way, an observability inequality on the Baouendi-Grushin implies an inequality on polynomials. But if we put Dirichlet boundary conditions on $x = \pm 1$, this link between solutions and polynomials is only approximate. It is still true that the observability inequality implies an inequality on polynomials, but we require a new tool to prove it. In this talk, I will present this tool: an estimate on "pseudo-differential-type" operators on polynomials. This estimate is proved essentially by a few elementary tools from complex analysis.

Optimal feedback controls for nonlinear systems are characterized by the solutions to a Hamilton Jacobi Bellmann (HJB) equation. In the deterministic case, this is a first order hyperbolic equation. Its dimension is that of the statespace of the nonlinear system. Thus solving the HJB equation is a formidable task and one is confronted with a curse of dimensionality. In practice, optimal feedback controls are frequently based on linearisation and subsequent treatment by efficient Riccati solvers. This can be effective, but it is local procedure, and it may fail or lead to erroneous results. In this talk, I give a brief survey of current solution strategies to partially cope with this challenging problem. Subsequently I describe three approaches in some detail. The first one is a data driven technique, which approximates the solution to the HJB equation and its gradient from an ensemble of open loop solves. The second one is based on Newton steps applied to the HJB equation. Combined with tensor calculus this allows to approximately solve HJB equations up to dimension 100. Results are shown for the control of discretized Fokker Planck equations. The third technique circumvents the direct solution of the HJB equation. Rather a neural network is trained by means of a succinctly chosen ansatz. It is proven that it approximates the optimal feedback gains as the dimension of the network is increased. This work relies on collaborations with B.Azmi, S.Dolgov, D.Kalise, Vasquez, and D.Walter.

Motivated by applications in the optimal control of phase separation and quantitative image processing, a class of optimization problems with data-driven differential equation (DE) constraints is considered. In this context, the data-driven DE may arise from coupling ab initial components with machine learned ones with a subsequent application of a solution scheme, or from learning the DE solver directly from data. For this setting and depending on the regularity of the activation functions of underlying deep neural networks, approximation results and stationarity conditions are derived. Moreover, numerical tests finally validate the theoretical findings.hod. Several numerical examples are provided and the efficiency of the algorithm is shown.

For given Banach spaces $X,U$ we consider abstract control systems of the form $\dot{x}(t) = -A x(t) + Bu(t)$ for $t\in (0,T]$ with $x(0) = x_0\in X$, where $-A$ is the generator of a strongly continuous semigroup on $X$ and $B\colon U\to X$ is a bounded linear operator. For such systems the question of null-controllability arises, i.e.\ whether for all initial conditions $x_0\in X$ there exists a control function $u\colon [0,T]\to U$ such that $x(T) = 0$. In this talk we will first review the classical duality result stating that this controllability question can be answered by showing a so-called final-state observability estimate of the form $\|x'(T)\|\leq C_{\mathrm{obs}} \|y\|$, where $y\colon [0,T]\to Y:=U'$ is the observation function of the dual system given by $\dot{x'}(t) = -A' x'(t)$ for $t\in (0,T]$ with $x'(0) = x_0'\in X'$, and $y(t) = B' x'(t)$ for $t\in (0,T]$. We then show sufficient conditions for obtaining such a final-state observability estimate, with explicit dependence of $C_{\mathrm{obs}}$ on all parameters. Having established this abstract result we show an application to heat-like evolution equations and comment on further generalisations, e.g.\ non-autonomous equations, weak versions of observability and controllability, as well as stabilizability. The talk is split into two parts, and both parts are based on joint works together with Clemens Bombach, Michela Egidi, Fabian Gabel and Dennis Gallaun.

For given Banach spaces $X,U$ we consider abstract control systems of the form $\dot{x}(t) = -A x(t) + Bu(t)$ for $t\in (0,T]$ with $x(0) = x_0\in X$, where $-A$ is the generator of a strongly continuous semigroup on $X$ and $B\colon U\to X$ is a bounded linear operator. For such systems the question of null-controllability arises, i.e.\ whether for all initial conditions $x_0\in X$ there exists a control function $u\colon [0,T]\to U$ such that $x(T) = 0$. In this talk we will first review the classical duality result stating that this controllability question can be answered by showing a so-called final-state observability estimate of the form $\|x'(T)\|\leq C_{\mathrm{obs}} \|y\|$, where $y\colon [0,T]\to Y:=U'$ is the observation function of the dual system given by $\dot{x'}(t) = -A' x'(t)$ for $t\in (0,T]$ with $x'(0) = x_0'\in X'$, and $y(t) = B' x'(t)$ for $t\in (0,T]$. We then show sufficient conditions for obtaining such a final-state observability estimate, with explicit dependence of $C_{\mathrm{obs}}$ on all parameters. Having established this abstract result we show an application to heat-like evolution equations and comment on further generalisations, e.g.\ non-autonomous equations, weak versions of observability and controllability, as well as stabilizability. The talk is split into two parts, and both parts are based on joint works together with Clemens Bombach, Michela Egidi, Fabian Gabel and Dennis Gallaun.

This talk will focus on the analysis of null controllability for some parabolic problems. I will first recall the classical moment method and exhibit some cases were its usual application is not optimal. These cases are related to spectral assumptions which can lead to a positive minimal null control time even if the problem is of parabolic nature. To deal with theses situations, when the control is scalar, I will present the block moment method we introduced with Assia Benabdallah and Franck Boyer (2020). It is a generalization of the work by Fattorini and Russell (1973, 1974) which roughly consists in treating simultaneously the moments equations associated with eigenvalues that are close. Then, I will present a recent generalization done with Franck Boyer (preprint) to deal with any non scalar admissible control operator. The presentation of this general methodology will come with some examples of applications to coupled systems of one dimensional linear parabolic equations.

This presentation focus on the null-controllability problem for the \emph{generalized Baouendi-Grushin equation} $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = \mathbf{1}_\omega u$ on a rectangular domain. Sharp controllability results already exist when the control domain $\omega$ is a vertical strip, or when $q(x) = x$. In a recent work with Armand Koenig and Julien Royer, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $\omega$. In some geometries for $\omega$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. I will present this new results and some key points of their proofs: known results when $\omega$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-\partial_x^2 + \nu^2 q(x)^2$ when $\Re(\nu)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.

This talk deals with a general method to prove final-state observability estimates based on the Lebeau-Robbiano strategy. Complementing previous results obtained in the context of Hilbert spaces, we derive observability estimates of operator semigroups in general Banach spaces by combining an uncertainty principle and a dissipation estimate. Thereby, we allow for generalized growth rates in the assumptions and use an integral expression to estimate the observability constant. We apply the results to subordinated semigroups where the extension of the growth rates naturally appears. The talk is based on joint work with Jan Meichsner and Christian Seifert.

In collaboration with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are selfadjoint hypoelliptic operators satisfying the Hörmander condition. Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl law. When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, Martinet) the Wey law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.

In this talk I will present results about the simultaneous interior control for heat and wave equation. i.e. the control of both equations by the same control function acting in the interior of the domain. I will show that while for the heat equation, null controllability is always achievable, for waves, it requires a geometric condition, stronger than Bardos-Lebeau-Rauch geometric control condition. This is a joint work with I. Moyano (université de Nice Cote d’Azur).

As is surely known to the audience, the Uncertainty Principle (UP) is a statement in Fourier analysis which can be summarized as "a function and its Fourier transform can not be both concentrated at the same time". There are many formulations of this principle depending on how concentration is measured (small dispersion, fast decrease, smallness of support,...). On the other hand, many PDEs (heat, Schrödinger,...) can be solved using the Fourier transform so that each UP can be reformulated in terms of solutions of those PDEs. This in turn raises new questions e.g. when adding a potential to the Schrödinger equation. In the opposite direction, some properties of PDEs can also bring new results that take the form of an uncertainty principle. The aim of this talk is to give a short survey of some of those interactions.

Starting with a presentation of simple models, we generalize the approach considering different models. We consider a problem of observability of square membranes [2]. We discuss the wave equation with memory modeling viscoelastic materials, the representation of the solution as a Fourier series and an application to glass relaxation [3]. [1] V. Komornik, P. Loreti: Fourier series in control theory, Springer Monographs in Mathematics, 2005. [2] V. Komornik, P. Loreti: Observability of Square Membranes by Fourier Series Methods Bollettin of the South Ural State University, Series Mathematical Modelling, Programming & Computer Software 8 (3), 2015. [3] P. Loreti, D. Sforza: Viscoelastic aspects of glass relaxation models, Physica A 526, 2019.

In this talk, we will present some stability results of a system corresponding to the coupling between a dissipative equation (set in an infinite dimensional space) and an ordinary differential equation. Namely we consider $U, P$ solution of the system \begin{equation*} \begin{cases}U_t=\mathcal{A} U +M P, \hbox{ in } H,\newline P_t=BP+ N U, \hbox{ in } X,\newline U(0)=U_0,P(0)=P_0, \end{cases} \end{equation*} where $\mathcal{A}$ is the generator of a $C_0$ semigroup in the Hilbert space $H$, $B$ is a bounded operator from another Hilbert space $X$, and $M$, $N$ are supposed to be bounded operators. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of {ODE}-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators that garantee stability properties of our system, i.e., strong stability, exponential stability or polynomial one. We will illustrate our general results by examples of dispersive medium models. Since our stability results are based on the frequency domain approach, a short introduction on this topic will be presented.

It is a well-known fact in control theory, that null-controllability in time $T$ of a control system given by an inhomogeneous Cauchy problem is equivalent to final-state-observability of the associated dual system. A key ingredient to show the latter property is the availability of a spectral inequality, that is an inequality of the form $\lVert f \rVert_{L^2(\Omega)}\leq C\lVert f\rVert_{L^2(\omega)}$ for all $f$ in the range of the spectral projection of an appropriate operator up to a certain energy value, where $\omega\subset \Omega$ is the considered observability set. In this talk we present a general framework based on complex analytical technique to obtain such an inequality in the case that $\omega$ is a thick set, that is well-distributed in some sense within the bigger space. The main ideas are a generalization of techniques first exploited by Kovrijkine to study functions with compactly supported Fourier Transform. This talk is based on a joint work with Albrecht Seelmann.

We consider a transport equation by a gradient vector field with a small viscous perturbation. We study uniform observability properties from a small subset in the (singular) vanishing viscosity limit. We prove with a series of examples that in general, the minimal time for uniform observability may be much larger than the minimal time needed for the observability of the limit equation. We also prove that the two minimal times coincide for positive solutions. This is a joint work with Camille Laurent.

We explain some observability results for subelliptic Schrödinger equations depending on the step of the subelliptic structure. The proofs are based on resolvent estimates, and the results shed light on the speed of propagation of these equations, notably in the ``degenerated directions'' of the subelliptic structure. As a corollary, we also obtain observability results for subelliptic heat-type equations and establish a decay rate for subelliptic damped wave equations. Joint work with Chenmin Sun.

We consider a bilevel optimization problem, where the lower level problem is given by the Kantorovich problem of optimal transportation. The upper level optimization variables are the optimal transport plan and one of the marginals. A possible application of the problem under consideration is the identification of a marginal based on measurements of the transportation process. Due to the curse of dimensionality associated with the Kantorovich problem, regularization techniques are frequently employed for its numerical solution. A prominent example is the entropic regularization leading to the well-known Sinkhorn algorithm. Here, we pursue a different approach and apply a quadratic regularization leading to transport plans in $L^2(\Omega_1 \times \Omega_2)$, where $\Omega_1$ and $\Omega_2$ are the domains of the marginals. The dual problem is a problem in $L^2(\Omega_1) \times L^2(\Omega_2)$, which yields the desired reduction of the dimension. We investigate the convergence behavior of the regularized bilevel problems (where the Kantorovich problem as lower level problem is replaced by its quadratic regularization) for regularization parameter tending to zero. It turns out that, under additional assumptions, weak-* accumulation points of sequences of optimal solutions of the regularized bilevel problems are solutions of the original bilevel Kantorovich problem.

We review some general connections between the observability, controllability and stabilizability of time reversible linear distributed systems. They apply among others to simple models of vibrating bodies, plates and to Maxwell's equations. We present in a historical context some basic methods to study such problems.

In this talk, we study the null-controllability of heat equations associated to fractional anisotropic Shubin operators. We consider these equations posed on the whole Euclidean space with a control supported on a measurable subset. Thanks to the Lebeau-Robbiano method, it is sufficient to obtain quantitative spectral inequalities for these operators. Such spectral estimates will be presented for different geometric conditions on the control support and will be obtained from recent uncertainty principles holding in Gelfand-Shilov spaces. In particular, these results generalize and improve known results about (an)harmonic oscillators.

We study the controllability of the Schrödinger equation, on a bounded interval, with Dirichlet boundary conditions and bilinear control. In general, to prove small-time local controllability (STLC), one can use the linear test: if the linearized system is controllable, one can hope to prove the STLC of the nonlinear system through a fixed-point theorem. Here, we study the controllability of the Schrödinger equation around the ground state when the linearized system is not controllable. More precisely, we study to what extent the quadratic and cubic terms of the expansion of the solution can help to recover the directions lost at the linear level. First, for any positive integer $n$, we formulate assumptions under which the quadratic term induces a drift in the nonlinear dynamics, quantified by the $H^{-n}$-norm of the control, preventing STLC for controls small in the $H^{2n-3}$-norm. Then, we prove on the contrary that for controls small in less regular spaces, the cubic term enables us to recover the controllability lost at the linear level, despite the drift. The proof is inspired by Sussmann's method to show the sufficiency of the $\mathcal{S}(\theta)$ condition for STLC in finite dimension. However, it uses a different global strategy relying on a new concept of tangent vector, better adapted to an infinite-dimensional framework.

In this talk, we present our new results on null controllability and inverse problems of the parabolic equation with dynamic boundary conditions and drift terms \[ \begin{cases} \partial_t y- d \Delta y+ B(x).\nabla y+c(x).y={1}_{\omega}u +f &\text{ in } \Omega_T, \newline \partial_t y_{\Gamma}-\delta \Delta_{\Gamma}y_{\Gamma}+d\partial_{\nu}y + b(x).\nabla_{\Gamma}y_{\Gamma}+\ell(x) y_{\Gamma}= {1}_{\Gamma_0}v+g &\text{ on } \Gamma_{T}, \newline y_{|\Gamma}(t, x)= y_{\Gamma}(t, x) \qquad &\text{ on } \Gamma_{T}, \newline y(0,\cdot)= y_{0} &\text{ in } \Omega, \newline y_{|\Gamma}(0,\cdot)= y_{0,\Gamma} &\text{ on } \Gamma, \end{cases} \] where $\Omega$ is a bounded domain of $\mathbb{R}^{N}$, with smooth boundary $\Gamma=\partial \Omega$ of class $C^{2}$, $\nu(x)$ is the outer unit normal field to $\Omega$ in the point $M(x)$ of $\Gamma$, $\partial_{\nu} y:= (\nu. \nabla y)_{| \Gamma}$, $ d $, $\delta$ are positive real numbers, $c\in L^{\infty}(\Omega)$, $\ell\in L^{\infty}(\Gamma)$, $B\in L^{\infty}(\Omega)^N$, $b\in L^{\infty}(\Gamma)^N$, $f\in L^2((0,T)\times \Omega)$ and $g\in L^2((0,T)\times \Gamma)$. The functions $u$ and $v$ are internal and boundary controls, acting on small regions $\omega$ and $\Gamma_0$, respectively. To obtain our aim, we show first some suitable Carleman estimates for the backward adjoint problems.

In this talk we revisit the classical connections between the long-time behaviour of the damped wave equation and the geometry of the support of the damping. As shown by Bardos, Lebeau, Rauch, Taylor and Burq among others, the geometric control condition is necessary to obtain exponential decay in this case, but slower decay can be expected otherwise. In particular, we show that for waves on a compact Riemannian manifold a rough damping function supported on any positive-measure set yields at least logarithmic decay in large time. These ideas can be extended to the context of linear Boltzmann operators under degenerate collisions, following (Han-Kwan and Léataud, Bernard and Salvarani). In the kinetic context, we get some quantitative estimates on a simple model using probabilistic tools. (This is based on joint works with Nicolas Burq and Josephine Evans).