Special Session 158: From PDE control to the qualitative study of (random) dynamical systems

Feedback stabilization of entropy solutions to the p-system at a junction

Nicola De Nitti University of Pisa Italy Co-Author(s):    G. M. Coclite, N. De Nitti, M. Garavello, F. Marcellini

Abstract: We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions in the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional.

Differentiability of transition semigroup of generalized Ornstein-Uhlenbeck process

Ben Goldys Sydney University Australia Co-Author(s):    Szymon Peszat

Abstract: Let $X$ be a Banach space-valued Ornstein-Uhlenbeck process driven by a possibly degenerate Brownian Motion and let $P_t$ stand for the transition operator of the process. We provide a simple probabilistic proof of the fact that null-controllability of the corresponding deterministic system implies infinite Fr\`echet differentiability of $P_tf$ for every bounded Borel function $f$. Moreover, using the Girsanov theorem we provide a pointwise formula for all the derivatives of $P_tf$ in terms of multiple stochastic integrals. Applications to the analysis of transition semigroups in finite and infinite dimensions are given. In particular we consider the heat equation with space-time white noise boundary conditions.

Robust control of linear systems under small parameter variation

Pierre Lissy CERMICS, Ecole nationales des ponts et chausses France Co-Author(s):    Sylvain Ervedoza and Yannick Privat

Abstract: In this talk, I will discuss insensitization problems for linear controlled evolutions under small parameter variations. The goal is to design a control which both drives the nominal system to a prescribed target and cancels the first-order sensitivity of the final state with respect to the parameter. I will present an abstract framework, in which this question is recast as a controllability problem for a coupled state-sensitivity cascade system. In finite dimension, I will give a complete characterization in terms of a Kalman-type rank condition. I will also illustrate the approach on PDE examples, including the heat equation with uncertain diffusion and the wave equation with uncertain potential.

Sensitivity analysis of colored-noise-driven interacting particle systems

Laurent Mertz City University of Hong Kong Hong Kong Co-Author(s):    Josselin Garnier, Harry Ip

Abstract: We propose an efficient sensitivity analysis framework for a broad class of interacting particle systems driven by colored noise. The method relies on unperturbed simulations and substantially extends the Malliavin weight sampling approach introduced by Szamel for systems driven by Ornstein--Uhlenbeck noise. It enables the computation of sensitivity indices, including linear response functions, in settings with general colored noise. We show that these sensitivities depend not only on effective quantities such as the noise variance and correlation time, but also explicitly on the full noise spectrum. For a single particle in a harmonic potential, we derive exact analytical expressions for two classes of linear response functions. We then apply the method to a many-particle system interacting through a repulsive screened Coulomb potential, where we compute transport properties such as mobility and effective temperature. Our results demonstrate that dynamical behavior depends on the spectral properties of the driving noise in a nontrivial manner.

Controllability as a key tool for studying long-time behaviour of random dynamical systems

Vahagn Nersesyan NYU Shanghai Peoples Rep of China Co-Author(s):

Abstract: This talk is devoted to the study of the long-time behaviour of random dynamical systems under highly degenerate non-Gaussian forcing. I will review recent results showing how controllability properties of the underlying deterministic dynamics can be used to handle the degeneracy and establish ergodic behaviour.

Stability of a KdV equation close to critical lengths

Jingrui Niu Institute for Advanced Study in Mathematics, Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Shengquan Xiang

Abstract: In this talk, we study the quantitative exponential stability of the KdV equation on a finite interval, focusing on the cases where the length is close to the critical set. Through a constructive PDE control framework, we obtain sharp decay estimates for such cases. Specifically, I will introduce a transition-stabilization approach that combines the Lebeau--Robbiano strategy with the moment method to establish quantitative null controllability for the KdV equation. Furthermore, based on a classification of these critical lengths, we show that the KdV equation exhibits different asymptotic behaviors near different types of critical lengths. This is a joint work with Shengquan Xiang.

Controllability of incompressible fluids and related systems driven by degenerate forcing

Manuel Rissel ShanghaiTech University Peoples Rep of China Co-Author(s):

Abstract: I will talk about global controllability properties of incompressible fluids and related models driven by degenerate forcing, that is, by controls acting only in few equations of the considered systems or by controls that have a particular finite-dimensional structure.

On the controllability of the semi-geostrophic equation. Part I

Franck Sueur University of Luxembourg France Co-Author(s):

Abstract: In the first part of this two-parts talk (second part would be done by Vincent Laheurte), I will present some controllability issues for the two-dimensional dual semi-geostrophic equation, a fully nonlinear counterpart of the incompressible Euler equations, where the linear Biot-Savart law is replaced by the nonlinear Monge-Amp\`ere equation.

Holomorphic regularity of processes generated by the heat equation with white noise boundary data

Marius Tucsnak University of Bordeaux France Co-Author(s):

Abstract: We consider the heat equation on a bounded interval with Dirichlet or Neumann boundary conditions driven by independent white noises at the endpoints. Although boundary white-noise forcing is usually associated with very low regularity, we show that the corresponding stochastic evolution in fact exhibits a sharp holomorphic regularity. More precisely, for every positive time the solution extends holomorphically to a rhombus in the complex plane having the original interval as one of its diagonals, and the resulting process admits a continuous version with values in a scale of weighted Bergman spaces on that domain. We determine the precise range of parameters for which this phenomenon holds, indexed by the parameters $\delta\in(0,1)$ and $\Theta\in\left(0,\frac{\pi}{4}\right)$, and prove that it is optimal: the conclusion fails at the critical values $\delta=0$ and $\Theta=\frac{\pi}{4}$. Our approach combines recent advances on reachability theory for boundary-controlled systems with techniques from Bergman-space theory. In this way, we extend to the stochastic setting the sharp regularity of time continuous trajectories recently developed for deterministic heat equations.

Symmetry and observability for wave equations on the torus

Shengquan Xiang Peking University Peoples Rep of China Co-Author(s):

Abstract: In this talk, I will discuss observability for wave equations, with particular emphasis on symmetry. I will focus on spacetime measurable observation sets on the one-dimensional torus, where a complete characterization is available. will first describe a counterexample showing that the Geometric Control Condition (GCC), while sufficient in many settings, does not guarantee observability when observation is restricted to certain spacetime measurable sets. Inspired by this counterexample, we introduce a symmetry condition, referred to as the Observable Symmetry Condition (OSC), which captures an additional structural constraint underlying observability. We show that OSC, together with GCC, provides a necessary and sufficient characterization of observability on spacetime measurable sets. Finally, motivated by OSC, we introduce a weak GCC and show that, when combined with OSC, it yields a necessary and sufficient condition for unique continuation.

Global Exponential Stabilization for a Simplified Fluid-Particle Interaction System

Zhuo XU University of Bordeaux France Co-Author(s):    Marius Tucsnak

Abstract: In this talk, we present a stabilization result for a simplified one-dimensional fluid--particle interaction system. First, without any smallness assumptions, we establish a non-collision property: the particle never reaches the fluid boundary in finite time, which in turn yields global well-posedness of the interaction system. Next, we study a stabilization problem for this model. In the absence of feedback control, the particle converges to an \emph{a priori} unknown limit position $h^\ast$ that cannot be described solely by the initial data. To regulate the final position of the particle to an arbitrary target $h_1 \in (-1,1)$, we employ a proportional feedback control acting on the particle, $u(t) = K\bigl(h_1 - h(t)\bigr), \qquad K>0$, and prove that both the position error $h(t)-h_1$ and the total kinetic energy of the closed-loop system decay exponentially to zero.

Level-3 large deviations for the white-forced 2D Navier--Stokes system in a bounded domain

Meng Zhao CY Cergy Paris University France Co-Author(s):

Abstract: We study the large deviations principle (LDP) of Donsker-Varadhan type for the white-forced Navier-Stokes system in a bounded domain. Under the assumption that the noise is non-degenerate, we establish level-2 and level-3 LDPs with rate functions given by the Donsker-Varadhan formulas. The proof relies on an improved version of Kifer`s criterion, a lift argument, an improved abstract result on the large-time asymptotics of generalized Markov semigroups, and a delicate approximation scheme utilizing the resolvent operators of the Markov semigroup.