Special Session 99: Emerging Trends in Analysis and Control of PDEs

Control and Energy Decay in Wave Networks with N-Inertial Interfaces.

Mohammad Akil Universite Polytechnique Hauts de France France Co-Author(s):    Ibtissam Issa, Ahmet Ozkan Ozer and Cristina Pignotti

Abstract: In this talk we study the asymptotic behavior and stability of a system of $N+1$ serially connected wave equations on adjacent intervals, coupled through $N$ inertia-generating effects at interior junctions. In contrast to earlier works limited to boundary feedback, we consider internal damping applied at different locations along the network and examine how the spatial distribution of these damping mechanisms influences the overall stability. The analysis reveals that both the presence and placement of damping significantly affect the rate of energy decay. The results establish new connections between damping configuration and asymptotic behavior, providing a comprehensive analytical approach to the stabilization of distributed systems with pointwise inertia.

Inverse problems for Kelvin--Voigt viscoelastic and Moore--Gibson--Thompson equations

Jone Apraiz University of the Basque Country Spain Co-Author(s):    Rodrigo Lecaros (Federico Santa Mar\`{i}a Technical University, Chile) and Sebasti\`{a}n Zamorano (University of Deusto, Spain).

Abstract: In this talk we will consider inverse coefficient problems for two evolution models with relaxation structure: a Kelvin--Voigt viscoelastic system and the Moore--Gibson--Thompson equation arising in nonlinear acoustics. In both cases, we will establish Lipschitz stability estimates for the recovery of a space-dependent damping coefficient from a single measurement. This research extends Carleman-based inverse theory to third-order-in-time acoustic models and provides a unified framework for inverse problems in relaxation-type evolution equations. This is a joint research project in collaboration with Rodrigo Lecaros (Federico Santa Mar\`{i}a Technical University, Chile) and Sebasti\`{a}n Zamorano (University of Deusto, Spain).

Analysis and Control in Poroelastic Systems with Applications to Biomedicine

Lorena Bociu NC State University USA Co-Author(s):

Abstract: In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of fluid flows through deformable porous media by means of poroelastic systems is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific region with an initial value problem for the rest of the circulatory system. We present new results on wellposedness analysis and control for this nonlinear multiscale interface coupling of PDEs and ODEs.

Spectral inequality for the n-dimensional Stokes operator with n-1 observation terms.

Felipe W. Chaves-Silva Federal University of Paraiba Brazil Co-Author(s):    M. G. Ferreira-Silva, D. A. Souza

Abstract: In this talk, we present a new spectral inequality for the low frequencies of the Stokes operator, obtained by observing only n-1 components of the velocity field. As a consequence, we show that the cost of driving the solution of the evolutionary Stokes system to zero at a given time, using controls acting on n-1 components, is of the same order as the cost corresponding to controls acting on all components of the system.

Reconstruction of degeneracy region and power for parabolic equations and systems

Veronica Danesi University of Rome Tor Vergata Italy Co-Author(s):

Abstract: In this talk, we will address the inverse problem of recovering a degeneracy region within the diffusion coefficient of a parabolic equation by observing the normal derivative at the boundary. The strongly degenerate case is analyzed and our method is based on a careful analysis of the spectral problem. In particular, we derive sufficient conditions on the initial data that guarantee stability and uniqueness of the solution obtained from a one-point measurement. Moreover, we present more general uniqueness theorems, which also cover the identification of the initial data and the degeneracy power, using measurements taken over time. Our investigations cover the case of real 1-D degenerate parabolic systems of equations with a specific coupling. Besides, possible extensions to multidimensional parabolic equations will be also analyzed. Theoretical results are also supported by numerical simulations. This talk is based on works in collaboration with P. Cannarsa and A. Doubova.

Controllability of a one-dimensional debonding model

Nicola De Nitti University of Pisa Italy Co-Author(s):    N. De Nitti, A. Shao

Abstract: We investigate a one-dimensional dynamic debonding model, introduced by Freund (1990), in which the wave equation is coupled with a Griffith criterion governing the propagation of the fracture. In particular, we study the boundary controllability of the system to a prescribed target state. Our main results provide precise characterizations of the reachable target states, in both $C^{0, 1}$ and $C^1$ regularity settings, and construct exact controls toward these target states.

Exponential stability of a generalised Burgers equation in a n-dimensional torus

Imene DJEBOUR CY Cergy Paris Universite France Co-Author(s):

Abstract: We study a generalized conservative Burgers equation on a $n$-dimensional torus. We first consider the associated linear system. By exploiting the Harnack inequality and the $L^1$ contraction property, we establish exponential stability in the $L^1$ norm. For the nonlinear problem, we derive a uniform-in-time estimate of the solution using the De Giorgi-Nash-Moser iteration method. Moreover, we show that the H\older norm of the solution remains uniformly bounded and decays exponentially.

Recent Advances in the Analysis of Energy Balance Climate Models (EBCMs)

Giuseppe Floridia Sapienza University of Rome Italy Co-Author(s):

Abstract: In this talk, we investigate the steady-state solutions of Energy Balance Climate Models (EBCMs). We first examine the one-dimensional case characterized by a degenerate diffusion coefficient that vanishes at the poles, before extending our analysis to the general two-dimensional framework. By employing a variational approach, we define a dedicated energy functional to establish critical qualitative properties of these solutions. Ultimately, these results provide a rigorous mathematical foundation for understanding temperature distribution and climate stability in EBCMs. This is joint work with Gianmarco Del Sarto (Technische Universit\at Darmstadt).

Null controllability for a degenerate Fokker-Plank equation with a drift term

GENNI FRAGNELLI University of Siena Italy Co-Author(s):    Dimitri Mugnai

Abstract: The Fokker-Planck equation describes the time evolution of the probability density function of the velocity for a particle under the influence of drag forces and random forces. In particular, this equation has multiple applications in information theory, graph theory, data science, finance, economics... In one spatial dimension the Fokker-Planck equation for the probability density $p(t,x)$ can be rewritten as \[ p_t(t,x) - (a(t,x)p(t.x))_{xx} + (\mu(t,x)p(t,x))_x= f(t,x) \] where $t\in [0, T]$, $T>$ is fixed and $x \in (0,1)$. In this talk we assume that $a$ is a function degenerating at $x=0$; the purpose is to study the null-controllability of the solution, namely the possibility to drive the solution $p$ to rest completely at time $T$.

Stability and Well-Posedness of Korteweg-de Vries-Burgers Equations with Delayed Feedback

Ibtissam Issa University of L`Aquila Italy Co-Author(s):

Abstract: This talk is devoted to the analysis of generalized Korteweg-de Vries-Burgers equations with delayed feedback and damping. Using semigroup methods and Lyapunov techniques, we establish well-posedness results and investigate the long-time behavior of solutions. In particular, we derive exponential decay estimates under suitable assumptions on the delay term and the system parameters. The analysis highlights how delay effects influence stability properties and provides conditions ensuring robust stabilization, even in challenging settings.

Fredholm backstepping for self-adjoint operators and applications to the rapid stabilization of parabolic and fractional parabolic equations in arbitrary dimensions

Nazim Kacher Sorbonne Universite France Co-Author(s):    Ludovick Gagnon, Hoai-Minh Nguyen

Abstract: In this talk, we address the Fredholm backstepping for self-adjoint operators in any dimension. This method of stabilization links an equation needing to be stabilized to an already stable equation, via an invertible transformation $T$. This method was until now restricted to 1D. This is because one had to deal with series that do not converge in dimension 2 or higher, due to the growth of the eigenvalues. In this talk, we will see how the introduction of a spectral projection gets rid of these difficulties, and allows us to apply this method to parabolic equations in all dimension. The growth condition of the spectrum and the gap condition is now relaxed. As an application, we establish the rapid stabilization of the heat and fractional heat equation $(-\Delta^s)$ with $s\in(0,1]$ from measurable or open sets in arbitrary dimensions

Exact output tracking of the 1-d heat equation

Pierre Lissy CERMICS, Ecole nationales des ponts et chausses France Co-Author(s):    Lucas Davron

Abstract: In this talk, I will address an exact output tracking problem (or sidewise control problem) for the one-dimensional heat equation on a bounded interval, with Neumann boundary conditions. The control is a Neumann control on the left of the interval, whereas the output is the Dirichlet trace at the right-hand side of the interval. We aim at characterizing all the possible outputs when the control $u$ lives in the space $L^2(0,T)$ for a finite or infinite $T$. In this setting, we obtain an exact characterization of all the trackable outputs. In infinite time, they form some kind of Gevrey class, whereas in finite time, some auxiliary power series involving the derivative of the signals in time T needs also to belong to the reachable space. The proof is based on applying the Laplace transform in time , and notably on some auxiliary lemmas on the Hardy spaces that might be interested by themselves, and some kind of Plancherel formula. If time permits, I will explain the link with some classical problems in real analysis, namely, the interpolation problems in some Gevrey classes.

Sharp geometric conditions for the contollability of magnetic Schr\odinger equations

Fabricio Macia Universidad Politecnica de Madrid Spain Co-Author(s):

Abstract: In this talk, we will present necessary and sufficient conditions for the controllability of Schr\odinger equations with electromagnetic potential. It is well known that such equations are controllable from an open set $\omega$ whenever it satisfies the geometric control condition; in that case, one has controllability for any time $T>0$. We show that this result can be improved on the sphere $\mathbb{S}^2$: in this setting, the system is controllable if $\omega$ satisfies a magnetic-field-dependent geometric control condition, which is weaker than the geometric control condition. Under this condition, there exists a minimum time, depending on the magnetic potential, beyond which the system is controllable. This geometric condition, and the minimum control time, are necessary for controllability as well. This is joint work with Gabriel Rivi\`ere (Nantes).

A method to determine the minimal null control time of one-dimensional linear hyperbolic systems

Guillaume Olive Jagiellonian University Poland Co-Author(s):

Abstract: In this talk, we address the problem of null controllability for one-dimensional first-order linear hyperbolic systems of the form: $$ \begin{cases} \frac{\partial y}{\partial t}(t,x)+\Lambda(x) \frac{\partial y}{\partial x}(t,x)=M(x) y(t,x), \ y_-(t,1)=u(t), \quad y_+(t,0)=Qy_-(t,0), \ y(0,x)=y^0(x), \end{cases} \quad (t,x) \in (0,T) \times (0,1), $$ where $u:(0,T) \to \mathbb{R}^m$ is the control. We present a method to find the minimal control time when the coefficients are regular enough. This presentation is based on a joint work with Long Hu (https://doi.org/10.1016/j.jde.2025.113455)

Time-delayed opinion dynamics with leader-follower interactions

Cristina Pignotti University of L`Aquila Italy Co-Author(s):    Young-Pil Choi, Chiara Cicolani

Abstract: We study time-delayed variants of the Hegselmann-Krause opinion formation model. In particular, we focus on a model featuring a small group of leaders and a large group of non-leaders. In our model, leaders influence all agents but only interact among themselves, while non-leaders update their opinions via interactions with both their peers and the leaders, with time delays accounting for communication and decision-making lags. We prove that the system achieves consensus with an exponential decay rate and establish uniform $l_\infty$-stability with exponentially decaying transients. Furthermore, we analyze the mean-field limit in two regimes: (i) with a fixed number of leaders and an infinite number of non-leaders, and (ii) with both populations tending to infinity, obtaining existence, uniqueness, and exponential decay estimates for the corresponding macroscopic models.

Steady Self-Propelled Motion of a Rigid Body in a Viscous Fluid with Navier-Slip Boundary Conditions

Arnab Roy Basque Center for Applied Mathematics (BCAM) Spain Co-Author(s):    S\`arka Ne\v casov\`a and Ana Leonor Silvestre

Abstract: In this talk, we consider the steady self-propelled motion of a rigid body immersed in a three-dimensional incompressible viscous fluid governed by the Navier-Stokes equations. Under suitable smallness assumptions on the boundary flux and on the normal component of the prescribed surface velocity, we establish the existence of weak steady solutions to the coupled fluid-structure system. Beyond existence, we provide a necessary and sufficient condition under which a prescribed slip velocity on the body surface induces nontrivial translational or rotational motion of the rigid body. This is achieved through the introduction of a finite-dimensional control thrust space, defined via auxiliary exterior Stokes problems with Navier boundary conditions, which captures the effective contribution of boundary-driven flows to the rigid-body motion. Our results clarify how boundary effects generate propulsion and extend the classical Dirichlet-based theory to the Navier-slip setting.

Controlling Klein-Gordon Chains and Lattices

Sarah Strikwerda University of Wisconsin - Madison USA Co-Author(s):    Hung Vinh Tran and Minh-Binh Tran

Abstract: The Klein-Gordon equation is a wave equation with a cubic nonlinearity. This system can be used to describe the interaction of particles on a lattice. I will describe a suitable choice of feedback control to drive the system to a collective flocking behavior in finite time. Finally, I will highlight the connection between our flocking problem and a minimal-time problem in the framework of nonlinear Hamilton-Jacobi equations and optimal control theory.

Controllability properties of coupled PDEs

Emmanuel TRELAT Sorbonne University France Co-Author(s):    Hugo Lhachemi, Christophe Prieur, Emmanuel Tr\`elat

Abstract: In a series of works with Hugo Lhachemi and Christophe Prieur, we investigate the controllability properties of some coupled PDEs, which can be of different natures. For a heat-wave PDE with coupling at the boundary, we establish exact, exact null and approximate controllability in appropriate Hilbert spaces, under sharp assumptions. Our approach relies on an Ingham-M\untz inequality, allowing us to establish an observability inequality for the dual problem. The resulting controllability space, which depends on the coupling function and is characterized in a spectral way, is not a usual functional space. I will also give some results for other cascade systems, providing new results for coupled heat equations where interesting changes happen at some specific times in the controllability properties.

Recent advances on approximate trajectory controllability. Application to the enhanced dissipation of a passive scalar.

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

Abstract: Given a continuous-time control system and $\varepsilon > 0$, we address the question of the existence of controls that maintain the corresponding state trajectories within the $\varepsilon$-neighborhood of any prescribed path in the state space. We first investigate this property, called approximate tracking controllability, in the linear finite-dimensional case, when our answers are negative: we demonstrate that approximate tracking controllability of the full state is impossible even in a certain weak sense, except for trivial situations. For finite-dimensional systems with quadratic nonlinearities, we prove approximate tracking controllability on any time horizon with respect to the relaxation metric. This approach is next generalized for a PDE system: the Euler equations with distributed control. We underline the relevance of this weak setting by developing applications to coupled systems and by remarking obstructions that would arise for natural stronger norms. The main application concerns obtaining the enhanced dissipation of a passive scalar using a controlled fluid flow.

Small-Time Approximate and Exact Controllability for Nonlinear Parabolic Equations with Bilinear Controls

Cristina Urbani Universitas Mercatorum Italy Co-Author(s):

Abstract: n this talk, we investigate the small-time controllability of a class of nonlinear parabolic evolution equations posed on a torus of arbitrary dimension and driven by bilinear control terms. Assuming an appropriate saturation condition on the potential, we prove a small-time approximate controllability result between states that share the same sign. In the one-dimensional setting, this result can be strengthened by coupling it with a local exact controllability property. This combined approach yields small-time exact controllability from any positive initial state to the ground state of the associated evolution operator.