Special Session 98: Control, Inverse problems and Long time dynamics of Evolutionary Systems

On the stability of a thermoelastic transmission problem with Kelvin-Voigt Damping

Maria Astudillo Federal University of Parana Brazil Co-Author(s):    Antonio Leite Guimaraes, Higidio Portillo Oquendo

Abstract: In this talk, we consider a transmission problem for a string composed of three different types of materials: an elastic material (without dissipation), a thermoelastic material and a Kelvin-Voigt viscoelastic material. We discuss how the position of the different components plays an important role in the study of the stabilization. In particular, when considering three distinct components, as expected, when the viscoelastic component is not in the middle of the material, then there exists exponential stability of the solution. On the other hand, when the viscoelastic part is in the middle of the material, the exponential stability of the system achieved through the dissipation given by the heat conduction is destroyed by the local Kelvin-Voigt damping with a discontinuous coefficient at the interface. In this case, the solution decays polynomially as $t^{-2}$.

Implications of a Novel Pressure Elimination Method for a Certain Nonlinear Fluid-Structure PDE Interaction

George Avalos University of Nebraska-Lincoln USA Co-Author(s):    Yuhao Mu

Abstract: In this talk we present some implications of our new technique for eliminating and recovering the pressure for a particular fluid-structure interaction model, a technique which is valid for general bounded Lipschitz domains. The specific fluid-structure interaction (FSI) that we consider is a well-known model of coupled Navier-Stokes flow with linear elasticity. The coupling between these two distinct PDE dynamics occurs across a boundary interface, with each of the components evolving on its own distinct geometry, and with the boundary interface being Lipschitz. Among other consequences, the new pressure elimination technique leads to a proof of well-posedness of the PDE system, globally in time in 2D (and 3D with small data assumptions), with again, this wellposedness being valid for general Lipschitz geometries.

Infinite Horizon Control Problems for Semilinear Parabolic Equations with Pointwise State Constraints

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

Abstract: We consider an optimal control problem for semilinear parabolic equations with infinite horizon, pointwise state contraints and two different types of control constraints (pointwise in space and time, and pointwise in time and $L^2$ in space). We present new results on first-order necessary conditions and second-order sufficient condition for local optimality. We also address the approximation of the infinite horizon control problem by finite horizon problems. We analyze the convergence of these approximations and provide error estimates.

Stabilization of the Critical Wave Equation via Energy-Dependent Nonlinear Damping

Marcelo Cavalcanti State University of Maringa Brazil Co-Author(s):    Valeria N. Domingos Cavalcanti, Josiane C. O. Faria , Cintya A. Okawa

Abstract: We investigate a semilinear wave equation with energy critical nonlinearity and a nonlinear damping mechanism driven by the total energy of the system. The model combines the quintic defocusing term with a time dependent dissipation of the form $E(t)u_t$, which introduces a nonstandard feedback structure coupling the dynamics and the energy functional. Weak solutions are constructed via Galerkin approximations, with the passage to the limit relying on uniform energy estimates and compactness arguments. Special attention is devoted to the critical nature of the nonlinearity, where concentration phenomena prevent purely energy-based methods from yielding refined spacetime control. This difficulty is resolved by incorporating nonhomogeneous Strichartz estimates together with smoothly truncated spectral approximations, ensuring uniform bounds at the dispersive level. Finally, we establish polynomial decay rates for the energy by adapting Nakao's method to the present nonlinear dissipative framework. The results highlight the stabilizing effect of the energy dependent damping and its interaction with the critical wave dynamics.

Finite Time Extinction and Stabilization for the Defocusing Schr\\{o}dinger Equation with Localized Damping

Wellington J Correa Federal Technological University of Parana Brazil Co-Author(s):

Abstract: In this paper, we study the existence and finite time extinction of solutions to a defocusing nonlinear Schr\"odinger equation on a smooth bounded domain as well as on the whole space and on an exterior domain. Initially, we construct approximate solutions within the framework of monotone operator theory. By employing multiplier methods and a unique continuation property, we establish that these approximations decay exponentially in the $L^2$-norm. Subsequently, through a limit passage using weak lower semicontinuity, we prove both global existence, $L^2$-decay, and finite time extinction for solutions of the original model. Furthermore, depending on the spatial dimension and the regularity of the solutions, we prove two distinct asymptotic behaviors: finite time extinction is achieved for weak solutions if $N=1$ employing a localized damping. Finite time extinction is also obtained for regular solutions if $N\lt 4$ with a distributive damping. In both situations, exponential stabilization is obtained in the remaining cases. To the best of our knowledge, the present paper is the first to establish finite time extinction for systems featuring a non-constant damping across bounded and unbounded domains, generalizing the works of \cite{Jesus1}, \cite{Jesus2}, \cite{Jesus3}, \cite{Gallo}, and \cite{Carles}. Finally, an efficient numerical algorithm is implemented to computationally verify the theoretical finite time extinction. The ensuing simulations confirm the efficacy of the proposed localized control design.

Asymptotic behaviour of the viscoelastic wave equation

Valeria Domingos Cavalcanti Universidade Estadual de Maringa - State University of Maringa Brazil Co-Author(s):

Abstract: We are concerned with the global existence of solutions as well as the decay rates for the viscoelastic wave equation in the past history framework.

Biological Concepts of Random Dynamical Systems in Infinite Dimension

Marcia Federson Universidade de Sao Paulo Brazil Co-Author(s):

Abstract: We investigate biological concepts such as persistence, permanence, and extinction within the framework of random dynamical systems in infinite-dimensional settings. Focusing on asymptotically random cocycles, we establish structural properties of the associated random attractors, including existence, compactness, invariance, minimality, and connectedness under suitable assumptions. These results provide a rigorous foundation for describing the long-term behavior of randomly evolving populations. In particular, we show how the qualitative features of random attractors can be used to characterize persistence and extinction phenomena, thereby linking dynamical properties with biologically meaningful outcomes. This approach offers a unified perspective for analyzing population dynamics under randomness in infinite-dimensional spaces.

Stability for non-autonomous degenerate wave equations.

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

Abstract: We consider a non-autonomous wave equation on $(0,1)$ with degeneracy at $x = 0$. Obviously, the presence of a non-autonomous term and of a degenerate function leads us to use different spaces with respect to the standard ones and it gives rise to some new difficulties. However, thanks to some suitable assumptions on the functions, one can prove some estimates on the associated energy that are crucial to obtain a uniform exponential decay.

Decay Properties of Higher-Order KdV-Type Systems with Dissipation and Time Delay

Fernando Gallego Universidad Nacional de Colombia Colombia Co-Author(s):    Roberto Capistrano, Vilmos Komornik

Abstract: In this talk, we analyze the decay properties of a class of higher-order nonlinear dispersive systems with time delay, extending classical models of KdV--Burgers type to a more general setting. The equations under consideration incorporate both dispersive and dissipative effects, together with a delayed feedback term acting on the dynamics. We show that the presence of delay can still lead to exponential decay of solutions under suitable structural conditions on the feedback coefficients. In particular, stabilization is achieved without imposing sign restrictions on the instantaneous damping term, highlighting the effective role of the delayed mechanism. In addition, we discuss the behavior of the system in the absence of delay and establish general decay results in Sobolev spaces, covering a broad range of regularity. The results provide a unified qualitative framework for understanding the interplay between dispersion, dissipation, and memory effects in higher-order evolution equations posed on unbounded domains.

Long-time dynamics of KdV-KdV systems

Victor Hugo Gonzalez Martinez Universidade Federal de Pernambuco Brazil Co-Author(s):

Abstract: The boundary stabilization problem of the Boussinesq KdV-KdV type system is investigated in this talk. An appropriate boundary feedback law consisting of a linear combination of a damping mechanism and a time-varying delay term is designed. Then, under a small restriction on the length of the spatial domain and the initial data, we show that the energy of the KdV-KdV system decays exponentially.

Supraconvergence and supercloseness in non-Fickian diffusion

Elias A Gudino Federal University of Parana Brazil Co-Author(s):

Abstract: Controlled Drug Delivery Devices (CDD) have demonstrated superior therapeutic performance compared to conventional drug delivery systems (DD). Progress in materials science and bio-nanotechnology has enabled the development of more advanced CDD platforms, enhancing targeted delivery while reducing adverse side effects. From the mathematical point of view, the description of the drug delivery from a CDD poses several challenges, including selecting appropriate materials and understanding their impact on release kinetics, as well as accurately describing drug transport within the target tissue. From a numerical perspective, the development of stable and accurate methods capable of simulating the drug delivery process also presents significant challenges. In this talk we will focus on the mathematical modeling, numerical simulation and numerical analysis of drug release kinetics from matrix systems. Specifically, the transport of small molecules through a polymeric material. In this context, we assume that a fluid diffuses into a swelling matrix and causes a deformation, which induces a stress-driven diffusion and consequently a non-Fickian mass flux. Other multiphysics scenarios will be considered where similar results are established for diffusion processes enhanced by temperature. Numerical results illustrating the behavior of the models and the obtained theoretical results will be presented.

Qualitative Properties of Composite-Structures and Fluid Interaction PDE Models

Pelin Guven Geredeli Clemson University USA Co-Author(s):    Robert Denk

Abstract: We consider the dynamics of composite structure-fluid interaction (FSI) PDE system. The coupling of 3D Stokes and 3D elastic dynamics is realized via an additional 2D elastic equation on the boundary interface. We analyze the (strong) stability and the exponential instability properties of the solution by way of analyzing the spectrum of the corresponding semigroup generator. We particularly appeal to Stone`s Theorem and linear perturbation theory for the lack of exponential decay of the said coupled PDE system. This is a joint work with Robert Denk (University of Konstanz).

Instability results for Volterra-Stieltjes integral equations

Jaqueline Mesquita Universidade Estadual de Campinas Brazil Co-Author(s):    Rogelio Grau, Jucileide dos Santos

Abstract: In this lecture, we will introduce the general class of functional Volterra Stieltjes integral equations and discuss how these equations can be connected to other types of equations. We will also present some results concerning instability for this class of equations.

On the Global Existence and Energy Decay of a Logarithmically Damped Wave Equation

Salim Messaoudi University of Sharjah United Arab Emirates Co-Author(s):    Muhammad Al-Gharabli

Abstract: This paper investigates a wave equation in the presence of nonlinear logarithmic damping. We present, for the first time, a comprehensive and rigorous proof of the global existence of weak solutions for this class of problems. The analysis is carried out using the Faedo-Galerkin approximation method. This result establishes a robust foundational framework for future studies involving logarithmic dampings. In addition, we prove that the energy of the system decays at a polynomial rate as time progresses, by employing a suitable multiplier method adapted to the structure of the logarithmic dissipation. Our findings contribute to the deeper understanding of stabilization mechanisms in nonlinear wave models and offer new insights into the role of logarithmic damping in the long-time behavior of solutions.

Energy decay for Korteweg-de Vries-Burgers type equations with delay feedback

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

Abstract: We consider generalized KdV-Burgers equations on the real line with indefinite damping and time delay. By combining semigroup methods with suitable Lyapunov functionals, we prove well-posedness and derive exponential decay estimates under different settings.

Exact Interior Controllability of Magnetoelastic Plates Using Purely Magnetic Actuation

Buddhika Priyasad University of Konstanz Germany Co-Author(s):    Reinhard Racke

Abstract: We establish exact interior controllability for a two-dimensional magnetoelastic plate system with control acting solely in the magnetic field equation. The main result shows that the coupled system is exactly controllable in arbitrarily small time T > 0, even though the control influences only the magnetic dynamics. This extends the principle of indirect control - previously observed in thermoelastic systems - to the magnetoelastic setting, demonstrating that the mechanical plate displacement can be steered using magnetic actuation alone. The analysis uses an operator-theoretic multiplier method adapted to handle the coupling between the mechanical and magnetic components. The proof consists of three main steps: establishing trace regularity for the adjoint system, deriving a suitable energy estimate, and applying a compactness-uniqueness argument to remove lower-order terms. This work provides the first exact controllability result for magnetoelastic systems and extends the indirect control framework from thermoelasticity to this setting. The techniques developed here are relevant for the control-theoretic study of magnetically coupled elastic structures, with potential applications in smart materials and electromagnetic actuators.

A Direct Approach for Detection of Bottom Topography in Shallow Water

Carole Rosier University of Littoral France Co-Author(s):    Noureddine Lamsahel

Abstract: We propose a fast, stable, and direct analytic method to detect underwater channel topography from surface wave measurements, based on one-dimensional shallow water equations. The technique requires knowledge of the free surface and its first two time derivatives at a single instant $t^{\star}$ above the fixed, bounded open segment of the domain. We first restructure the forward shallow water equations to obtain an inverse model in which the bottom profile is the only unknown, and then discretize this model using a second-order finite-difference scheme to infer the floor topography. We demonstrate that the approach satisfies a Lipschitz stability estimate and is independent of the initial conditions of the forward problem. The well-posedness of this inverse model requires that, at the chosen measurement time $t^{\star}$, the discharge be strictly positive across the fixed portion of the open channel, which is automatically satisfied for steady and supercritical flows. For unsteady subcritical and transcritical flows, we derive two empirically validated sufficient conditions ensuring strict positivity after a sufficiently large time. The proposed methodology is tested on a range of scenarios, including classical benchmarks and different types of inlet discharges and bathymetries.

Null controllability of strongly degenerate parabolic equations

Lionel ROSIER Universite du Littoral Cote d`Opale France Co-Author(s):    Antoine Benoit and Romain Loyer

Abstract: We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficients $a=a(x)$ in the principal part of the parabolic equation is such that the function $a->x/a(x)$ is in $L^p(0,1)$ for some $p>1$. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.

Interacting flexible structures with localized strong damping: Semigroup stability and regularity

LOUIS TEBOU Florida International University USA Co-Author(s):    Irena Lasiecka, Kais Ammari, Fathi Hassine and Souleymane Kadri Harouna

Abstract: Thanks, first to a conjecture of Goong Chen and David Russell (1982), then to results of Roberto Triggiani and Shuping Chen (1989, 1990), responding to that conjecture, it has been known since the late eighties that the Euler-Bernoulli plate with structural or Kelvin-Voigt damping exhibits an analytic and exponentially stable semigroup. Then came the natural question: What happens to these semigroup properties when the structural or Kelvin-Voigt damping is localized? In the late nineties, Kangsheng Liu and Zhuangyi Liu (1998) showed that the semigroup corresponding to an Euler-Bernoulli beam with localized Kelvin-Voigt damping is exponentially stable, but not analytic; in the same paper, those authors proved that the string equation with localized Kelvin-Voigt damping is not exponentially stable when the damping coefficient is discontinuous. In this talk, I'll give a brief historical account of what is known in this framework, then share recent findings with my collaborators Irena Lasiecka (case of Euler-Bernoulli plate with localized structural or Kelvin-Voigt damping), and Kais Ammari, Fathi Hassine and Souleymane Kadri Harouna (case of coupled wave equations with localized singular Kelvin-Voigt damping).

On the Long-time Stability of the Implicit Euler Scheme for a Two-phase Flow Model

Florentina Tone University of West Florida USA Co-Author(s):    Theodore Tachim

Abstract: We present results on the stability for all positive time of the fully implicit Euler scheme for an incompressible two-phase flow model. More precisely, we consider the time discretisation scheme and with the aid of the discrete Gronwall lemma and of the discrete uniform Gronwall lemma we prove that the numerical scheme is stable.

Stationary solutions for the wave equation with hyperbolic boundary conditions

Enzo Vitillaro Dipartimento di Matematica e Informatica Universit\`a degli Studi di Perugia Italy Co-Author(s):

Abstract: We deal with standing waves for the wave equation with hyperbolic boundary conditions, posed in a bounded domain with regular boundary. These problems possess a wide literature, including the papers on Arch. Rat. Mech. Anal. (2017), J.D.E. (2018) and DCDS-S (2021) by the author. Stationary solutions of these evolution problems turn out to be solutions of a doubly elliptic problem posed in the domain. This problem involved the Laplace operator inside the domain, the Laplace--Beltrami operator at the boundary, and up to two nonlinear sources, one inside the domain, the other one at the boundary. This type of problem has been studied by the author in a paper on Comm. Anal. Mech (2023) in presence of a single homogeneous boundary source. In this talk we discuss the more involved case of two nonlinear, possibly vanishing and nonhomogeneous, sources, which can also involve linear terms. In particular, we shall characterize in several ways the potential--well depth, by also proving the existence of solutions at this level. Finally, we shall also give multiplicity results for stationary solutions.