Abstracts of the Poster Session 1

Abstract: Let $T$ be an exponentially stable strongly continuous semigroup, $A$ its generator, and $C$ an admissible observation operator. We consider the modified Russell–Weiss conjecture: If $\left(A,C\right)$ satisfies the infinite-dimensional Hautus test and $T$ is similar to a contraction semigroup, then $\left(A,C\right)$ is exactly observable. We disprove this conjecture by constructing a counterexample. In contrast, we prove that the Hautus test implies exact observability when $A$ is self-adjoint.

Abstract: We study the asymptotic behavior of a coupled wave-heat system, which consists of a wave equation and a heat equation (with non-trivial heat coefficients) on two adjacent Lipschitz domains coupled by a common interface. We first establish asymptotic stability of the system, with the of aim later improving this by establishing specific, non-uniform rates of energy decay. Secondly, we analyze how decay rates of the coupled wave-heat system depend on the domain and the heat coefficients. This allows us to extend results of the existing literature to non-trivial, spatially dependent heat coefficients on less regular (Lipschitz) domains. Thirdly, we discuss three new geometric conditions on the wave domain and how they alone can be sufficient to prove or disprove the existence of polynomially fast decay rates.

Abstract: In this poster, we treat the strong stability radius of infinite-dimensional port-Hamiltonian (pH) systems under structure-preserving perturbations. In particular, we consider the operator $\mathrm{A}=(\mathrm{J}-\mathrm{R})\mathrm{Q}$ and analyze perturbations of the form ${\mathrm{A}}_{\mathrm{∆}}=(\mathrm{J}-(\mathrm{R}\mathrm{B}\mathrm{∆}{\mathrm{B}}^{\mathrm{*}}\left)\right)\mathrm{Q}$. We also address the exponential stability radius and perturbations of the operators $\mathrm{J}$ and $\mathrm{Q}$.

Abstract: We study the anisotropic, quasilinear Maxwell system on a bounded domain $\mathrm{\Omega }$ with perfectly conducting boundary conditions. It is damped via a conductivity $\mathrm{\sigma }$ which is strictly positive on a collar at the boundary. We prove that solutions decay exponentially to 0, if the fields have no magnetic charges on $\mathrm{\Omega }$, no electric charges off the support of $\mathrm{\sigma }$, and initial data are small. Our approach is adapted from the linear case and relies on a splitting of the solution via Helmholtz decompositions and an observability-type estimate for a related system without charges, shown using Morawetz multipliers. For simply connected domains, scalar coefficients, and a uniformly positive conductivity this problem was considered in [1] and [2]. For simply connected domains, anisotropic coefficients, and uniformly positive $\mathrm{\sigma }$ the problem was considered in [3].

References:

[1]

S. Nicaise and C. Pignotti, Internal stabilization of Maxwell’s equations in heterogeneous media, Abstr. Appl. Anal. 7 (2005), pp. 791–811.

[2]

K. D. Phung, Contrôle et stabilisation d’ondes électromagnétiques, ESAIM Control Optim. Calc. Var. 5 (2000), pp. 87–137.

[3]

I. Lasiecka, M. Pokojovy and R. Schnaubelt, Exponential decay of quasilinear Maxwell equations with interior conductivity, NoDEA Nonlinear Differential Equations Appl. 26 (2019) Paper No. 51, 34.

Abstract: We study the stabilization of elliptic and parabolic partial differential equations through the notion of coercivity-inducing feedback stabilizability. This property requires the existence of a feedback law that renders the closed-loop operator coercive and thereby provides exponential stability without overshoot. We consider this mechanism on the abstract operator level and give several characterizations. Based on these results we investigate the robustness of optimal control on large networks of elliptic equations with regard to spatially localized perturbations. By combining coercivity-inducing feedback stabilizability with suitable coercivity estimates and coupling assumptions that are uniform in the domain size, we are able to show, that the influence of the perturbation on the optimal control and state trajectory decays exponentially with the distance from the source of the perturbation. The theoretical results are supported by numerical simulations illustrating this behaviour.

Abstract: This poster presents a rigorous abstract framework ensuring the well-posedness of a class of integrodifferential input-output systems in Hilbert spaces. Motivated by the second-order equations governing thermal transport in linear thermoelastic materials, we specifically investigate the following system:

$$\{\begin{array}{l}\stackrel{‥}{x}\left(t\right)+\mathrm{\nu }\dot{x}\left(t\right)+{\int }_0^{t}a(t-s)\dot{x}\left(s\right)ds=Ax(t)+\mathrm{\gamma }A\dot{x}(t)+{\int }_0^{t}b(t-s\left)Kx\right(s)ds+Bu(t),\; t\ge 0,\\ \left[\begin{array}{c}x\left(0\right)\\ \dot{x}\left(0\right)\end{array}\right]=\left[\begin{array}{c}{x}_0\\ x_1\end{array}\right],\end{array}$$
augmented by the observation equation

$$y\left(t\right)=C\dot{x}\left(\mathrm{t}\right),\; t\ge 0.$$
Our primary contribution is demonstrating that such systems give rise to well-posed linear systems in the sense of Weiss and Salamon, and providing an explicit form for their corresponding transfer functions. The theoretical findings are ultimately validated through a detailed application to a model of rigid heat conductors.

Abstract: We characterize stability and its robustness for a class of boundary controlled, boundary observed hyperbolic partial differential equations. In particular, we show that asymptotic and exponential stability coincide for this class. Furthermore, we introduce the real and complex stability radii for which we give explicit formulas.

based on arXiv:2607.12812

The Koopman operator framework has become a powerful tool for analyzing nonlinear dynamical systems by transforming their complex behaviors into linear evolution in a function space. While widely applied in engineering, physics, and data science, theoretical foundations for time-dependent systems remain underdeveloped.

Abstract: In this poster, we analyze time-dependent nonlinear control systems $\dot{x}\left(t\right)=f\left(t,x\right)$ on a bounded domain $\mathrm{\Omega }\subseteq {\mathrm{ℝ}}^n.$ We introduce a Perron-Frobenius-like operator $S^{\mathrm{(}t,t_0)}$ acting on densities in $L^p\left(\Omega \right)$, which captures the transport of densities under the flow ${\mathrm{\Phi }}^{t,t_0}$. The dual concept, a Koopman-like operator acting on observables, arises naturally in this framework.

We characterize the infinitesimal generators of these families, showing that they provide unique weak solutions to associated infinite-dimensional linear systems. Additionally, we define a quadratic cost functional and the corresponding observability Gramian, which satisfies a Lyapunov equation.

This framework bridges finite-dimensional control theory and infinite-dimensional linear systems, enabling new theoretical and computational tools for analyzing time-dependent nonlinear dynamics.