Abstracts of the Poster Session 2

Abstract: Weak convergence is fundamental to many averaged algorithms in Hilbert spaces, yet it falls outside classical converse Lyapunov theory, which relies on norm convergence. We bridge this gap using stability with respect to two measures. We first extend to arbitrary Banach spaces the equivalence between $\mathrm{𝒦}\mathrm{ℒ}$-stability and the existence of continuous Lyapunov functions with respect to two measures. We then identify measures for averaged operators that characterize weak convergence, yielding a converse Lyapunov theorem for $\mathrm{\kappa }$-averaged operators on Hilbert spaces. Under a natural ${\mathrm{𝒦}}_{\mathrm{\infty }}$-equivalence assumption, the Lyapunov function can be chosen Lipschitz continuous and weak convergence is strengthened to strong convergence. The framework applies to projection algorithms, including alternating projections and the Douglas–Rachford algorithm, and reveals a connection between converse Lyapunov theory and nonlinear error bounds.

Abstract: Our work studies output regulation for parabolic distributed-parameter systems modeled as self-adjoint operators with compact inverse on a Hilbert space, of which the multi-dimensional heat equation is taken as an example. Although it was proven that the infinite-dimensional regulator equations admit a solution, the resulting regulator is itself infinite-dimensional and not implementable. Existing finite-dimensional approaches achieve only robust regulation via internal-model-based controllers obtained through numerical model reduction. This paper instead solves the regulator equations directly for an $N$-mode modal truncation of the plant, yielding an explicit finite-dimensional feedback/feedforward controller. It is shown that the truncated transfer operator converges in norm to its infinite-dimensional counterpart and that the control effort required can be bounded uniformly in the truncation order $N$. In addition, the asymptotic tracking error vanishes as $N\to \mathrm{\infty ,}$ yielding the first convergence result of this kind for finite-dimensional exact output regulation of parabolic systems. The approach is illustrated on an one-dimensional heat equation with a spatially distributed patch actuator and an averaged temperature measurement, where the regulator solvability condition is verified analytically and the predicted convergence is confirmed numerically.

Abstract: We study the age-structured equation

$${\mathrm{\partial }}_{t}u\left(t,a\right)+{\mathrm{\partial }}_{a}u\left(t,a\right)=-\mathrm{\mu }\left(a\right)u\left(t,a\right),\; t\ge \mathrm{0,} a\in \left[0,a_m\right),$$
with delayed boundary condition

$$u\left(t,0\right)={\int }_0^{a_m}\beta \left(a\right)u(t-r,a)da,\; u(s,·)=u_0(s,·),\; -r\le s\le \mathrm{0,}$$
where $0<a_m\le \mathrm{\infty }$, $\mathrm{\mu }\in L_{\mathrm{loc}}^{\infty }\left(0,a_m\right)$, $\mathrm{\beta }\in L^{\infty }\left(0,a_m\right)$ and $r>0$ is a discrete delay. The delay turns the birth term into an unbounded operator acting at the boundary, which puts the model outside the reach of bounded-perturbation arguments.

Working on a product space with an $L^p$ history component, we obtain well-posedness from the Weiss–Staffans perturbation theorem for regular linear systems, together with a variation of constants formula. The difficulty is then the essential spectral radius of the solution semigroup ${\left(T\right(t\left)\right)}_{t\ge 0}$, which is not eventually compact. We approximate the convolution term in the variation of constants formula by compact operators converging in the operator norm and deduce ${\mathrm{\omega }}_{\mathrm{ess}}\left(T\right)<{\mathrm{\omega }}_0\left(T\right)$. Asynchronous exponential growth follows, with an explicit spectral projection and limiting age profile, valid for every sign of ${\mathrm{\omega }}_0\left(T\right),$ and settles the discrete delay case left open by Piazzera and Tonetto. Numerical simulations based on a discretization of the model complete the picture.

Joint work with Said Boulite and Lahcen Maniar.

Abstract: We study exponential output-to-state stability (eOSS) of linear infinite-dimensional systems in Banach spaces with bounded output operators. It is shown that eOSS is equivalent to the existence of a coercive eOSS Lyapunov function in implication form. Also, exponential detectability guarantees the existence of a coercive eOSS Lyapunov function in dissipation form and therefore eOSS. Counterexamples demonstrate that the converse implications fail in general: exponential zero-detectability does not imply eOSS, eOSS does not imply the existence of an eOSS Lyapunov function in dissipation form, which, in turn, does not imply exponential detectability. If the unstable subspace is finite-dimensional, zero-detectability implies exponential detectability and yields equivalent characterizations of eOSS. The results are illustrated with a parabolic equation.

Abstract: The input-to-state stability of a class of infinite-dimensional second-order port-Hamiltonian systems on a one-dimensional spatial domain is analyzed under nonlinear dynamic boundary feedback and boundary disturbances. Using energy methods in the port-Hamiltonian framework, we obtain sufficient conditions that guarantee uniform input-to-state stability of the closed-loop system.

Abstract: Analysing reachability for a control system is a subtle issue, especially for infinite-dimensional dynamics under bounded control constraints. We develop a computer-assisted framework for establishing non-reachability in linear parabolic PDEs governed by strongly elliptic operators, extending recent finite-dimensional techniques to the PDE setting.

For a closed convex reachable set and a convex target, non-reachability is equivalent to finding a dual certificate at which a suitable functional is negative. Linearity makes this functional computable by pulling the support function of the reachable set back through the endpoint map and reducing its evaluation to a control-independent adjoint equation. Our certification combines numerical search and regularisation with explicit convergence estimates for space–time discretisations of the adjoint equation and interval arithmetic for round-off errors.

For the one-dimensional heat equation with local nonnegative controls, we certify non-reachability of a target with a corner, whose rough dual certificate must first be regularised. The resulting interval enclosure proves non-reachability with tight, balanced error bounds.

Abstract: In this paper, we investigate the feedback stabilization of a class of semilinear non-homogeneous systems with distributed delay by means of a parametrized inequality. Based on the choice of this parameter, the proposed feedback law ensures either exponential or polynomial stabilization, providing a unified stabilization framework. For the bilinear case, we study the feedback stabilization problem by employing a suitable decomposition of the state space and reducing the stabilization problem to its finite-dimensional unstable component. Finally, illustrative examples are provided to support the theoretical findings.

Abstract: Building on the feedback stabilization framework that we developed for McKean-Vlasov PDEs, we treat a class of entropy-regularized free energies $\mathrm{ℱ}=\mathrm{ℰ}+\mathrm{\sigma }\mathrm{Ent}$ on the flat torus, with $\mathrm{ℰ}$ possibly nonconvex and nonlocal, and show that the resulting controlled system admits a gradient flow formulation in Wasserstein space. Specifically, we realize the Wasserstein Hessian at a target stationary measure $\overline{\mathrm{\mu }}$ as a self-adjoint operator with compact resolvent and show that its negative is unitarily equivalent to the generator of the linearized dynamics. The resulting feedback stabilization can then be interpreted as a local convexification of the free energy landscape: for any prescribed threshold $\mathrm{\delta }>0,$ the control, obtained from an algebraic Riccati equation for the linearized problem, induces a finite-rank perturbation of the Wasserstein Hessian that lifts its spectrum above $\mathrm{\delta },$ yielding local displacement convexity on a Hölder neighborhood of $\overline{\mathrm{\mu }}$ in regimes where the uncontrolled energy is nonconvex or slowly contracting. Under a local well-posedness assumption, $\overline{\mathrm{\mu }}$ is moreover a locally exponentially stable equilibrium of the closed-loop dynamics, with rate at least $\mathrm{\delta }.$ The framework covers McKean-Vlasov models and extends to moment-constrained Fokker-Planck dynamics and Fokker-Planck equations on closed Riemannian manifolds.

Joint work with Dante Kalise and Greg Pavliotis.

Abstract: Positive linear systems constitute a significant class of dynamical systems due to their occurrence in a wide range of applications. In many control design problems, the preservation of positivity is a fundamental requirement. This poster presents an analytical study of the observer-based positive stabilization problem for a generic 1D diffusion system with boundary control and observation. It is shown that the considered system cannot be positively stabilized using an observer-based state feedback of the considered form and related to a positive state observer. Motivated by this limitation, an alternative control design methodology is developed. Beyond ensuring closed-loop stability, the proposed framework focuses on guaranteeing the nonnegativity of the real state trajectory for all nonnegative initial conditions, rather than that of the estimated state. This objective is achieved by characterizing nontrivial and readily implementable admissible sets of initial estimation errors.

Abstract: The concepts of input-to-state stability (ISS) and integral input-to-state stability (iISS), introduced by Sontag in the 1980s and 1990s, are known to be equivalent for finite-dimensional linear systems in a straightforward manner. A more delicate situation emerges in the infinite-dimensional case: for bounded control operators, the finite-dimensional proof can be adapted; for unbounded control operators, arising, e.g., from boundary control systems, the question of equivalence of ISS and iISS has been open for the last decade.

In specific cases, under assumptions on the semigroup, e.g., for diagonal systems, the equivalence has previously been shown using a characterization of ISS by admissibility with respect to Orlicz spaces. We present a proof for general linear systems on large classes of Banach spaces, including reflexive spaces. The results are based on the Orlicz-admissibility criterion and a general weak compactness argument. The proof makes use of sun-duality theory and a classical result due to de la Vallee Poussin.

Moreover, we provide an overview of applications of the general proof technique in related areas. Among these are admissibility results, simplifications of well-known perturbation theorems and a new proof of Guerre-Delabriere’s theorem on $L^1$-maximal regularity.

This is joint work with Felix L. Schwenninger (University of Twente).

Abstract: We propose a control-by-interconnection scheme in which the controller is a continuous-time primal-dual gradient flow associated with a finite-horizon, input-constrained optimal control problem. Exploiting the port-Hamiltonian structure of this optimization flow allows a power-preserving interconnection with the nonlinear plant. A time-scale parameter renders the closed loop singularly perturbed: the reduced dynamics recover the optimal finite-horizon feedback, while the boundary layer describes convergence of the primal-dual variables to the optimality system. Via a composite Lyapunov function, we establish asymptotic stability of the coupled plant-controller dynamics – local for sufficiently fast optimization, and global under additional assumptions. A nonlinear port-Hamiltonian oscillator illustrates the approach.

Abstract: ISS-like properties of infinite-dimensional discrete-time systems are studied and selected properties are compared with their continuous-time counterparts. Characterizations of these properties in the form of superposition theorems are developed and relations between them are discussed, e.g. ISS is equivalent to UBIBS and UAG or iISS implies 0-UGAS. We also compare some results with their finite-dimensional counterparts. Furthermore, results regarding Lyapunov functions are presented. Illustrative examples are given to demonstrate selected relations.