Abstracts of the Contributed Talks

Abstract: For linear periodic finite-dimensional systems, it is well-known that, first, exponential stability is equivalent to the existence of a unique periodic positive definite solution to the Lyapunov equation, and second, the Lyapunov equation admits a unique periodic solution, if and only if the monodromy matrix has no reciprocal eigenvalues. In this talk, we discuss generalization of these results to the case of periodic evolution families on Hilbert spaces, with application to the stability theory of linear periodic time delay systems with constant delays. An explicit connection between the infinite-dimensional periodic Lyapunov equation on a Hilbert space and the delay Lyapunov matrix framework which stems from the Lyapunov—Krasovskii stability conditions for linear periodic time delay systems is established. Based on this connection, we prove the uniqueness theorem of the delay Lyapunov matrix, where the uniqueness condition — that the monodromy operator has no reciprocal eigenvalues — naturally extends the corresponding condition for the ODE case. The delay Lyapunov matrix is defined in the literature as a solution of the delayed hyperbolic matrix PDE with a non-standard boundary condition along with the symmetry and periodicity properties, which makes its construction a nontrivial problem. In the second part of the talk, we address the construction issue for a particular case when the period of the system matrices is equal to the delay. In particular, the explicit construction for the case of a scalar equation is given.

This is a joint work with Prof. Dr. Juan J.L. Velázquez at the University of Bonn.

Abstract: In control theory, a central problem is the construction of a Lyapunov functional for a linear dynamical system. The associated Lyapunov operator plays a fundamental role in stability analysis: its positivity characterizes exponential stability and provides valuable information on the dissipative properties and dynamics of the system. While existence and uniqueness results for Lyapunov operators are well established, their explicit computation becomes challenging in infinite-dimensional spaces, where classical finite-dimensional techniques are no longer directly applicable. This motivates the development of computational approaches capable of approximating Lyapunov operators directly from the underlying operator equation. In this talk, we focus on a Physics-Informed Neural Network (PINN) approach for the approximation of Lyapunov operators. The Lyapunov equation is incorporated into the neural network loss function, allowing the operator to be approximated while enforcing the governing operator equation and relevant structural properties, such as symmetry and positivity. This provides a mesh-free framework that is particularly well suited to infinite-dimensional systems and partial differential equations. We present the formulation and numerical implementation of the PINN methodology and discuss its ability to approximate Lyapunov operators without relying on an explicit finite-dimensional representation. Finally, we apply the proposed approach to hyperbolic partial differential equations, illustrating how PINNs can be used to construct and approximate Lyapunov functionals for distributed-parameter systems and to assess their stability and dissipative properties.

Abstract: In this talk I will present recent work bringing a complete answer to Hautus type conditions for the exact controllability of linear controlled difference equations in $L^r$ spaces, $r\in \left(\mathrm{1,}\mathrm{\infty }\right)$, completing earlier work in the case $r=1$. As a corollary, we provide rapid stabilization results and show how to extend that work to classes of hyperbolic systems.

This is a joint work with S. Fueyo, P. Lissy and G. Mazanti.

Abstract: In this talk, the author will present recent results for consensus-based and flocking models, with non-universal interaction, time delays, and possible communication failures. In particular, extending previous results, they allow the amount of interaction between the agents over fixed-length time intervals to decay polynomially. For the first-order Hegselmann-Krause type model, they prove asymptotic convergence to consensus, and for the second-order Cucker-Smale model, the authors establish asymptotic flocking under an additional assumption on the decay of the influence function. These results show that collective behavior can still emerge under progressively weakening communication and without requiring strong connectivity of the interaction graph.

Abstract: We present the newly introduced notion of twofold control, in which the initial state and the control input are optimised simultaneously. The main idea is to distribute the available control effort between preparing a suitable initial state and applying the subsequent control input. This additional degree of freedom can substantially reduce the required control energy and enlarge the set of reachable states.

Special attention is devoted to the case of a cost functional penalising the deviation of the state from a prescribed constant-in-time trajectory, which is the classical setting in which the turnpike property arises. We analyse the asymptotic behaviour of optimal twofold controls as the time horizon tends to infinity and investigate whether, and under which conditions, the turnpike property persists in this new control framework.

Abstract: In this talk, I will develop a general framework for the flatness of parabolic linear systems with boundary controls. The approach considers an abstract boundary control system where the underlying operator generates an analytic semigroup and possesses a spectrum of isolated eigenvalues with finite multiplicities. By assuming the existence of an entire function of order lower than one that satisfies the eigenvalue equation for the system operator, we establish a general null-controllability result. For any time greater than a specific threshold – determined by the condensation of the spectrum and the growth of the eigenfunctions – there exists a control and a corresponding state with Gevrey regularity. Using this framework, we recover various results in the literature but also show some general flatness properties for cascade systems and the simultaneous controllability of some parabolic systems. Applications to the degenerate heat equation and coupled heat equations will be given.

This work has been done in collaboration with Takéo Takahashi.

Abstract: The complexity of differential equations with distributed time delays limits their use in modeling, estimation, control, and optimization of time delay systems. Therefore, we propose to approximate the memory kernel in the convolution in the distributed time delay by an exponentially weighted polynomial, referred to as an Erlang mixture approximation. The reason is that the approximate set of delay differential equations (DDEs) can be transformed analytically to ordinary differential equations (ODEs). We refer to this as the Erlang ODE approximation of the DDEs, and we show that under mild conditions on the state space and the memory kernel, the solution and steady state of the ODE approximation converge to those of the original DDEs. The utility and convergence rate of the approximation are illustrated with numerical examples.

Abstract: Bounded-input bounded-output stability (BIBO), that is, the property of an input-output system that inputs uniformly bounded in time give rise to outputs uniformly bounded in time, has applications, for instance, in the design of novel control techniques. While this notion is a classical and well-studied one for linear finite-dimensional systems, when turning to infinite-dimensional systems such as boundary-controlled PDEs, the situation becomes more complex.

In this talk, we want to give some examples where this difference becomes apparent, and answer questions on generalizing results from the finite to the infinite-dimensional case. This includes, for instance, the characterization of BIBO stable systems by convolution operators, and the relation to other stability concepts and the notions of observability and controllability.

This talk is based upon joint work with Sahiba Arora, Jochen Glück and Felix L. Schwenninger.