Well-Posedness of Time-Varying Linear Systems

Abstract

In this paper, we give easily verifiable sufficient conditions for two classes of perturbed linear, passive PDE systems to be well-posed, and we provide an energy inequality for the perturbed systems. Our conditions are in terms of smoothness of the operator functions that describe the multiplicative and additive perturbation, and here well-posedness essentially means that the time-varying systems have strongly continuous Lax-Phillips evolution families. A time-varying wave equation with a bounded multi-dimensional Lipschitz domain is used as illustration, and as a part of the example, we show that the timeinvariant wave equation is a "physically motivated" scatteringpassive system in the sense of Staffans and Weiss. The theory also applies to time-varying port-Hamiltonian systems.

Contraction Theory for Dynamical Systems on Hilbert Spaces

“…Theorem 5.1 (Partial and semi-contraction of reaction-diffusion systems). Consider the reactiondiffusion system (16) with the standard assumptions on f , Γ, and over a bounded and convex set domain Ω ⊂ R m . Suppose that there exists a positive definite matrix P ∈ R n×n such that µ 2 (P Γ) ≥ 0 and µ 2 (P (Df (x) − λ 2 Γ)) ≤ −c for all x ∈ Ω and some constant c > 0.…”

“…. , n}, Ω ∇ • ( n j=1 ũi (P Γ) ij ∇ũ j )dx = ∂Ω n j=1 ũi (P Γ) ij ∇ũ j • n dS = 0 where the last equality follows from the boundary condition in (16). Then, from the identity ũ⊤ P Γ∇ 2 ũ = n Since µ 2 (P Γ) ≥ 0, there exists a positive semi-definite matrix Q ∈ R n×n such that Q ⊤ Q = 1 2 (P Γ + Γ ⊤ P ⊤ ).…”

“…In the controls community, the recent works [31,16] have considered dynamical systems on Banach and Hilbert spaces and their applications to PDEs. Other recent interests in dynamical systems on these abstract spaces include controller design [28], event-triggered control [34], observability studies [12], optimal control [32], and stability characterizations [22].…”

Controller design for the closed loop system with non-interaction condition

“…Closed loop performance monitoring and diagnosis comprise a crucial step in the maintenance of model-based control system. In the event of performance degradation, diagnostic tools allow us to verify if the unsatisfactory closed loop operation results from the idea of plant-model mismatch (Kurula, 2020). In all these subjects, control system performance assessment gives the analysis of the existing controllers to tell when the control performance is inadequate, as controller performance assessment is important in assuring the process control effectiveness.…”

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

“…In the literature most attention has been devoted to autonomous control systems. However, in view of applications, the interest in non-autonomous systems has been rapidly growing in recent years, see e.g., [22,13,27,7,31,19,6,18,30] and the references therein. In particular, a class of scattering passive linear non-autonomous linear systems of the forṁ…”