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

Abstract

We study a class of non-autonomous linear boundary control and observation systems that are governed by non-autonomous multiplicative perturbations. This class is motivated by fundamental partial differential equations, such as controlled wave equations and Timoshenko beams. Our main results give sufficient condition for well-posedness, existence and uniqueness of classical and mild solutions.

Well-Posedness of Time-Varying Linear Systems

“…The theory in the present paper applies to timevarying port-Hamiltonian systems, in the same way as it applies to the wave equation in §V; see in particular [6, §11.3] and [7, (3.1) and Thm 4.6]. In fact, in closely related indpendent work [8], Jacob and Laasri consider well-posedness (defined sligthly differently) of time-varying boundary control systems, using port-Hamiltonian systems as motivating example. There is a considerable methodical overlap between the present paper and [8], and the theory in [8] can likely also cover the example in §V.…”

“…In fact, in closely related indpendent work [8], Jacob and Laasri consider well-posedness (defined sligthly differently) of time-varying boundary control systems, using port-Hamiltonian systems as motivating example. There is a considerable methodical overlap between the present paper and [8], and the theory in [8] can likely also cover the example in §V. Finally, we mention that Paunonen [9] and Pohjolainen [10] have studied robust output regulation of periodically time-varying distributed parameter systems.…”

Well-posedness of non-autonomous semilinear systems

“…A simple sufficient condition for the above assumption to hold is provided by the following lemma. See Example 2.6 of [9]. Lemma 2.2.…”

“…As our open-loop system, we consider a non-autonomous linear port-Hamiltonian system of order N ∈ N on a bounded interval (a, b) with control and observation at the boundary [9]. Such a system evolves according to the differential equation…”

“…Since t → x(t) ∈ X and t → A(t)x(t) = ẋ(t) ∈ X are continuous and t → H(t) is strongly continuous, it follows from (5.28) that the classical outputt → y(t, s, x s , u) = C(t)x(t) = W C (H(t)x(t))| ∂is continuous and hence measurable, as desired.Example 5.3. Consider a vibrating string with possibly time-dependent material coefficients[9], that is, the transverse displacement w(t, ζ) of the string at position ζ ∈ (a, b) evolves according to the partial differential equation∂ t ρ(t, ζ)∂ t w(t, ζ) = ∂ ζ T (t, ζ)∂ ζ w(t, ζ) (t ∈ [0, ∞), ζ ∈ (a, b)) (5.29)…”