Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems

Abstract

We study the non-autonomous version of an infinite-dimensional linear port-Hamiltonian system on an interval [a, b]. Employing abstract results on evolution families, we show C 1 -well-posedness of the corresponding Cauchy problem, and thereby existence and uniqueness of classical solutions for sufficiently regular initial data. Further, we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non-autonomous setting.

Well-posedness of a class of hyperbolic partial differential equations on the semi-axis

“…In this section we first introduce weighted L 2 -spaces of scalar valued functions in one variable. Notice, that in contrast to previous results on port-Hamiltonian partial differential equations (1) in this article the weighted spaces will not necessarily be isomorphic to the unweighted L 2 -space. Secondly we will repeat well-known facts about the relation of absolutely continuous functions and functions with an integrable weak derivative.…”

“…Also under the standard assumption of, e.g., [1,7,10,11], namely ∃ m, M > 0 ∀ ξ ∈ [0, ∞), ζ ∈ C n : m|ζ| 2 ζ * H(ξ)ζ M |ζ| 2 (24) one can see that Theorem 4.10 is applicable, if we additionally assume that ρ ′ and T ′ are bounded. Notice that in [1,7,10,11] the estimate in (24) comes "for free" if H is continuous, since in these papers ξ ∈ [a, b] is considered. In our setting of a non-compact spacial domain this is not the case.…”

“…Once the question of characterizing the generator property is settled, we add to the partial differential equation (1) an input u = u(t) and an output y = y(t), i.e., we consider…”

“…where we assume W C ∈ C q×n and W B = [W B,1 W B,2 ] T with W B,1 ∈ C p×n to allow that not all boundary conditions are subject to a control but some have just zero input. In Section 5 we show that (1) and (6) give rise to a boundary control system which is well-posed. That is, for every τ > 0 there exists m τ > 0 such that for every x 0 ∈ L 2 H (0, ∞) with (Hx) ′ ∈ L 2 H (0, ∞) and every u ∈ C 2 ([0, τ ], C p ) with u(0) = W B,1 H(0)x 0 (0) there is a unique classical solution x = x(ξ, t) such that…”

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

“…Different variations around these first results can be found in (Villegas, 2007) and in (Jacob & Zwart, 2012). Well-posedness and stability have been investigated in open-loop and for static boundary feedback control in (Zwart et al, 2010) and (Villegas et al, 2005;Villegas et al, 2009) respectively, and linear dynamic boundary control has been studied in (Ramirez et al, 2014;Augner & Jacob, 2014;Villegas, 2007).…”