Perturbations of superstable linear hyperbolic systems

Abstract

The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L 2 as well as in C 1 under small bounded perturbations. To show this for C 1 , we prove a general smoothing result implying that the solutions to the perturbed problems become eventually C 1 -smooth for any L 2 -initial data.

Classical bounded and almost periodic solutions to quasilinear first-order hyperbolic systems in a strip

“…The proof of this theorem repeats the proof of [20, Theorem 2.3]. As it follows from the results of [17,18,20], the problems (…”

“…As usual, by L(X, Y ) we denote the space of linear bounded operators from X into Y , and write L(X) for L(X, X). Note that the assumption (H1) (especially, ( (1.6) Theorem 1.2 [20] Suppose that the coefficients a and b of the system (1.4) have bounded and continuous partial derivatives up to the first order in (x, t) ∈ Π. If the inequalities (1.6) are fulfilled, then, given s ∈ R and ϕ ∈ L 2 ((0, 1); R n ), there exists a unique L 2 -generalized solution u : R 2 → R n to the problem (1.4), (1.2), (1.5).…”

“…We will consider boundary conditions ensuring that the regularity of solutions to the initial boundary value problem for the linearized system increases in a finite time. In other words, we assume that the system (1.4), (1.2), (1.5) has a smoothing property of the following kind, see [17,18,20]. Definition 1.3 Let Y ֒→ Z be continuously embedded Banach spaces and, for each s and t ≥ s, V (t, s) ∈ L(Z).…”

“…Problems (1.1), (1.2) satisfying Assumption (H3) occur in chemical kinetics [34,35], population dynamics [28,29], boundary feedback control theory [1,6,13,29], and inverse problems [33]. A collection of examples from these areas can be found in [20].…”

“…We will use the following variant of the existence-uniqueness result stated in [20,Theorem 2.3 ], for the case of the non-homogeneous system (3.1). Theorem 3.3 Suppose that the coefficients a, b, and f of the system (3.1) have bounded and continuous partial derivatives up to the first order in (x, t) ∈ Π.…”