On Stability of Switched Linear Hyperbolic Conservation Laws with Reflecting Boundaries

We consider stability of an infinite dimensional switching system, posed as a system of linear hyperbolic partial differential equations (PDEs) with reflecting boundaries, where the system parameters and the boundary conditions switch in time. Asymptotic stability of the solution for arbitrary switching is proved under commutativity of the advective velocity matrices and a joint spectral radius condition involving the boundary data. Problem Formulation. Motivated by applications [2], we consider hybrid dynamics governed by linear hyperbolic PDE systems and a discrete set of modes: ∂tu(t, s) +A ∂su(t, s) = 0 C j Lu(t, a) = 0, C j Ru(t, b) = 0 , j ∈ Q ≃ {1, . . . , N}, (1) where the matrices A ∈ R specify the advective velocities and the matrices C L ∈ R (n−mj)×n and C R ∈ R mj×n specify the boundary data for the unknown vector function u(t, s) = (u(t, s), . . . , u(t, s)) on the space-time strip Ω([t1, t2]) := {(t, s) | t ∈ [t1, t2], s ∈ [a, b]}. We assume that (H)1 the subsystems for fixed j are strictly hyperbolic, i. e. A j has mj negative and (n−mj) positive eigenvalues λ j i with n corresponding linearly independent left (right) eigenvectors l i (r j i ); (H)2 the switching signals in time T = {t ≥ 0} are piecewise constant functions σ(·): T → Q with switching times τk (k ∈ N) such that there are only finitely many switches j y j in each finite time interval of T . We consider the switched system in the space of piecewise continuously differentiable functions, denoted as PC = PC([a, b],R), setting u(t) := u(t, ·), and say that for an initial condition ū(·) ∈ PC, the function u(·): T → PC is a solution of the switched system (1) if u|t=0 = ū ∧ { u|τk+ := u|τk− for all switching times τk of σ(·), u|(τk+,τk+1−) solves (1) with j = σ(t) = const. (2) Under the above assumptions, it is easy to see that the system is well-posed, if and only if it is well-posed in each mode, i. e., following [1]: rank [ (C L) ⊤ ∣