Exponential Stability of Switched Linear Hyperbolic Initial-Boundary Value Problems

We consider the initial-boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes. The switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time. We show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching. We also provide an explicit dwell-time bound for guaranteeing exponential stability of the switching system when, for each mode, the system is exponentially stable. Our stability conditions only depend on the system parameters and boundary data. These conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption. We present tutorial examples to illustrate the instabilities that can result from switching.