Stability Analysis for a Class of Linear $2\times 2$ Hyperbolic PDEs Using a Backstepping Transform

Abstract

In this paper, we develop a sufficient stability condition for a class of coupled first-order linear hyperbolic partial differential equations (PDEs) with constant coefficients that appear when considering target systems for backstepping boundary control.Using a backstepping transform, the problem is reformulated as a stability problem for a difference equation with distributed delay. Finding the explicit solution to the backstepping kernels, we derive an explicit sufficient condition depending on the plant coefficients. This stability condition is compared to an existing stability result based on a Lyapunov analysis. Both the proposed and existing sufficient conditions are then contrasted in some examples to a (computationally expensive) numerical approximation of a necessary and sufficient condition for exponential stability to illustrate their conservatism.