An explicit mapping from linear first order hyperbolic PDEs to difference systems

Abstract

In this paper, we prove that the space generated by the solutions of a general class of first-order hyperbolic PDEs is isomorphic to the space generated by the solutions of a difference equation with distributed delays. This difference equation is obtained using a backstepping approach (combining a Volterra transformation of the second kind and an invertible Fredholm transformation) and the method of characteristics. Moreover, we prove that the stability properties are equivalent between the two systems. An important by-product is the design of a delay-robust stabilizing control law.