Exponential stabilization of a class of 1-D hyperbolic PDEs

The problem of exponential stabilization of 1-D hyperbolic system with spatially varying coefficients is investigated. The main strategy reposes on mapping the original system into a target one by an invertible Volterra transformation with a kernel satisfying an appropriate PDE. This enables to convert a multiplicative perturbation exerted from the whole domain to a boundary perturbation in the target system. The problem is reformulated in the context of semigroups theory and solved via a quadratic Lyapunov functional. The stabilizer is explicitly constructed by means of a collocated-type controller of the auxiliary system combined with a term containing the solution of the kernel PDE. The technics of the feedback law construction also offer information about the stabilization mechanism which makes the proposed controller realizable in concrete situations.