Well-Posedness of Time-Varying Linear Systems

In this article, we give easily verifiable sufficient conditions for two classes of perturbed linear, passive partial differential equation (PDE) systems to be well-posed, and we provide an energy inequality for the perturbed systems. Our conditions are in terms of smoothness of the operator functions that describe the multiplicative and additive perturbations, and here, well-posedness essentially means that the time-varying systems have strongly continuous Lax–Phillips evolution families. A time-varying wave equation with a bounded multidimensional Lipschitz domain is used as illustration, and as a part of the example, we show that the time-invariant wave equation is a “physically motivated” scattering-passive system in the sense of Staffans and Weiss. The theory also applies to time-varying port-Hamiltonian systems.