Two dual classes of time-varying well-posed linear systems constructed from passive systems

Abstract

We introduce two classes of well-posed linear time-varying (LTV) systems. Each of these can be constructed easily, starting from a time-invariant scattering-passive system, by introducing a time-dependent inner product on the state space and modifying some of the generating operators. These classes of LTV systems are motivated by physical examples, such as an electromagnetic field around a moving object. To prove the well-posedness of these LTV systems, we use the Lax-Phillips semigroup induced by a well-posed linear system, as in scattering theory. We modify this semigroup to obtain a Lax-Phillips type evolution family.