Hybrid control and observation systems in symmetric form

The paper considers several models of linear hybrid systems described by discrete-difference and difference-differential equations with control. Special attention is paid to the difference-differential hybrid systems in symmetric form. For solutions of such systems, a variation-of-constants formula is proposed and the relative controllability-observability principle is established. For stationary systems, we introduce the determining equations and present solutions in the form of series of their determining equation solutions. Then algebraic properties of solutions of determining equation are investigated, in particular, the well-known Hamilton-Cayley matrix theorem is extended to the solutions of the system of determining equations. As a result, parametric criteria for the relative controllability and relative observability are given.