Hybrid discrete-continuous control systems: I. Representation of solutions

We consider the Cauchy problem for linear control hybrid discrete-continuous systems in symmetric and normal forms. We obtain a solution of the problem in the form of definite integrals, where the solutions of special adjoint systems are used as the kernels of integral transform; this generalizes the representation of solutions by the Cauchy formula for ordinary linear dynamical systems to hybrid discrete-continuous systems. We discuss some other approaches to finding solutions of hybrid discrete-continuous systems. The results are used for deriving criteria for the relative controllability of considered systems and are refined in the time-independent case. We consider an example illustrating the obtained results.