Controllability of a Singular Hybrid System

Abstract

We consider the linear hybrid system with constant coefficients that is not resolved with respect to the derivative of the continuous component of the unknown function. In Russian literature such systems are also called discrete-continuous. Hybrid systems usually appear as mathematical models of a various technical processes. For example, they describe digital control and switching systems, heating and cooling systems, the functioning of a automobile transmissions, dynamical systems with collisions or Coulomb friction, and many others. There are many papers devoted to the qualitative theory of such systems, but most of them deal with nonsingular cases in various directions. The analysis of the note is essentially based on the methodology for studying singular systems of ordinary differential equations and is carried out under the assumptions of the existence of an equivalent structural form. This structural form is equivalent to the nominal system in the sense of solutions, and the operator which transformes the investigated system into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. Necessary and sufficient conditions for R–controllability (controllability in the reachable set) of the hybrid systems are obtained.