On the Maximum Principle for Impulsive Hybrid Systems

Hybrid systems consist of dynamical systems where both continuous and discrete event dynamics are interacting. These are widely accepted as realistic models of diverse technical systems and processes, for instance, industrial electronics, power systems, maneuvering aircrafts, automotive control systems, chemical processes and communication networks. Recently optimization problems and variants of the Maximum Principle (MP) for hybrid systems have attracted a great deal of attention. Both theoretical results and computational techniques were developed see e.g.,[1]-[14]. These results are extended here to a general class of hybrid systems with state jumps and the corresponding constrained optimal control problems. The family of hybrid optimization problems under consideration (called Impulsive Hybrid Optimal Control Problems IHOCP) include dynamical systems with internally forced switchings and continuous control signals. The discrete state transitions are triggered by the continuous state and are accompanied by a discontinuous change in the latter variable. This class captures phenomena arising e.g., in cyclically operated batch processes and certain epidemic propagation models. Using the mathematical techniques of distributional derivatives and impulsive differential equations, we extend the necessary optimality conditions to the above class of problems (IHOCPs). We obtain specific elements of the Impulsive Hybrid MP (IHMP), namely, the corresponding boundary-value problem and some additional relations. As in the classical case, the proposed IHMP provides a basis for diverse computational algorithms for the treatment of IHOCPs