On the Maximum Principle for Impulsive Hybrid Systems

Abstract

Abstract. In this contribution, we consider a class of hybrid systems with continuous dynamics and jumps in the continuous state (impulsive hybrid systems). By using a newly elaborated version of the Pontryagintype Maximum Principle (MP) for optimal control processes governed by hybrid dynamics with autonomous location transitions, we extend the necessary optimality conditions to a class of Impulsive Hybrid Optimal Control Problems (IHOCPs). For these problems, we obtain a concise characterization 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.

On Optimization Techniques for a Class of Hybrid Mechanical Systems

“…Using the Hybrid Maximum Principle (see [4]), we conclude that the admissible control u opt (t) ≡ 0.5 is an optimal solution of the given OCP. Note that this result is also consistent with the Bauer Maximum Principle (see e.g., [2] ).…”

“…We refer to [3,4,7,8,13,17,23,26,27] for some concrete examples of hybrid systems with the above dynamic structure. Consider a class of hybrid mechanical systems that can be represented by n generalized configuration coordinates q 1 , ..., q n .…”

“…Several classes of general hybrid and switched dynamic systems have been extensively studied, both in theory and practice [3,4,7,11,14,17,19,26,27,30]. In particular, driven by engineering requirements, there has been increasing interest in optimal design for hybrid control systems [3,4,7,8,13,17,23,26,27].…”

“…In particular, driven by engineering requirements, there has been increasing interest in optimal design for hybrid control systems [3,4,7,8,13,17,23,26,27]. In this paper, we investigate some specific types of hybrid systems, namely hybrid systems of mechanical nature, and study the corresponding hybrid OCPs.…”

“…There is a number of results scattered in the literature on numerical methods for optimal control problems. One can find a fairly complete review in [3,4,8,24,25,29]. The main idea of our investigations is to use the variational structure of the solution to the specific two point boundary-value problem for the controllable hybrid-type mechanical systems in the form of Euler-Lagrange or Hamilton equation and to propose a new computational algorithm for the associated OCP.…”