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## 7.4.1 Hybrid optimal control problem

We begin by defining the class of hybrid control systems and associated optimal control problems that we want to study. The first ingredient of our hybrid control system is a collection of (time-invariant) control systems (7.32)

where is a finite index set; for simplicity, we assume that all the systems in the collection (7.32) share the same state space and control set . The second ingredient is a collection of switching surfaces (or guards) , one for each pair . A function is an admissible trajectory of our hybrid system corresponding to a control if there exist time instants and indices such that satisfies (7.33)

and Here and are the values of right before and right after , respectively, and the value is taken to be equal to one of these one-sided limits, depending on the desired convention. At each possible discontinuity , a discrete transition (or switching event) is said to occur. The function defined by for describes the evolution of along the trajectory; is often called the discrete state of the hybrid system. We can use it to rewrite (7.33) more concisely as for , .

We consider cost functionals of the form (7.34)

where is the usual running cost and is the switching cost, for . For simplicity, we do not include a terminal cost (this is no loss of generality, as terminal cost can be easily incorporated into the above running-plus-switching cost along the lines of Section 3.3.2). We also introduce an endpoint constraint, characterized by a set for each pair , according to which the trajectory must satisfy (7.35)

(Here plays the same role as at the end of Section 4.3.1.) Then, the hybrid optimal control problem consists in finding a control that minimizes the cost (7.34) subject to the endpoint constraint (7.35). We emphasize that the choice of a control is accompanied by the choice of two finite sequences and , to which we henceforth refer as the time sequence and switching sequence, respectively.     Next: 7.4.2 Hybrid maximum principle Up: 7.4 Maximum principle for Previous: 7.4 Maximum principle for   Contents   Index
Daniel 2010-12-20