Optimization of discrete systems (Q1062007)

Let g be a real-valued function and let \(F_ k\) \((k=0,1,...,N-1)\) be N set-valued maps, all defined on the same space X. The problem we shall consider is that of minimizing \(g(x_ N)\) over the trajectories \((x_ 0,x_ 1,...,x_ N)\) of the discrete inclusion \(x_{k+1}\in F_ k(x_ k)\), \(k=0,1,...,N-1\). The paper is organized as follows. In Section 1 we recall the notion of derivative of set-valued maps and give an auxiliary result. In Section 2 we discuss the conditions guaranteeing the existence of a derivative of a set-valued map. Combining the results of Section 2 and Theorem 3.1 we are able to obtain necessary optimality conditions for the given problem under various assumptions on g and \(F_ k\). In particular, we can recover the support principle for the case where the maps \(F_ k\) have local sections as well as for the case where \(F_ k\) have smooth support functions. Section 4 is devoted to the proof of Theorem 3.1. In the last Section 5 we show that Theorem 3.1 remains valid for the infinite dimensional case provided that g and \(F_ k\) are locally Lipschitzian.