On local controllability conditions of discrete systems (Q1390491)

Consider the discrete inclusion \(x_{t+1}\in F_t (x_t)\), \(t=0,1,2,\dots, N-1\), where \(F_t\) is a closed multifunction on \({\mathbb R}^n\) with \(0\in F_t (0)\). The authors prove the following Theorem: The discrete inclusion is locally \(N\)-controllable if the adjoint discrete inclusion \(p_t \in D^* F_t (0)p_{t+1}\), \(t=N-1,\dots, 1,0\) has no non-zero solution. Here \(D^* F_t (0)p\) is the coderivative of the multifunction \(F_t\) at the point \((0,0)\) in the direction \(p\). No Lipschitzian or convex conditions are needed. An example is given as an application. The approach is based on results of nonsmooth analysis.