THE BELLMAN EQUATION FOR TIME-OPTIMAL CONTROL OF NONCONTROLLABLE, NONLINEAR-SYSTEMS

For a general nonlinear system and closed target set T we study the value functions nu and nu of the control problems of reaching T and, respectively, its interior, in minimum time. Under no controllability assumptions on the system, we characterize them as, respectively, the minimal viscosity supersolution and the maximal viscosity subsolution of the Bellman equation with appropriate boundary conditions. Then we prove that nu is the unique upper semicontinuous 'complete solution' of such a boundary value problem, which means in particular that the (completed) graph of nu contains the graph of any solution, as well as all the limits of reasonable approximating sequences. We give some applications to verifications theorems and to the stability of the minimum time function with respect to general perturbations.