The Bellman equation for time-optimal control of noncontrollable,nonlinear systems

For a general nonlinear system and closed target set we study the value functions and of the control problems of reaching 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 is the unique upper semicontinuous complete solution of such a boundary value problem, which means in particular that the (completed) graph of 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.The authors are partially supported by the Italian National Projects Equazioni di evoluzione e applicazioni fisico-matematiche and Equazioni differenziali e calcolo delle variazioni, respectively.