| ¡@¡@It is known that the value function associated with a nonlinear optimal control problem is often undifferentiable. At each singular point, both the classic gradient and the classic optimal control are undefined. However, the undifferentiable value function is a viscosity solution of the Hamilton-Jacobi-Bellman equation associated with the problem; we may resolve theis problem by adopting a kind of generalization of gradient which we call the clusterdifferential of the value function, and compute a numerical approximation of the value function by using an upwind Essentially NonOscillatory (ENO) finite difference method. This numerical method determines the upwind direction and the numerical viscosity by checking all possible covecting velocities associated with the super- or the sub-differentials, and gives an optimal controls. The advantage of this alternative is that when the value function is locally Lipschitz, the clusterdifferential is never empty, and every element of the clusterdifferential defines an optimal control. Some numerical results will be presented. |