NUMERICAL-METHODS FOR FINDING CLUSTERSOLUTIONS OF OPTIMAL-CONTROL PROBLEMS

Obtaining the value function associated with an optimal control proble m is an important part of solving the control problem. It is known tha t the value function may be not a classical solution but is always a v iscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation associ ated with the control problem. This means that the value function is n ot obtained from the HJB equation directly, but may be obtained by sol ving two viscosity HJB inequalities. Essentially nonoscillatory (ENO) upwind finite difference methods for solving the HJB inequalities have been developed. They determine the upwind direction and the numerical viscosity by checking all possible convecting velocities associated w ith the super- or subdifferentials. This leads to high computational c ost. In order to lower the computational cost, we develop ENO-upwind f inite difference methods for finding a generalized solution called a c lustersolution of the HJB equation. These numerical methods determine the upwind direction and the numerical viscosity more simply by checki ng the mean value of a generalized gradient called the clusterdifferen tial at a point. Clustersolutions and clusterdifferentials have been d efined in earlier papers. Some numerical results are presented.