Numerical Methods for Finding Clustersolutions of Optimal Control Problems

Abstract

Obtaining the value function associated with an optimal control problem is an important part of solving the control problem. It is known that the value function may be not a classical solution but is always a viscosity solution of the Hamilton--Jacobi--Bellman (HJB) equation associated with the control problem. This means that the value function is not obtained from the HJB equation directly, but may be obtained by solving 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 with the super- or subdifferentials. This leads to high computational cost. In order to lower the computational cost, we develop ENO-upwind finite difference methods for finding a generalized solution called a clustersolution of the HJB equation. These numerical methods determine the upwind direction and the numerical viscosity more simply by checking the mean value of a generalized gradient called the clusterdifferential at a point. Clustersolutions and clusterdifferentials have been defined in earlier papers. Some numerical results are presented.