Application of dynamic programming to high-dimensional non-linear optimal control problems

In using dynamic programming, by taking only accessible states for the x-grid and using an iterative procedure employing region contraction, only a small number of grid points are required at each iteration to yield very good accuracy even if the dimension of the system is high. The effect of the number of grid points and the choice of the contraction factor are analysed by considering a non-linear system consisting of eight ordinary differential equations and four control variables. No difficulties were encountered in convergence to the optimal solution in no more than 20 iterations. The proposed procedure overcomes the curse of dimensionality that has discouraged the use of dynamic programming in the past to solve high-dimensional non-linear optimal control problems, and provides an attractive means of solving optimal control problems in general