Nested Sparse Successive Galerkin Approximation for Nonlinear Optimal Control Problems

Solving the Hamilton-Jacobi-Bellman (HJB) equation for nonlinear optimal control problems usually suffers from the so-called curse of dimensionality. In this letter, a nested sparse successive Galerkin method is presented for HJB equations, and the computational cost only grows polynomially with the dimension. Based on successive approximation techniques, the nonlinear HJB partial differential equation (PDE) is transformed into a sequence of linear PDEs. Then the nested sparse grid methods are employed to solve the resultant linear PDEs. The designed method is sparse in two aspects. Firstly, the solution of the linear PDE is constructed based on sparse combinations of nested basis functions. Secondly, the multi-dimensional integrals in the Galerkin method are efficiently calculated using nested sparse grid quadrature rules. Once the successive approximation process is finished, the optimal controller can be analytically given based on the sparse basis functions and coefficients. Numerical results also demonstrate the accuracy and efficiency of the designed nested sparse successive Galerkin method.