Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions with a Quadratic Cost Function with Respect to the Input

An algorithm to approximate a solution to the Hamilton-Jacobi equation for a generating function for a nonlinear optimal control problem with a quadratic cost function with respect to the input is proposed in this paper. An approximate generating function based on Taylor series up to the order N is obtained by solving (N + 2)(N − 1)/2 linear first-order ordinary differential equations recursively. A single generating function is effective to generate optimal trajectories to the same nonlinear optimal control problem for any different boundary conditions. It is useful to online trajectory generation problems. Numerical examples illustrate the effectiveness of the proposed algorithm.