Optimal control for a class of affine nonlinear systems: an improved successive approximation approach

This paper considers an optimal control for a class of affine nonlinear systems with respect to linear quadratic functional. An improved successive approximation approach is proposed. By introducing the successive approximation approach of the nonlinear differential equation theory, a linear two-point boundary value problem sequence is constructed to approximate the nonlinear two-point boundary value problem that is the necessary condition of the nonlinear control problem. A technique is employed to make sure the constructed linear two-point boundary value problems of the approximating sequence homogeneous. Iteratively solving this approximating sequence, optimal control law of linear closed-loop feedback form is designed. An algorithm is presented to design certain approximate optimal control law by truncating the infinite iteration. An illustrative example shows the validity of the algorithm.