This paper develops the generating function method for the discrete-time nonlinear optimal control problem. This method can analytically give the optimal input as state feedforward control in terms of the generating functions. Since the generating functions are nonlinear, we also develop numerical implementations to find their Taylor series expressions. This finally gives optimal solutions expressed only in terms of the pre-computed generating function coefficients and state boundary conditions, such that it is useful for the on-demand optimal solution generation for different boundary conditions. Examples demonstrate the effectiveness of the developed method.

This paper develops the double generating function method for the discrete-time linear quadratic optimal control problem. This method can give generators for optimal solutions only in terms of pre-computed coefficients and boundary conditions, which is useful for the on-line repetitive computation for different boundary conditions. Moreover, since each generator contains inverse terms, the invertibility analysis is also performed to conclude that the terms in the generators constructed by double generating functions with opposite time directions are invertible under some mild conditions, while the terms with the same time directions will become singular when the time goes infinity which may cause instability in numerical computations. Examples demonstrate the effectiveness of the developed method.