Algebraic approach to nonlinear finite-horizon optimal control problems of discrete-time systems with terminal constraints

This paper proposes a method to solve nonlinear finite-horizon optimal control problems of discrete-time polynomial systems with polynomial terminal constraints. Algebraic equations with all variables at each time step, which are independent of variables at other time steps, are derived from the necessary conditions for optimality by eliminating variables recursively. The candidates of the optimal solution are obtained by solving these equations, and algorithms to find all of these candidates are also proposed. Because of its structure, the proposed method is suitable for nonlinear model predictive control that needs only the initial optimal control law. A simple example to illustrate the methodology and a practical example with the nonlinear model predictive control framework are provided.