Closed-form solution of discrete-time optimal control and its convergence

The convergence of a new closed-form solution for the discrete-time optimal control is presented. First, a new time optimal control law with simple structure is constructed in the form of the state feedback for a discrete-time double-integral system by using the state backstepping approach. The control signal sequence in this approach is determined by the linearised criterion according to the position of the initial state point on the phase plane. This closed-form non-linear state feedback control law clearly shows that time optimal control in discrete time is not necessarily the bang-bang control. Second, the convergence of the time optimal control law is proved by demonstrating the convergence path of the state point sequence driven by the corresponding control signal sequence. Finally, numerical simulation results demonstrate the effectiveness of this new discrete-time optimal control law.