A Construction of a Sequence of Feedback Controls for Discrete Control Problems Using Asymptotics

This paper explores the possibility of obtaining a symbolic description of a parametric family of synthesizing controls for nonlinear systems without control constraints using the Padé approximation method and the SDRE-based approach for control synthesis. On an example of a discrete optimal control problem with a positive parameter in the system dynamic equations an algorithm for constructing the first terms of a minimizing sequence of controls is proposed, where each term of the obtained control sequence is represented in the state feedback form. This construction is based on a set of neurons by which we understand various asymptotic approximations, Padé approximations (PA), and functions derived from them. Depending on a particular dynamic system a certain number of neurons will be used to get good quality approximation for the control. Asymptotic approximations and the PAs constructed on their basis capture the qualitative structure of the exact solution. Moreover, PAs can be improved through additional optimization procedures and modifications. The process of neurons selection, or activation, can be interpreted as a reinforcement learning algorithm in a certain neural network.