Mathematical Approach of Cost Function Constructed by Hermite Interpolation to Obtain Optimal Signal by Using DP Algorithm

ABSTRACT In this article, we investigate an optimal control problem characterized by fixed terminal time and a partially constrained terminal state, for which we construct the cost function by using Hermite interpolation. The formulation of this cost function constitutes the principal methodological contribution of the work which is a novel technique in our article. To demonstrate its effectiveness, we present an illustrative dynamic model in which the proposed cost function is employed to determine admissible ranges of control inputs between prescribed fixed points. Leveraging the proposed framework, we obtain the optimal control signal using the Dynamic Programming (DP) algorithm and satisfying the reachable set between the range of admissible controls corresponding to the original range of state variables with respect to time step for the defined discrete‐time model. Then we formulate a discrete‐time optimal control problem to minimize the cost‐to‐go function with respect to control inputs across time intervals related to each time step between the final time range , while the system evolves between specified minimum initial and terminal sizes within the original state range . Finally, that example on the dynamic time model demonstrates the validity and applicability of our proposed technique.