A direct probabilistic global search method for the solution of constrained optimal control problems

Abstract

This research focuses on the development of a new direct stochastic algorithm to address the global optimization of the constrained optimal control problem where the interaction between state and control variables is governed by a system of ordinary differential equations. The objective of this method is to localize a globally optimal control curve in the feasible control space of the problem in such a way that the performance index attains its minimum value. The stochastic methodology is used on the development of the method. Thus, the resulting method is still effective when the complexity of the arising problems prohibits applying gradient-based methods. In this approach, the aforementioned control problem has first to be transformed into a nonlinear programming problem via a suitable discretization technique. The resulting problem is then solved using a stochastic method called Probabilistic Global Search Johor (PGSJ). The idea underpinning the PGSJ is to intelligently sample among potential solutions while no recombination or mutation operator is used. The sampling procedure is performed in accordance with some probability density functions (pdf) which are first initialized uniformly and then iteratively biased towards a globally optimal solution using the information obtained by evaluating the sampling points. After the PGSJ has been successfully implemented, it is found that it is able to arrive at an acceptable solution of the applied optimal control problems. The algorithm is also furnished with some theoretical supports verifying its convergence in probabilistic sense. In addition, some existing global stochastic methods which are based on using pdf are also applied on the optimal control problems where simulations reveal that the PGSJ method is superior to its competitors in terms of computation time and solution quality. These investigations lead to the extension of PGSJ into PGSJ-LS where LS indicates a line search operator added to the original method. These are then assessed and compared by applying them to a practical problem of controlling avian influenza H5N1 where it is verified that the PGSJ-LS performs slightly better than PGSJ