A Numerical Algorithm for the Coupled PDEs Control Problem

Abstract

For the coupled PDE control problem, at time \(t_i\) with the ith point, the standard algorithm will first obtain the two space variables \((z_i,v_i)\) and then obtain the control variables \((\varsigma _i^{opt},\mu _i^{opt})\) from the given initial points \((\varsigma _i^0,\mu _i^0)\). How many points i are determined by the facts of the case? We usually believe that the largest i defined by n is big because the small step size \(\tau =\frac{T-t_0}{n}\) will generate a good approximation, where T denotes the terminal time. Thus, the solution process is very tedious, and much CPU time is required. In this paper, we present a new method to overcome this drawback. This presented method, which fully utilizes the first-order conditions, simultaneously considers the two space variables \((z_i,v_i)\) and the control variables \((\varsigma _i^{opt},\mu _i^{opt})\) with \(t_i\) at i. The computational complexity of the new algorithm is \(O(N^3)\), whereas that of the normal algorithm is \(O(N^3+N^3K)\). The performance of the proposed algorithm is tested using an example.