A Numerical Algorithm for the Coupled PDEs Control Problem

For the coupled PDE control problem, at time $$t_i$$ t i with the ith point, the standard algorithm will first obtain the two space variables $$(z_i,v_i)$$ ( z i , v i ) and then obtain the control variables $$(\varsigma _i^{opt},\mu _i^{opt})$$ ( ς i o p t , μ i o p t ) from the given initial points $$(\varsigma _i^0,\mu _i^0)$$ ( ς i 0 , μ 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 $$\tau =\frac{T-t_0}{n}$$ τ = 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 $$(z_i,v_i)$$ ( z i , v i ) and the control variables $$(\varsigma _i^{opt},\mu _i^{opt})$$ ( ς i o p t , μ i o p t ) with $$t_i$$ t i at i. The computational complexity of the new algorithm is $$O(N^3)$$ O ( N 3 ) , whereas that of the normal algorithm is $$O(N^3+N^3K)$$ O ( N 3 + N 3 K ) . The performance of the proposed algorithm is tested using an example.