Müntz–Legendre wavelet collocation method for loaded optimal control problem

In this paper, a Müntz–Legendre wavelet collocation method is proposed for solving optimal control problems with constraints as the loaded differential equations, called loaded optimal control problems. Necessary optimality conditions as the two-point boundary value problem are derived for the loaded optimal control problem by using the method of calculus of variations and integration by parts formula. The left and right operational matrices of integration have been utilised to transform the necessary optimality conditions into a system of algebraic equations using the collocation method. Convergence of the approximated cost functional to the exact optimal cost has been proved, and the stability of the proposed algorithm has been investigated. Finally, four test problems have been taken to show the convergence of the proposed method to the true solution via the $ L_2 $ L2-errors in approximation, and the results have been compared with methods derived in previous literature.