Solving optimal control problems of the time-delayed systems by Haar wavelet

We consider an approximation scheme using Haar wavelets for solving time-delayed optimal control problems with terminal inequality constraints. The problem is first transformed, using a Pade approximation, to one without a time-delayed argument. Terminal inequality constraints, if they exist, are converted to equality constraints via Valentine-type unknown parameters. A computational method based on Haar wavelets in the time domain is then proposed for solving the obtained nondelay optimal control problem. The Haar wavelets integral operational matrix and direct collocation method are utilized to find the approximated optimal trajectory and the optimal control law of the original problem. Numerical results are also given for several test examples to demonstrate the applicability and the efficiency of the method.