ýspline interpolationý

ýHereý, ýwe give a two phases algorithm based on integrating differential evolution (DE) algorithm with modified hybrid genetic algorithm (MHGA) for solving the associated nonlinear programming problem of a nonlinear optimal control problemý. ýIn the first phaseý, ýDE starts with a completely random initial population where each individualý, ýor solutioný, ýis a random matrix of control input values in time nodesý. ýAfter phase 1ý, ýto achieve more accurate solutionsý, ýwe increase the number of time nodesý. ýThe values of the associated new control inputs are estimated by linear or spline interpolations using the curves computed in the phase 1ý. ýIn additioný, ýto maintain the diversity in the populationý, ýsome additional individuals are added randomlyý. ýNextý, ýin the second phaseý, ýMHGA starts by the new population constructed by the above procedure and tries to improve the obtained solutions at the end of phase 1ý. ýWe implement our proposed algorithm on some well-known nonlinear optimal control problemsý. ýThe numerical results show the proposed algorithm can find almost better solution than other proposed algorithmsý.