Optimal control for quasi-linear systems with small parameters

We consider the controlled systems where the non-linear term is multiplied by a small scalar parameter epsilon. In the class of these quasi-linear systems, we shall determine the control and optimal trajectory which minimizes the index of performance represented by quadratics functionals. The initial and final conditions are specified and the final time is free. The presence of the small parameter leads to an approximate solution of the formulated problem of optimum. Thus, the zeroth-order solution is obtained for epsilon = 0. The first order solution results by using the sweep method which determines the perturbation of the control and of the state variable on the optimal neighboring trajectory.