Optimality conditions in control problems within the class of generalized solutions

The problem of the optimality conditions derivation is the basic one in optimal control. Well-known necessary optimality conditions in the form of the maximum principle have been obtained at the beginning of 60-ths, and hereafter are widely used in the practice of the optimal control as a powerful tool for the solution of applied problems and development of the optimization algorithms and software. In its typical form the maximum principle reduces the infinitely dimensional optimization problem to some boundary-value problem for the system of differential equations. However, in view of the specific of systems with impulse control, the problem of optimality conditions did not have an adequate solution especially for nonlinear systems. As follows from the results of the above chapters the optimal solutions in systems with impulse control require the special class of equations, namely, the differential equations with measures. Meanwhile, the general methods of the necessary optimality condition derivation, based on the classical Dubovitskii-Milytin scheme [47], [63], can not be directly applied to this class of equations, particularly in the case when the measure itself serves as an additional control component. Indeed, as was shown in the Introduction the “small” variations of measure (or impulse control) might generate the “strong” variations of the paths, thereby the application of the general scheme, based on the linear correspondence between control-paths variation would be inapplicable.