L. S. Pontryagin maximum principle for some optimal control problems by trajectories pencils

In this paper we consider some optimal problems for pencils of trajectories of nonlinear control systems, when integral functional of general type is minimized. For these problems an initial state of control system belongs to some compact set with positive Lebesgue measure. Such control systems are connected, for example, with study of control pencils of charged particles in physics (D. A. Ovsyannikov and other) and in problems of control when initial state of control system is known with error. An importance problem in this field is proof of Pontryagin's maximum principle. In the paper we continue research of Ovsyannikov on this problem. We have proved the Pontryagin's maximum principle for the case of integral functional and instantaneous geometric restrictions on control for Lebesgue measured control functions (previously piecewise continuous control functions were considered in literature). We used classical techniques of variations for measured optimal control function with some modifications. We note that our form of Pontryagin's maximum principle is distinguished from some another forms. In the end of our paper there is a remark of D. A. Ovsyannikov about the connections of different forms of Pontryagn's principle maximum. As some illustration, we consider a control object with linear dynamics. For this case our maximum principle can be written in more simple form than in general nonlinear case. Refs 8.