Zurab Tsintsadze

The paper elaborates a general method for studying smooth-convex conditional minimization problems that allows one to obtain necessary conditions for solutions of these problems in the case where the image of the mapping corresponding to the constraints of the problem considered can be of infinite codimension.

On the basis of the elaborated method, the author proves necessary optimality conditions in the form of an analog of the Pontryagin maximum principle in various classes of quasilinear optimal control problems with mixed constraints; moreover, the author succeeds in preserving a unified approach to obtaining necessary optimality conditions for control systems without delays, as well as for systems with incommensurable delays in state coordinates and control parameters. The obtained necessary optimality conditions are of a constructive character, which allows one to construct optimal processes in practical problems (from biology, economics, social sciences, electric technology, metallurgy, etc.), in which it is necessary to take into account an interrelation between the control parameters and the state coordinates of the control object considered. The result referring to systems with aftereffect allows one to successfully study many-branch product processes, in particular, processes with constraints of the “bottle-neck” type, which were considered by R. Bellman, and also those modern problems of flight dynamics, space navigation, building, etc. in which, along with mixed constraints, it is necessary to take into account the delay effect.

The author suggests a general scheme for studying optimal process with free right endpoint based on the application of the obtained necessary optimality conditions, which allows one to find optimal processes in those control systems in which no singular cases arise.

The author gives an effective procedure for studying the singular case (the procedure for calculating a singular control in quasilinear systems with mixed constraints.

Using the obtained necessary optimality conditions, the author constructs optimal processes in concrete control systems.

Using the Lagrange Principle of taking restrictions off the necessary conditions of optimality for quasilinear systems with mixed restrictions and delays are given. As against before executed works necessary conditions of optimality in case of continuous initial function, that is most natural for processes with delays, are proved.