On variational problems of optimization of control processes

Abstract

The problems of optimization of control processes in recent years attract considerable attention of researchers. The classical apparatus of the calculus of variations [ 1–5 ] as well as newer methods are employed for their solution. Certain results, important for the theory of optimum systems, have been obtained with the use of the maximum principle of Pontriagin [ 6–9 ], the methods of functional analysis [ 10 ], and the method of dynamic programming [ 11 ]. Numerous questions of optimization of control processes can be formulated in the form of the Lagrange problem [ 1, 12–15 ], the Mayer problem [ 6,8 ], and the Mayer-Bolza problem of the calculus of variation. Here, the most general of them, the Mayer-Bolza problem, is discussed, with the modification introduced by the questions of optimization [ 2–4, 13, 14 ] and with the limitations imposed on the controls being taken into account. For this case the necessary conditions of minimum are established. The author expresses his gratitude to A.I. Lur'e for his attention and help in carrying out this work.