THE MATHEMATICAL PROBLEM OF OPTIMAL CONTROL OF THE STRING

Abstract

It is generally accepted that optimal control problems or system design problems determine for a given object or system of control objects a law or a certain control sequence of actions that provide a maximum or minimum of a given set of system quality criteria. In this case, the speed problem can be considered, i.e. the problem of bringing the system to a given state in the shortest time. We also study the problems of minimizing a given functional for a fixed time of system management. Optimal control is closely related to the choice of the most rational modes for managing complex objects. A lot of works has been devoted to the problem of control, in addition, wellknown mathematical schools are currently engaged in such research. In problems with concentrated parameters, the systems under study are described by ordinary differential equations or their systems. In this case, the Pontryagin maximum principle plays an important role in this study. For partial differential equations, we talk about systems with distributed parameters. In thispaper, we investigate the possibility of synthesizing optimal control of a single system with distributed parameters. A model of string oscillation under the influence of control functions under boundary conditions is considered. The role of the choice of the functional to be minimized in creating opportunities for the synthesis of optimal control. In this case, the control action is searched for at each point of the time interval, which leads to the possibility of constructing it explicitly. The conditions for the existence of optimal control everywhere in the corresponding functional spaces are formulated. In a specific statement of the problem, everywhere optimal control is explicitly constructed.