On One Control Problem of a Linear Dynamical System Subject to Integral-Type Phase Constraints

Adequate mathematical modeling of various dynamical control processes leads to control problems with nonlocal conditions, which are described by both partial differential equations and ordinary differential equations. Control problems with nonlocal conditions arise, on the one hand, as mathematical models of real processes, and on the other hand, as a result of the inability of formulating problems correctly using local conditions. Nonlocal problems in the form of integral conditions, in particular, occur in the mathematical modeling of phenomena of various natures. Research into problems with nonlocal conditions, in various formulations, is important not only for theoretical purposes but also for practical necessity. This paper investigates the problem of controlling a linear dynamical system subject to a nonlocal integral-type condition. The control problem for a linear dynamical system is formulated with an integral condition imposed on the components of the phase vector over a certain portion of the time interval during the system’s operation. Conditions are obtained under which a solution to the system of ordinary linear differential equations with the given integral condition exists and this solution is constructed. Control functions are developed for the control problem, under the influence of which the phase trajectories are realized, satisfying the integral conditions of the problem. The continuity and nonuniqueness of the control functions are demonstrated. A criterion of complete controllability of a linear system with phase constraints of the integral type is obtained. An example is given as an illustration.