Approximate solution to the resource consumption minimization problem. II. Estimates for the proximity of controls

Abstract

The proximity is estimated of a resource quasioptimal control to the optimal control. We give a method for subdividing the bounded region of initial conditions into subregions and bringing the quasioptimal control closer to the resource optimal control. This article continues [1]. We keep all notation and definitions of [1] and also continue the numbering of sections, propositions, formulas, and figures.In the first part of the article for the linear systeṁwith bounded control |u j | ≤ M j , j = 1, m, and fixed transition time we proposed some approximate solution method for the resource consumption minimization problem.Problem. Find an admissible control u 0 (t) taking the linear system from an initial state x(t 0 ) = x 0 to the origin x(t k ) = 0 in a fixed time T = t k − t 0 , T ≥ T 0 , and minimizing the functionalHere T 0 is the time of a reaction-speed-optimal control. For the maximal values |x i (t 0 )| max , i = 1, n, in a bounded region of initial conditions D, we find the resource optimal controls. We construct n "axial" quasioptimal controls: we take the switching moments equal to those of the optimal control for the boundary point on this axis, while the control amplitude is directly proportional to the initial condition x i (t 0 ) on the ith axis taken with some weight. We proved the independence of the switching moments of the initial conditions and their constancy for systems with constant parameters. The combined control with respect to all axes takes a linear system from an arbitrary initial point in the controllability region V of the phase space into the origin in a specified time T and approximately minimizes resource consumption. We found the region of the initial conditions for which the constraints on the control are satisfied. *