On the Rod Heating Problem by Sources on the Class of Zonal Controls Using the Current and Past State at Measurement Points

Abstract

We study an optimal feedback control problem for the rod heating process by means of lumped sources. The control actions are the powers of the sources, the values of which are defined on the class of zonal controls. The values of the parameters of zonal control actions are determined by subsets of the state space, to which belong the values of the process state at the measurement points at the current and past time moments. The posed problem is reduced to a parametric optimal control problem on determining a finite-dimensional vector of values of the parameters of zonal control actions. We have obtained optimality conditions for the values of the parameters of zonal control actions. These conditions contain formulas for the gradient of the objective functional with respect to the optimizable parameters. They make it possible to solve the reduced problem numerically with the application of efficient first-order optimization methods.