Iterative Solution of Mesh Constrained Optimal Control Problems with Two-Level Mesh Approximations of Parabolic State Equation

Abstract

We consider a linear-quadratical optimal control problem of a system governed by parabolic equation with distributed in right-hand side control and control and state constraints. We construct a mesh approximation of this problem using different two-level approximations of the state equation, ADI and fractional steps approximations in time among others. Iterative solution methods are investigated for all constructed approximations of the optimal control problem. Their implementation can be carried out in parallel manner.There are few results in the area of numerical solution methods for the constrained parabolic optimal control problems. One of the approaches to overcome the difficulty connected with low regularity of the solutions is using Lavrentiev regularization. This approach has been used in [6] [7]. In [8] [9] [10] [11] a priori error estimates for space-time discretizations of linear-quadratic parabolic optimal control problems have been obtained for problems.

The Problem of Boundary Control of the Thermal Process in a Rod

“…In the study of controllable thermal processes, there are problems of controlling thermal processes whose mathematical models are described by partial differential equations of the parabolic type [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These control problems find their application, for example, in various fields of science and technology.…”

“…In practice, there are often problems of thermal diffusion in a rod, the ends of which are exposed to variable controlled temperatures. These problems are reduced to the study of the thermal conduction equation, the boundary conditions in which are expressed through control functions [6,[9][10][11][12][13][14][15].…”

“…A theoretical study of the above problems, as well as various formulations of problems of control and optimal control of processes described by parabolic equations, were given, in particular, in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. The papers [12][13][14][15][16] consider problems of control and optimal control of a thermal (parabolic) equation using distributed control, in particular, allowing taking into account of restrictions on the structure of the solution and control [16], restrictions on the control and states [13], and point restrictions for control [14]. Often, there are boundary control problems of a thermal process in which it is necessary to generate the desired temperature state within a given time interval.…”