Optimal control of the temperature field in a rod with a programmed change in the control zone

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

“…conditions. Thus, the solution to the original problem is reduced to the problem of controlling a thermal process described by the non-homogeneous Equation (10) with homogeneous boundary conditions (7), which is stated as follows: find such temperature conditions µ(t) and ν(t), t 0 t T that change the temperature state as described by the nonhomogeneous Equation (10) with homogeneous boundary conditions (7) from a given initial state (12) to a final state (13) within a given time interval.…”

“…Given that the boundary conditions (7) are homogeneous, and the consistency conditions are met, the solution to Equation (10) is sought in the form…”

“…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.…”

“…The relevance of issues of developing temperature control conditions of thermal processes is well-established. 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].…”