ISSUE 180, article 6


Kibern. vyčisl. teh., 2015, Issue 179, pp 66-82.

Kotsur Maksim P., Assistant of the Department of Mathematical Control Problems and Cybernetics of Yuriy Fedkovych Chernivtsi National University, Ul. Kotsyubynsky, 2, Chernivtsi, 58012, Ukraine, e-mail:

Nakonechniy Аleksandr G., Dr (Phys. and Math.), Prof., Academician-Secretary of the Department of Cybernetics and Systems Analysis, Head of the Department of System Analysis and Decision Theory of the Taras Shevchenko Kiev National University, Pr. Glushkov, 4 D, Kyiv, 03187, Ukraine, e-mail:


Introduction. A large variety of optimization problems is related to distributed parameter systems (DPSs), whose behavior is described by partial differential equations. The optimal control theory is widely employed for optimization of lumped parameter systems (LPSs), which are modeled by ordinary differential equations. Development of the theory and methods for obtaining optimal control functions for DPSs is much more difficult as compared with solving such problem for LPSs.

The purpose of the present paper is to obtain optimality conditions and to develop numerical methods for solving the optimization problem of an unsteady one-dimensional process with distributed parameters, as well as their application to optimization of transient thermoelectric cooling.

Methods. A method is proposed for solving of optimal control problem for DPS described by nonlinear partial differential equations of parabolic type with nonstandard boundary conditions. A method consists in coordinate discretization of distributed object and transition to the problem for LPS. Optimal control theory based on the Pontryagin maximum principle can be used for such system optimization.

Results. This method is applied for optimization of transient thermoelectric cooling process. Optimal dependences of current on time have been calculated for stage thermoelectric cooler power supply with the purpose of minimizing the cooling temperature within a preset time interval. Results of computer experiment for one- and two-stage coolers are presented.

Conclusion. The obtained results show that the implementation of optimal functions for cooler power supply can provide the cooling temperature in transient mode by 10 K lower than minimum temperature reached at direct current.

Keywords: distributed parameter system, optimal control, Pontryagin maximum principle, optimization, transient thermoelectric cooling.

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27 One-stage thermoelectric modules. Available at: Two-stage thermoelectric modules. Available at:

Received 18.02.2015