Geodesic
Time optimal control problem with integral constraint for the heat transfer process
Eurasian mathematical journal, Tome 15 (2024) no. 1, pp. 8-22.

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In the present paper a mathematical model of thermocontrol processes is studied. Several convectors are installed on the disjoint subsets $\Gamma_k$ of the wall $\partial\Omega$ of a volume $\Omega$ and each convector produces a hot or cold flow with magnitude equal to $\mu_k(t)$, which are control functions, and on the surface $\partial\Omega\setminus\Gamma$, $\Gamma=\bigcup\Gamma_k$, a heat exchange occurs by the Newton law. The control functions $\mu_k(t)$ are subjected to an integral constraint. The problem is to find control functions to transfer the state of the process to a given state. A necessary and sufficient condition is found for solvability of this problem. An equation for the optimal transfer time is found, and an optimal control function is constructed explicitly. 
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Sh. A. Alimov; G. I. Ibragimov. Time optimal control problem with integral constraint for the heat transfer process. Eurasian mathematical journal, Tome 15 (2024) no. 1, pp. 8-22. http://geodesic.mathdoc.fr/item/EMJ_2024_15_1_a0/

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