Optimal multivariable flux control of heat transfer in a metal bar

An optimal control problem for heat conduction in a metal bar heated and cooled by two Peltier elements attached to its bottom surface is investigated. The bar has the shape of an elongated cuboid, the vertical sides of which are insulated, whereas the heat exchange with the environment takes place on the upper surface. The problem is to find the time history of the elements' heating powers transferring the system from its initial thermal state to a stationary terminal state and simultaneously minimizing a quadratic cost function.

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