On the time-optimal control problem for a pseudo-parabolic equation with variable thermal conductivity in a bounded domain

Abstract

This paper studied a time-optimal control problem for a pseudo-parabolic equation in a bounded multidimensional domain. The equation involved a uniformly elliptic operator and a small parameter characterizing memory effects. The control acted through a distributed source term, where the spatial part was fixed and the time-dependent part was a bounded measurable function. The goal was to steer the system from a zero initial state to a prescribed spatially averaged temperature in minimal time and maintain it thereafter. Using spectral decomposition, we derived an explicit solution representation and reduced the control problem to a Volterra integral equation. Under natural positivity assumptions on the Fourier coefficients, we established that the kernel was continuous, positive, and monotone. These properties yielded an explicit upper bound for the minimal time in terms of the first eigenvalue, the relaxation parameter, and the system constraints. A comparison with the parabolic case showed that the pseudo-parabolic term led to longer minimal times, reflecting the memory effects inherent in such models.

Keywords

pseudo-parabolic equation / thermal conductivity / Volterra integral equation / admissible control / minimal time / memory effect

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