Time-optimal control of heat conduction with memory effects under a nonlocal boundary condition

Abstract This paper studies a time-optimal control problem for heat conduction in a thin rod governed by a pseudo-parabolic equation with memory effects. The control function enters as a distributed heat source, and the goal is to raise the spatially averaged temperature to a prescribed level in minimal time under a bounded control constraint. The problem includes a nonlocal Bitsadze-Samarskii boundary condition coupling the temperatures at the right endpoint and the midpoint of the rod, which arises in feedback control systems and temperature equalization problems. Using spectral decomposition, the problem is reduced to a Volterra integral equation of the first kind. The kernel of this equation is proved to be positive and continuous. An explicit upper bound for the minimal time is obtained in terms of the first eigenvalue and the Fourier coefficients. The results reveal how memory effects (characterized by the parameter α) and the nonlocal boundary coupling influence the controllability of the system. For small target temperatures, the minimal time admits a linear approximation independent of α, while the memory parameter explicitly appears in the estimate for larger targets.

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