We study the problem of stabilizing reaction-diffusion equations within a prescribed time. We employ the backstepping method and select a target equation whose state and its spatial derivative converge to zero within the prescribed time. This stability is related back to the plant by characterizing the growth-in-time of the kernel associated to the backstepping transformation. We extract from the backstepping transformation a time-varying boundary feedback controller which stabilizes the plant in the prescribed time. Using the same technique, we then propose an observer that incorporates flux measurements at the boundary which are scaled by a time-varying observer gain; we show that the state estimate error and its spatial derivative converge to zero within the prescribed time. We combine the prescribed-time stabilizing controller and observer results to develop output feedback, for which we verify the separation principle. Finally, a simulation is provided where output feedback is utilized to stabilize an unstable reaction-diffusion equation within a prescribed time.
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Steeves et al. (2019) studied this question.