Robustness of constant-delay predictor feedback for in-domain stabilization of reaction-diffusion PDEs with time- and spatially-varying input delays

This paper discusses the in-domain feedback stabilization of reaction-diffusion PDEs with Robin boundary conditions in the presence of an uncertain timeand spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finitedimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a C-0-semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent. (c) 2020 Elsevier Ltd. All rights reserved.

Automated summary

This paper discusses in-domain feedback stabilization of reaction-diffusion PDEs with Robin boundary conditions in the presence of uncertain time and spatially-varying delay. The proposed control design strategy involves a constant-delay predictor feedback based on the known nominal value of the delay, synthesized on a finite-dimensional truncated model capturing unstable modes. The resulting closed-loop system is shown to be exponentially stable with small enough variations of the delay around its nominal value.