Maximal Lp-regularity of an abstract evolution equation: Application to closed-loop feedback problems, with boundary controls and boundary sensors

Abstract

We present an abstract maximal $L^p$-regularity result up to $T=\infty$ on a Banach space, that is tuned to capture (linear) PDEs of parabolic type, defined on a bounded domain and subject to finite dimensional, boundary controls and boundary sensors, in feedback form. It improves Lasiecka et al. (2021), which covered boundary controls and interior sensors. The present proof must necessarily be completely different from the one in Lasiecka et al. (2021). In applications (Section 3), the case $T<\infty$ requires no further assumptions on the boundary control/sensor vectors. Instead, the case $T=\infty$ requires, of course, the property of uniform stabilization for suitable boundary control/sensor vectors, which is geometrically sensitive.