Output feedback stabilization for heat equations with sampled-data controls

TL;DR

This paper develops an explicit output feedback law for stabilizing heat equations with sampled-data controls, allowing flexible sampling periods and based on new controllability and observability results.

Contribution

It introduces a novel feedback law for heat equations with sampled-data control, with explicit dependence on sampling period and new controllability inequalities.

Findings

Feedback law is explicit and adaptable to any sampling period.

Stability behavior is well-understood as sampling period varies.

New observability inequality for heat equations was established.

Abstract

In this paper, we build up an output feedback law to stabilize a sampled-data controlled heat equation (with a potential) in a bounded domain $\Omega$. The feedback law abides the following rules: First, we divide equally the time interval $[0,+\infty)$ into infinitely many disjoint time periods, and divide each time period into three disjoint subintervals. Second, for each time period, we observe a solution over an open subset of $\Omega$ in the first subinterval, take sample from outputs at one time point of the first subinterval, add a time-invariant output feedback control over another open subset of $\Omega$ in the second subinterval; let the equation evolve free in the last subinterval. Thus, the corresponding feedback control is of sampled-data. Our feedback law has the following advantages: the sampling period (which is the length of the above time period) can be arbitrarily taken; the feedback law has an explicit expression in terms of the sampling period; the behaviors of the norm of the feedback law, when the sampling period goes to zero or infinity, are clear. The construction of the feedback law is based on two kinds of approximate null-controllability for heat equations. One has time-invariant controls, while another has impulse controls. The studies of the aforementioned controllability with time-invariant controls need a new observability inequality for heat equations built up in the current work.