Output feedback stabilization for heat equations with sampled-data controls

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 Ω. The feedback law abides the following rules: First, we divide equally the time interval [0, +∞) 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 Ω 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 Ω in the second subinterval; let the equation evolve free in the last subinterval. Thus, the corresponding feedback control is of sampleddata. 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.

Boundary sampled-data feedback stabilization for parabolic equations

“…The adapting of the sampled-data control becomes more and more popular with the developing of digital technique. Sampled-data feedback stabilization for linear and nonlinear parabolic equations were studied in [14,15,16,17]. Among these works, [14] showed that two kinds of sampled-data boundary feedback, which emulate the reduced model design and the backstepping design respectively, can stabilize the 1-D linear parabolic equations when the length of the sampling interval is small enough; [15,16] concerned about the length of sampling interval that preserves the stability of the closed-loop system with proportional feedback; while [17] provided a way to construct an output feedback control to stabilize the heat equations for arbitrarily given sampling period.…”

Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations

“…As the time-discretization of control systems, periodic sampling and control-updating are widely used. Various problems on periodic sampleddata control have been studied for infinite-dimensional systems; for example, stabilization [14,15,21,22,24,29,34,38], robustness analysis of continuous-time stabilization with respect to periodic sampling [23,30,31], and output regulation [17][18][19]25,37]. Event/self-triggering mechanisms are other time-discretization methods, which send measurements and update control inputs when they are needed.…”

“…Moreover, by looking at the update behavior on [0.3, 0.5] in Figure 4, we find that the self-triggering mechanism in the large perturbation case is conservative, i.e., the control input is updated even when the difference F x(t k+1 ) − F x(t k ) is small. It is worthwhile to mention that if τ max > 0.91, then (r 1 , r 2 , ε) = (0.1, 0.1, 0.29) does not satisfy the sufficient condition (21). Figure 5 illustrates inter-event times t k+1 −t k in the large perturbation case (blue circle) and the small perturbation case (red squire).…”

Strong Stability of Sampled-data Riesz-spectral Systems

“…This result has been extended to the exponential stability of some classes of infinite-dimensional systems in [18,31], but even exponential stability is much more delicate in the infinite-dimensional case [30]. Sampled-data systems are ubiquitous in computer-based control systems, and various sampled-data control problems have been studied for infinite-dimensional systems; for example, stabilization [10,11,15,17,19,29,34,37] and output regulation [12-14, 20, 36]. Robustness of strong stability with respect to sampling has been posed as an open problem in [32], and it has not been solved yet.…”