A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

Abstract

The present work is concerned with the stabilization of a general class of time-varying linear parabolic equations by means of a finite-dimensional receding horizon control (RHC). The stability and suboptimality of the unconstrained receding horizon framework is studied. The analysis allows the choice of the squared 1-norm as control cost. This leads to a nonsmooth infinite-horizon problem which provides stabilizing optimal controls with a low number of active actuators over time. Numerical experiments are given which validate the theoretical results and illustrate the qualitative differences between the 1-and 2-control costs.

Saturated feedback stabilizability to trajectories for the Schlögl parabolic equation

“…Verification of (3.4a): Since the stabilizability result for the time-varying system (3.1) holds globally, the proof follows with similar arguments given in [1,Thm. 2.6].…”

“…for every t ≥ s, where C h := C inf (D 0 + 2C B )C J , and (3.12) and (3.14) were used. The rest of proof follows the same lines as in the proof of [1,Thm. 6.4] based on (3.15) and (3.12).…”

“…In this work, we also continue the investigation on the receding horizon framework initiated in [1] for the stabilization (to zero) of linear nonautonomous (time-varying) systems. In this framework, no terminal cost or constraints is needed and, the stability is obtained by an appropriate concatenation scheme on a sequence of overlapping temporal intervals.…”

“…Recall that, in theory, stabilizing system (1.1) to a given time-dependent trajectory y = y(t) is equivalent to stabilizing the nonautonomous error dynamics (1.10) to zero. We adapt the analysis given in [1] for (1.11), with the differences that here, firstly, the dynamics is nonlinear, secondly, control constraints are imposed and, finally, numerically, instead of stabilizing (1.10) to zero we stabilize the original system (1.1) to the trajectory y.…”

Moment-Driven Predictive Control of Mean-Field Collective Dynamics

“…Based on dynamic estimates of the moments decay, we are able to perform a forward error analysis to estimate the next point in time where we need to update the linearization the dynamics and its feedback law. This strategy can be seen as model predictive control (MPC) technique [57,20,45,10], where an open-loop control signal is applied only up to a subsequent point in time, after which the optimization is repeated. Moreover, the proposed control strategy is capable of treating efficiently high-dimensional control problems, and is robust in the case of limited access to the state and can be implemented with a small number of Left: the classical control loop for nonlinear dynamics.…”