Rapid stabilizability of delayed infinite-dimensional control systems
Yaxing Ma, Lijuan Wang, Huaiqiang Yu
TL;DR
The paper addresses rapid stabilization of linear infinite-dimensional control systems with constant time delay. It develops an abstract framework where the state evolves under $y_t=Ay+\kappa y(t-\tau)+Bu$, extends the system to a coupled state-space $(\mathcal{A},\mathcal{B})$, and employs semigroup and spectral analysis to prove that delay does not affect rapid stabilizability and that history is unnecessary for instantaneous-valued feedback. The main contributions include a frequency-domain characterization and a constructive approach to stabilize delayed systems by stabilizing a finite-dimensional critical part, with applications to delayed heat equations under Neumann boundary and internal controls. These results provide robust feedback design principles for delayed infinite-dimensional systems and validate the theory in PDE contexts.
Abstract
In this paper, the rapid stabilizability of linear infinite-dimensional control system with constant-valued delay is studied. Under assumptions that the state operator generates an immediately compact semigroup and the coefficient of the delay term is constant, we mainly prove the following two results: (i) the delay does not affect rapid stabilizability of the control system; (ii) from the perspective of observation-feedback, it is not necessary to use historical information to stabilize the control system when the system is rapidly stabilizable. Some applications are given.