Feedback law to stabilize linear infinite-dimensional systems

TL;DR

This paper introduces a novel feedback law for stabilizing linear infinite-dimensional systems with unbounded control operators, leveraging a mutated Gramian operator and weak observability, avoiding the need for exact controllability.

Contribution

The paper presents a new feedback law based on a mutated Gramian operator that stabilizes systems under weak observability, expanding applicability beyond exact controllability.

Findings

Successfully stabilizes systems with unbounded control operators.

Uses weak observability inequality instead of exact controllability.

Provides a new approach to feedback design in infinite-dimensional systems.

Abstract

We design a new feedback law to stabilize a linear infinite-dimensional control system, where the state operator generates a C0-group and the control operator is unbounded. Our feedback law is based on the integration of a mutated Gramian operator-valued function. In the structure of the aforementioned mutated Gramian operator, we utilize the weak observability inequality in [21, 14] and borrow some idea used to construct generalized Gramian operators in [11, 23, 24]. Unlike most related works where the exact controllability is required, we only assume the above-mentioned weak observability inequality which is equivalent to the stabilizability of the system.