Characterizations of Complete Stabilizability

Feedback law to stabilize linear infinite-dimensional systems

“…Stabilization is one of the most important objectives in control theory. There are two important subjects in stabilization for linear control systems: The first one aims to find sufficient conditions/equivalent conditions on stabilizability, such as resolvent conditions (see, e.g., [7,8,14,16]) and weak observability inequalities (see [13,21,26]). The second one aims to design feedback laws (see, e.g., [6,10,11,12,18,19,23,24,27]).…”

“…However, when researching how to design feedback laws, it is more natural to use stabilizability instead of controllability as an assumption. Indeed, there are many systems which are not controllable but are stabilizable, even completely stabilizable (see examples in [20,Chapters 5] for finite-dimensional settings and in [9,13,21] for infinite-dimensional settings). However, the stabilizability of a linear control system is equivalent to some weak observability inequality for its dual system (see [13,21]).…”

“…U ]dt implies the stabilizability in the sense of Definition 1.1. Note that the above stabilizability is equivalent to the above finite cost condition (see [13,Proposition 3.9]). (ii) In general, A + BK is not the generator of the semigroup S(•) except for the case B ∈ L(U ; H).…”

“…The detailed explanation is given in the proof of our main theorem. (iii) It was proved in [21,Section 3.3] (see also [13,Theorem 3.4] for the complete stabilizability) that the stabilizability of the system (1) is equivalent to the following weak observability inequality for the dual system of (1): There exist δ ∈ (0, 1), T > 0 and C(δ, T ) ≥ 0 such that…”

“…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 [21,13] 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.…”