Weak and strong stabilisation of bilinear systems in a Banach space

Robust stabilization for affine control systems in Banach spaces

“…Thus (see Pazy [29], p. 185), system () admits a unique mild solution y ( t ) which depends continuously on y 0 and is defined for all t ≥ 0 by (). Moreover, following the same techniques as in previous research [14, 17], one can show that y satisfies the estimate for all 0 ≤ s ≤ t ≤ T and for all y 0 ∈ X . In particular, we have …”

“…Thus (see Pazy [29], p. 185), system (1) admits a unique mild solution y(t) which depends continuously on y 0 and is defined for all t ≥ 0 by (9). Moreover, following the same techniques as in previous research [14,17], one can show that y satisfies the estimate…”

“…Proof. Since dim X 1 < ∞, we have J = id X 1 and the duality pairing can be identified with the inner product ⟨., .⟩ of the Euclidean space X 1 (with an associate norm denoted by ‖.‖), so that inequality (17) reads as follows:…”

“…Notice that Γ is a linear space and that any P belonging to Γ is admissible w.r.t S(t). Moreover, using the superposition property of the semigroup S(t), we can show that if L P is finite for some T > 0, then it is finite for all T > 0, so it is possible to take the same T as in the inequality (17), which we will consider in the sequel.…”

“…[13]) (see also Khlebnikov [16] for the case of finite dimensional). Recently, the question of feedback stabilization of system () in a reflexive Banach state space has been tackled in Benzaza and Ouzahra [17] and the case of general Banach state space (not necessary reflexive) in Ouzahra [18]. The case of unbounded bilinear systems (i.e., B is an unbounded linear operator) has been considered in Ouzahra [19].…”