Feedback stabilization of linear and bilinear unbounded systems in Banach space

Optimal control problem for unbounded bilinear systems and applications

“…It is worth noting that there are several works that are interested in the question U ad ̸ = ∅ (see for instance [3,4,14,24]).…”

“…there exists h ∈ X such that ξ(y) = ⟨y, h⟩ and b ∈ X −1 \X (for example b = A −1 1 [0,1] ), we can observe that B * y = ⟨b, y⟩ X −1 ,X ′ −1 h. Such systems serve as useful models for the practical description of various real problems such as air pollution or traffic flow (see [2,19,28]). It is clear that B is a bounded operator from X to X −1 and it is (p, q)-admissible for p, q ≥ 2 (see [4]). Now, let us consider the following quadratic cost function J…”

“…However, the control operator B considered in this work is defined from X to some larger Banach space Y ⊃ X where B is (p, q)-admissible. As an example, one can mention the fractional diffusion equation where the control operator B is (−∆) α , 0 < α < 1 (see [4,7]). This type of unbounded control operator may also occur in many situations of PDEs when the control is applied on a part of the boundary or at a point of the geometrical domain.…”

Exponential stability of linear systems under a class of Desch–Schappacher perturbations

“…In other words, the 1−admissibility condition excludes several applications that are also available in Hilbert space. Moreover, in Ammari et al [9], it was assumed that D ((…”

“…This problem has been considered in Liu et al [7] for a bounded operator $$$ B $$$, and the case of a Miyadera–Voigt type operator has been investigated in Ouzahra et al [8]. Moreover, in Ammari et al [9], the case of 1−admissibility in Banach space has been considered. However, the 1−admissibility assumption prevents us to consider the case of Hilbert state space as in this case, the operator $$$ B $$$ will be necessary bounded (see Weiss [10]).…”