Strong Stabilization of Distributed Bilinear Systems with Time Delay

Feedback stabilization of non‐homogeneous bilinear systems with a finite time delay

“…29 2. In the delayed case, with 𝛾 b u = 2, we retrieve the result of Hamidi et al 24 3. It follows from (23) that if 𝛾 b u = 1, 𝜌 < 𝛼 M‖B s ‖e 𝛼r , and b s = 0 , then the system (1) is exponentially stable.…”

“…Remark In the undelayed homogeneous case, with $$$ {\gamma}_{b&#x0005E;u}&#x0003D;2 $$$, we can see that the component $$$ {y}&#x0005E;s(t) $$$ of $$$ y(t) $$$ decays exponentially, retrieving thus the result of Ouzahra 29 In the delayed case, with $$$ {\gamma}_{b&#x0005E;u}&#x0003D;2 $$$, we retrieve the result of Hamidi et al 24 It follows from () that if $$$ {\gamma}_{b&#x0005E;u}&#x0003D;1 $$$, $$$ \kern0.1em \rho &lt;\frac{\alpha }{M\left\Vert {B}&#x0005E;s\right\Vert {e}&#x0005E;{\alpha r}} $$$, and $$$ {b}&#x0005E;s&#x0003D;0 $$$ , then the system () is exponentially stable. If $$$ {\gamma}_{b&#x0005E;u}&#x0003D;1 $$$ and $$$ {B}&#x0005E;s&#x0003D;0 $$$, then the corresponding solution of system () is exponentially stabilizable without any condition on $$$ \rho $$$. …”

“…Many authors have recently established stability results for delayed bilinear systems (see, for example, Hamidi et al 24 ). In Hamidi et al, 24 using an approach based on a conditioned decomposition method of the state space and the system, the authors have proved the asymptotic stability via a quadratic control under a suitable smallness condition on the delay. This article aims to contribute to the study of the stabilization of the non-homogeneous bilinear system (1).…”

“…This article aims to contribute to the study of the stabilization of the non-homogeneous bilinear system (1). We consider the same stabilization problem as in Hamidi et al 24 without any condition on the delay. Our approach is based on the unconditioned decomposition method and an infinite-dimensional Lyapunov's method.…”

Robust stabilization for affine control systems in Banach spaces

“…The case of unbounded bilinear systems (i.e., B is an unbounded linear operator) has been considered in Ouzahra [19]. Moreover, the above study has been adopted to delayed bilinear systems in Hamidi et al [20]. Furthermore, using an adequate decomposition of system (1), it has been showed in previous research [12,13] that stabilizing a parabolic bilinear system like (1) in a Hilbert space turns out to stabilizing a finite dimensional subsystem of (1).…”