Feedback Stabilization of Semilinear System With Distributed Delay

Abstract

This work is concerned with the feedback stabilization of the following semilinear system with distributed delay <disp-formula id="deqn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> \begin{equation*} \begin{array}{c}\dot{w}(t)=\mathcal {A} w(t)+v(t)F(w_{t})\ \text{and}\ w_{0}=\phi \in \mathcal {C}:=C([-r,0]; \mathcal {X}), \end{array} \tag{1} \end{equation*} </tex-math></disp-formula> where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {A}$</tex-math></inline-formula> is the infinitesimal generator of a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C_{0}-$</tex-math></inline-formula> semigroup <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(S(t))_{t\geq 0}$</tex-math></inline-formula> on a Hilbet space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> . <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F$</tex-math></inline-formula> is a locally Lipschitz function from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F(0)=0$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {C}$</tex-math></inline-formula> is the space of all continuous function from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[-r,0]$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> which is equipped with the following norm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Vert \varphi \Vert _{\mathcal {C}}=\displaystyle \sup _{-r\leq \theta \leq 0} \Vert \varphi (\theta)\Vert _{\mathcal {X}}$</tex-math></inline-formula> for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi \in \mathcal {C}$</tex-math></inline-formula> . The contro l <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$v$</tex-math></inline-formula> , of type feedback, is a defined function from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{+}$</tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}.$</tex-math></inline-formula> For <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$t\geq 0$</tex-math></inline-formula> , the history function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$w_{t} \in \mathcal {C}$</tex-math></inline-formula> is defined by <disp-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> \begin{equation*} w_{t}(\theta)=w(t+\theta)\ \; \text{for all}\; \theta \in [-r,0]. \end{equation*} </tex-math></disp-formula> Here, we consider an observation condition in terms of the semigroup solution of system ( <xref ref-type="disp-formula" rid="deqn1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> ) with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$F=0$</tex-math></inline-formula> and parameterized by a positive constant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma$</tex-math></inline-formula> , it means that there exist <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T,\,\delta &gt;0$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma \geq 2$</tex-math></inline-formula> such that <disp-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> \begin{equation*} \int _{0}^{T}|\langle F(U(s)\xi), (U(s)\xi)(0)\rangle | ds \geq \delta \Vert \xi (0)\Vert _{\mathcal {X}}^{\gamma } \;\;\;\; \text{for all}\; \xi \; \in \;\mathcal {C}. \end{equation*} </tex-math> </disp-formula> Some sufficient conditions are given with respect to a bounded feedback control and the parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma$</tex-math></inline-formula> to guarantee the feedback stabilization of the semilinear system with distributed delay. Moreover, for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma \geq 2,$</tex-math></inline-formula> an explicit polynomial decay rate of the stabilization state is estimated in the strong stabilization case, more precisely, we show that <disp-formula id="deqn2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX"> \begin{equation*} \left\Vert w(t)\right\Vert _{\mathcal {X}}=O\left(t ^{-\dfrac{1}{2(\gamma -1)}}\right) \text{ as } t \longrightarrow +\infty. \, \tag{2} \end{equation*} </tex-math></disp-formula> In the bilinear case with distributed delay and by using the decomposition method of the state space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {X}$</tex-math></inline-formula> , we investigate the feedback stabilization of system (1) using some suitable conditions like observability assumption. In the case of strong stabilization, we obtain the same explicit decay estimate (2) of the stabilized state. Furthermore, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\gamma = 2$</tex-math></inline-formula> , we show that the normalized feedback control exponentially stabilizes the semilinear system (1). The obtained results are illustrated by three examples and numerical simulations for wave equation with distributed delay.