On the stability of Bresse system with one discontinuous local internal Kelvin–Voigt damping on the axial force

Abstract

In this paper, we investigate the stabilization of a linear Bresse system with one discontinuous local internal viscoelastic damping of Kelvin–Voigt type acting on the axial force, under fully Dirichlet boundary conditions. First, using a general criteria of Arendt–Batty, we prove the strong stability of our system. Finally, using a frequency domain approach combined with the multiplier method, we prove that the energy of our system decays polynomially with different rates.

Appendix A. Some notions and stability theorems

In order to make this paper more self-contained, we recall in this short appendix some notions and stability results used in this work.

Definition A.1

Assume that A is the generator of $C_0$-semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on a Hilbert space H. The $C_0$-semigroup $\left( e^{tA}\right) _{t\ge 0}$ is said to be

• (1) Strongly stable if

  $$\begin{aligned} \lim _{t\rightarrow +\infty } \Vert e^{tA}x_0\Vert _H=0,\quad \forall \, x_0\in H. \end{aligned}$$

• (2) Exponentially (or uniformly) stable if there exists two positive constants M and $\varepsilon$ such that

  $$\begin{aligned} \Vert e^{tA}x_0\Vert _{H}\le Me^{-\varepsilon t}\Vert x_0\Vert _{H},\quad \forall \, t>0,\ \forall \, x_0\in H. \end{aligned}$$

• (3) Polynomially stable if there exists two positive constants C and $\alpha$ such that

  $$\begin{aligned} \Vert e^{tA}x_0\Vert _{H}\le Ct^{-\alpha }\Vert A x_0\Vert _{H},\quad \forall \, t>0,\ \forall \, x_0\in D(A). \end{aligned}$$

  $\square$

To show the strong stability of the $C_0$-semigroup $\left( e^{tA}\right) _{t\ge 0}$, we rely on the following result due to Arendt–Batty [8].

Theorem A.2

Assume that A is the generator of a C$_0$-semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on a Hilbert space H. If A has no pure imaginary eigenvalues and $\sigma \left( A\right) \cap i\mathbb {R}$ is countable, where $\sigma \left( A\right)$ denotes the spectrum of A, then the $C_0$-semigroup $\left( e^{tA}\right) _{t\ge 0}$ is strongly stable. $\square$

Concerning the characterization of polynomial stability stability of a $C_0$-semigroup of contraction $\left( e^{tA}\right) _{t\ge 0}$, we rely on the following result due to Borichev and Tomilov [12] (see also [11, 31])

Theorem A.3

Assume that A is the generator of a strongly continuous semigroup of contractions $\left( e^{tA}\right) _{t\ge 0}$ on $\mathcal {H}$. If $i\mathbb {R}\subset \rho (\mathcal {A})$, then for a fixed $\ell >0$ the following conditions are equivalent

$$\begin{aligned} \sup _{\lambda \in \mathbb {R}}\left\| \left( i\lambda I-\mathcal {A}\right) ^{-1}\right\| _{\mathcal {L}\left( \mathcal {H}\right) }= & {} O\left( |\lambda |^\ell \right) , \end{aligned}$$

(5.1)

$$\begin{aligned} \Vert e^{t\mathcal {A}}U_{0}\Vert ^2_{{\mathcal {H}}}\le & {} \frac{C}{t^{\frac{2}{\ell }}}\Vert U_0\Vert ^2_{D({\mathcal {A}})},\forall t>0, U_0\in D({\mathcal {A}}), \text {for some} C>0. \end{aligned}$$

(5.2)

$\square$