| Volume 14, Number 6, 2024, Pages - DOI：10.11948/JAAC-2024-0081 |
| General decay of solutions of a weakly coupled abstract evolution equations with one finite memory control |
| Mostafa Zahri |
| Keywords:Coupled system, finite memory control, general decay, fractional operator. |
| Abstract: |
| In this work, we consider the following abstract evolution system:
\begin{equation*}
\left\{
\begin{array}{cc}
u_{tt}(t)+Au(t)-\displaystyle\int_{0}^{t}g(t-s)A^{\theta }u(s)ds+\alpha
v(t)=0, & t>0 \v_{tt}(t)+Av(t)+\alpha u(t)=0, & t>0 \u(0)=u_{0},\text{ }u_{t}(0)=u_{1},\text{ }v(0)=v_{0},\text{ }v_{t}(0)=v_{1},
&
\end{array}%
\right.
\end{equation*}%
where $A:\mathcal{D}(A)\subset H\longrightarrow H$ is a linear positive definite self-adjoint operator, $H$ is a Hilbert space, $g$ is a positive
nonincreasing function with some general decay rate, $\theta\in \lbrack 0,1]$, $\alpha$ is a positive constant and $u_0 ,u_1 ,v_0$ and $v_1$ are fixed initial data. Under appropriate conditions on $g,\,\alpha$ and the regularity of the initial data, we establish a general decay rate of the solution energy which generalizes some earlier results in the literature. We, also, illustrated our results by performing several numerical tests. |
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