Stabilization of fractional-evolution systems

This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation $$\partial^{\alpha,\eta}_{t} u(t)=\mathcal{A}u(t)-\frac{\eta}{\Gamma (1-\alpha)}\int_{0}^{t}(t-s)^{-\alpha} \, e^{-\eta(t-s)}u(s)\, ds,\; t > 0,$$ with the initial data $u(0)=u^{0}$, where $\mathcal{A}$ is a unbounded operator in Hilbert space and $\partial_{t}^{\alpha,\eta}$ stands for the fractional derivative. We provide two main results concerning the behavior of the solutions when $t\longrightarrow+\infty$. We look first to the case $\eta>0$ where we prove that the solution of this problem is exponential stable then we consider the case $\eta=0$ when we prove under some consideration on the resolvent that the energy of the solution goes to $0$ as $t$ goes to the infinity as $1/t^\alpha$.