Exponential stabilization for spatial multiple-fractional advection-diffusion-reaction system

The exponential stabilization for spatial multiple-fractional advection-diffusion-reaction system (SMFADRS) is considered, and for the disturbed SMFADRS, the finite-time H_∞ stabilization is investigated. To ensure the considered system to achieve the desired performance, a distributed controller is designed to be located in the sub-intervals of the whole spatial domain. Then, by deriving an improved fractional Poincare's inequality and resorting to the Lyapunov functional method, the sufficient criteria of exponential stability and finite-time H_∞ performance are obtained. Besides, we also explore the effect of the space domain and its division, the control gain, the distribution of controller and the fractional order on the stability. Moreover, we apply the obtained theoretical results to address the control problem of the groundwater pollution, and the corresponding numerical simulations are performed to show the effectiveness of our results.