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An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the stabilization of the flow modeled by the Saint-Venant equations. We have solved two problems in this study. The first, we have proved that the operator associated to the Saint-Venant system has a finite number of unstable eigenvalues. Consequently, the system is not exponentially stable on the space L 2 (Ω) × L 2 (Ω) , but is exponentially stable on a subspace of the space L 2 (Ω) × L 2 (Ω) , (Ω is a given domain). The second problem, if the advection is dominant, the natural stabilization is very slow. To solve these problems, we have used an extension method due to Russel (1974) and Fursikov (2002). Thanks to this method, we have determined a boundary Dirichlet control able to accelerate the stabilization of the flow. Also, the boundary Dirichlet control is able to kill all the unstable eigenvalues to get an exponentially stable solution on the space L 2 (Ω) × L 2 (Ω). Then, we extend this method to the finite difference equations analog of the continuous Saint-Venant equations. Also, in this case, we obtained similar results of stabilization. A finite difference scheme is used to compute the control and several numerical experiments are performed to illustrate the efficiency of the control. [ABSTRACT FROM AUTHOR]