Stabilization of fractional-evolution systems

2019/02/07 by Kaïs Ammari, Ammari, Kaïs, Fathi Hassine +3
Computer Science · Engineering · Mathematics · #35A01 #35A02 #35M33 #93D20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1902.02558

openalex publication_date 2019/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is devoted to the analysis of the problem of stabilization of fractional (in time) partial differential equations. We consider the following equation ∂^α,η_t u(t)=Au(t)-\fracηΓ(1-α)∫₀^t(t-s)^-α e^-η(t-s)u(s) ds, t gt; 0, with the initial data u(0)=u⁰, where A is a unbounded operator in Hilbert space and ∂_t^α,η stands for the fractional derivative. We provide two main results concerning the behavior of the solutions when t\longrightarrow+∞. We look first to the case η>0 where we prove that the solution of this problem is exponential stable then we consider the case η=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^α.

Citations

• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian
• Fractional Differential Equations

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