2014/11/03 by N. S. Hoang, Hoang, N. S.
Computer Science · Engineering · Mathematics · #34G20 #37L05 #44J05 #47J35 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations #math.DS #msc:34G20 #msc:37L05 #msc:44J05 #msc:47J35
paper · pdf · doi:10.48550/arxiv.1411.0552
arxiv created 2014/11/03 · openalex publication_date 2014/11/03 · arxiv updated 2014/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The stability of the solution to the equation u = A(t)u + G(t,u)+f(t), t≥ 0, u(0)=u0 is studied. Here A(t) is a linear operator in a Hilbert space H and G(t,u) is a nonlinear operator in H for any fixed t≥ 0. We assume that ‖G(t,u)‖≤ α(t)‖u‖p, p>1, and the spectrum of A(t) lies in the half-plane \Real λ≤ γ(t) where γ(t) can take positive and negative values. We proved that the equilibrium solution u=0 to the equation is Lyapunov stable under persistantly acting perturbations f(t) if supt≥ 0∫0t γ(ξ) dξ<∞ and ∫0^∞ α(ξ) dξ<∞. In addition, if ∫0t γ(ξ) dξ→ -∞ as t→∞, then we proved that the equilibrium solution u=0 is asymptotically stable under persistantly acting perturbations f(t). Sufficient conditions for the solution u(t) to be bounded and for limt→∞u(t) = 0 are proposed and justified.