Stability results of some abstract evolution equations

TL;DR

This paper investigates the stability of solutions to a class of abstract evolution equations in Hilbert spaces, providing conditions for Lyapunov and asymptotic stability of the zero solution under persistent perturbations.

Contribution

It establishes new stability criteria for abstract evolution equations with time-dependent operators and nonlinearities, including conditions for boundedness and convergence to zero.

Findings

Lyapunov stability under bounded integral of spectrum function

Asymptotic stability when spectrum integral tends to negative infinity

Conditions for solution boundedness and convergence to zero

Abstract

The stability of the solution to the equation $\dot{u} = A(t)u + G(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ 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 $t\ge 0$. We assume that $\|G(t,u)\|\le \alpha(t)\|u\|^p$, $p>1$, and the spectrum of $A(t)$ lies in the half-plane $\Real \lambda \le \gamma(t)$ where $\gamma(t)$ 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 $\sup_{t\ge 0}\int_0^t \gamma(\xi)\, d\xi <\infty$ and $\int_0^\infty \alpha(\xi)\, d\xi<\infty$. In addition, if $\int_0^t \gamma(\xi)\, d\xi \to -\infty$ as $t\to\infty$, 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 $\lim_{t\to\infty}u(t) = 0$ are proposed and justified.