Stability property for an integrodifferential equation
The paper discusses the relationship between stability under disturbances from the hull and total stability is an almost periodic system of integro-differential equations of the form \[ x'(t)=f\bigl( t,x(t) \bigr)+\int^ 0_{-\infty} F \bigl( t,s,x(t+s),x(t) \bigr) ds, \] where \(f:\mathbb{R} \times \mathbb{R}^ n \to \mathbb{R}^ n\) is continuous and almost periodic in \(t\) uniformly for \(x \in \mathbb{R}^ n\), and \(F(t,s,x,y)\) is continuous on \(\mathbb{R} \times (-\infty,0] \times \mathbb{R}^ n \times \mathbb{R}^ n\) and almost periodic in \(t\) uniformly for \((s,x,y) \in (-\infty,0] \times \mathbb{R}^ n \times \mathbb{R}^ n\). The author also shows that for a periodic integrodifferential equation, uniform stability and stability under disturbances from the hull are equivalent.
stability under disturbances from the hull
scientific article; zbMATH DE number 459211