Lipschitz stability for nonlinear Volterra integrodifferential systems
Consider the nonlinear integro-differential system \[ (1)\quad x'(t)=f(t,x(t))+\int^{t}_{t_ 0}g(t,s,x(s))ds,\quad x(t_ 0)=x_ 0,\quad t_ 0\geq 0, \] and the variational systems \[ y'(t)=f_ x(t,x(t,t_ 0,x_ 0))y(t)+\int^{t}_{t_ 0}g_ x(t,s,x(s,t_ 0,x_ 0))y(s)ds, \] where \(f\in C[J\times R^ n,R^ n]\), \(g\in C[J\times J\times R^ n,R^ n]\), \(f(t,0)=g(t,s,0)=0,\) \(J=[t_ 0,\infty)\), and f and g are continuously differentiable with respect to the components of x on \(J\times R^ n\) and \(J\times J\times R^ n\), respectively. By using the linearization technique and the technique of utilizing the variational system around an arbitrary solution, the authors investigate sufficient conditions for uniform Lipschitz stability of system (1) through their associated variational systems. Examples are given to demonstrate the relationship between the classical notions of stability and the new concept of Lipschitz stability.
scientific article; zbMATH DE number 4065648
scientific article