Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation.
This paper addresses the nonlinear Volterra equation \[ x(t) = a(t,x(t))+\lambda \int_{t_0}^t K(t,s,x(s),u(s))\,ds, \] where \(u(t)\) is a known function from the closed ball centered at the origin with radius \(\mu\) in \(L_p\) and the kernel is assumed to be Lipschitz continuous. It is proved that the equation has Lipschitz continuous solutions with respect to \(\mu\) for each fixed \(p\), and is continuous with respect to \(p\) for each fixed \(\mu\). For the theory of the nonlinear Volterra equation, readers may also refer to the monograph of \textit{D. Sidorov} [Integral dynamical models. Singularities, signals and control. Hackensack, NJ: World Scientific (2015; Zbl 1311.45012)].
scientific article; zbMATH DE number 6362228