Control system defined by some integral operator

Abstract

In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control $u$ there exists a uniquely defined trajectory $x_{u}$ which continuously depends on control $u$ and the operator $u\mapsto x_{u}$ is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.