Control system defined by some integral operator

Abstract

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 xu which continuously depends on control u and the operator u → xu 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.

Global invertibility of mappings between Banach spaces and applications to nonlinear equations

“…Since the proofs do not differ that much apart from some estimation, we provide only the main differences referring to [9] for the more detailed reasoning. We were inspired by [20] to come up with these results. Prior to formulating the problem under consideration we introduce some required function space setting.…”

Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

“…Moreover, it allows for obtaining uniqueness of a solutions without any notion of convexity, again contrary to what is known in the application of a direct method, see for example [13,Corollary 1.3]. However up to now Theorem 1 and related global implicit function theorem from [10] have been applied to various first order integro-differential problems which cover also the so called fractional case (with the fractional derivative) and correspond to Urysohn and Volterra type equations, see [4,11,12]. Some comments on the global invertibility results from [9], relation with other approaches and possible applications are contained in [8].…”

Global diffeomorphism theorem applied to the solvability of discrete and continuous boundary value problems

“…We would like to mention that in this work it is the first attempt to prove the existence of solutions to a second order problem by a global diffeomorphism theorem. This theorem has been previously applied to the solvability of first order integro-differential systems, see for example [5], [8]. There is also a related research which allows for obtaining the existence of unique solutions to second order ODE contained in [12].…”