Opuscula Math. 37, no. 2 (2017), 313-325
http://dx.doi.org/10.7494/OpMath.2017.37.2.313
Opuscula Mathematica
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.
Keywords: Volterra equation, implicit function theorem, sensitivity.
Mathematics Subject Classification: 45D05, 34A12, 47J07, 46T20.
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- Marek Majewski
- Department of Mathematics and Computer Science, University of Łódź, Banacha 22, 90-238 Łódź, Poland
- Communicated by Zdzisław Jackiewicz.
- Received: 2016-06-21.
- Revised: 2016-07-19.
- Accepted: 2016-07-22.
- Published online: 2017-01-03.
