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

Marek Majewski

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.
• Revised: 2016-07-19.
• Accepted: 2016-07-22.
• Published online: 2017-01-03.