On solution set associated with a class of fractional variational control problems with semi-infinite constraints

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Abstract

In this paper, we extend semi-infinite variational programming problems to a class of fractional variational control problems using Caputo-Fabrizio fractional derivatives. In this context, by using integration by parts, the necessary optimality conditions for these problems are obtained, incorporating complete Karush-Kuhn-Tucker (KKT) criteria which include both state and control stationarity conditions. Then, we build a Lagrange functional, where duality properties are derived under suitable assumptions of the involved functions. Furthermore, saddle point criteria connected to the considered Lagrange functional are developed. The main contribution of the study is the extension of semi-infinite fractional variational programming to a controlled setting, leading to a unified framework that combines state and control stationarity conditions, Lagrange duality, and saddle-point characterization. An illustrative numerical example is also provided to demonstrate the applicability of the developed framework to a specific state-control pair.

Keywords

Caputo-Fabrizio fractional derivative / optimal solution / semi-infinite constraints / Lagrange dual / saddle point criteria / KKT conditions

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