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arXiv · 2504.09611

An Operator-Theoretic Framework for the Optimal Control Problem of Nonlinear Caputo Fractional Systems

Dev Prakash Jha·Raju K. George
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Abstract

This paper addresses the optimal control problem for a class of nonlinear fractional systems involving Caputo derivatives and nonlocal initial conditions. The system is reformulated as an abstract Hammerstein-type operator equation, enabling the application of operator-theoretic techniques. Sufficient conditions are established to guarantee the existence of mild solutions and optimal control-state pairs. The analysis covers both convex and non-convex scenarios through various sets of assumptions on the involved operators. An optimality system is derived for quadratic cost functionals using the G\^ateaux derivative, and the connection with Pontryagin-type minimum principles is discussed. Illustrative examples demonstrate the effectiveness of the proposed theoretical framework.

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Dev Prakash Jha, Raju K. George. 2025-04-13. An Operator-Theoretic Framework for the Optimal Control Problem of Nonlinear Caputo Fractional Systems. https://arxiv.org/abs/2504.09611

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