An investigation on the optimality condition of Caputo fractional time delay system

Abstract

Optimal control problem of a Caputo fractional state-dependent delay system is discussed in this paper. Both Dirichlet and Neumann fractional optimal control problems are studied. Using a linear continuous operator, the delay system is converted to an equivalent system not involving explicit delay term. The existing results for the unique solution of the fractional system associated with the optimal control problem are attained by the application of Lax-Milgram Theorem. Optimality conditions, both necessary and sufficient for the fractional Dirichlet and Neumann problems with the quadratic objective function, are obtained. Interpreting the first-order optimality condition of Euler-Lagrange along with the corresponding adjoint system involving the right Caputo derivative, the optimality system is derived. Initially, the first-order Euler-Lagrange optimality condition is used along with the corresponding adjoint system to derive the optimality system. Subsequently, adjoint equations and Hamiltonian maximization conditions are derived using duality and variational analysis.

Keywords

Optimal control / State dependent delay / Dirichlet & Neumann conditions / Caputo fractional derivative

Funding

This work has financial support of Farhangian University (Contract No. 500.17474.120).

Declaration of competing interest

The author declare that they have no conflict of interest regarding the publication of this article.

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