Page 173 - IJOCTA-15-2

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.2, pp.368-378 (2025)
https://doi.org/10.36922/ijocta.1689

RESEARCH ARTICLE

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

Sanjukta Das *

Department of Mathematics, Mahindra University, India
sanjukta.das@mahindrauniversity.edu.in

ARTICLE INFO ABSTRACT

Article History: Optimal control problem of a Caputo fractional state-dependent delay system
Received: November 17, 2024 is discussed in this paper. Both Dirichlet and Neumann fractional optimal con-
Accepted: March 12, 2025 trol problems are studied. Using a linear continuous operator, the delay system
Published Online: April 28, 2025 is converted to an equivalent system not involving explicit delay term. The ex-
Keywords: isting results for the unique solution of the fractional system associated with the
Optimal control optimal control problem are attained by the application of Lax-Milgram The-
State dependent delay orem. Optimality conditions, both necessary and sufficient for the fractional
Dirichlet and Neumann problems with the quadratic objective function, are
Dirichlet & Neumann conditions
obtained. Interpreting the first-order optimality condition of Euler-Lagrange
Caputo fractional derivative
along with the corresponding adjoint system involving the right Caputo deriv-
AMS Classification:
ative, the optimality system is derived. Initially, the first-order Euler-Lagrange
46C05; 49J20; 93C20; 49K20;
optimality condition is used along with the corresponding adjoint system to
34K05; 34A12; 26A33
derive the optimality system. Subsequently, adjoint equations and Hamilton-
ian maximization conditions are derived using duality and variational analysis.

1. Introduction systems. For instance, fractional-order systems
render adequate design flexibility for optimal con-
Optimal control involves optimizing an objective trol problems. Since fractional derivatives can
function or performance index subject to state be adjusted to obtain the required performance,
constraints modeled by a dynamical system in as in lesser response times, better stability, and
order to improve its performance. See refs. 1–3 improved disturbance rejection. Fractional or-
If the system is modeled by fractional order dy- der systems exhibit more accuracy in modeling
namics then it is known an optimal control prob- problems involving non-integer behavior, which is
lem of fractional order. These problems have re- prevalent in the real world. 7–12 Fractional-order
cently gained significant attention due to their problems achieve better robust optimal control
many advantages over traditional integer-order in systems involving parameter disturbances and
control problems. Fractional order systems can uncertainties compared to integer-order counter-
be modeled easily by several definitions of frac- parts, as the former captures systems behavior
tional derivatives, but generally, Riemann and more appropriately. Fractional order models aid
Caputo are mostly used definitions. Readers can in reducing energy consumption by optimizing
refer to refs. 4–6 for more details. Optimal control control efficiently, which in turn leads to lesser
of fractional order systems exhibit many advan- energy consumption and enhanced sustainabil-
tages that are lagging in classical integer-order ity. Fractional order models aid in increasing
*Corresponding Author
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