Page 112 - IJOCTA-15-3

U. Arora, S. Singh, V. Vijayakumar, A. Shukla / IJOCTA, Vol.15, No.3, pp.483-492 (2025)
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Kunze, Debnath, George, Klamka, 6–8 Lions, 9 principle, T-controllability ensures that an ap-
George et al., 10 Nandakumaran and George, 11 propriate control function can be used to lead a
Miller, 12 Cardetti and Gordina 13 Muslim and dynamical system from an arbitrary start state to
Kumar, 14 Shukla et al., 15 Shukla et al., 16 Wang et a final state with a predefined trajectory. As a re-
al., 17 Jajarmi and Baleanu, 18 Kumar and Malik, 19 sult, T-controllability is a more powerful concept
Liu and Li, 20 Muslim and Kumar, 14 Dineshkumar of controllability. The T-controllability of inte-
et al., 21 Mahmudov et al. 22 Mahmudov et al., 23 ger order semilinear integrodifferential systems
Shukla et al., 24 Vijayakumar. 25–27 in finite and infinite dimensional settings was es-
tablished by Chalishajar et al. 37 and Chalishajar
In the setting of fractional systems having or- et al. 38 In 39 the authors established approximate
der ν ∈ (1, 2], Matar, 28 Wang and Zhou, 29 Zhou controllability of Sobolev-type fuzzy Hilfer frac-
and Jiao 30 established sufficient criteria for the tional systems using fractional calculus, Clarke
fractional-order evolution equations in the case subdifferentials, and Dhage’s theorem. 40 This
of fractional systems of order ν ∈ (0, 1]. In manuscript studied the optimal control problem
Kumar and Sukavanam, 31 Sukavanam and Ku- for coupled semilinear wave systems, proved mild
mar, 32 the authors provided adequate conditions solution existence, optimal pairs, and time opti-
for the approximate controllability of fractional- mal control. However, because of their nonlocal
order semilinear delay control systems utilizing nature, fractional differential equations are more
fixed point techniques, based on existence re- efficient in representing real-life processes than
sults for mild solutions. In the setting of frac- classical differential equations. As a result, inves-
tional systems having order ν ∈ (1, 2], Kexue tigating T-controllability discussion for several
et al. 33 discussed the existence and exact con- types of semilinear fractional dynamical systems
trollability of nonlocal fractional systems of or- is fascinating. The goal of our research is to attain
der ν ∈ (1, 2] in infinite-dimensional spaces by the T-controllability of fractional semilinear inte-
employing Sadovskii’s fixed point theorem. In 34 grodifferential symmetries, which is motivated by
explored mild solutions and approximate control- the previous work. This study explores trajectory
lability of semilinear neutral integro-differential controllability (T-controllability) for fractional-
equations in Banach spaces, utilizing resolvent order semilinear integro-differential systems, ad-
families, optimal control, and sufficient conditions dressing both ν ∈ (0, 1] and ν ∈ (1, 2]. By lever-
under Lipschitz-type assumptions. In 35 estab- aging Caputo derivatives, monotone nonlinearity,
lished mild solution existence, uniqueness, and and coercivity conditions, it establishes explicit
trajectory controllability of conformable Hilfer control functions using Gronwall’s inequality and
fractional stochastic systems using semigroup the- fixed-point methods. Unlike traditional control-
ory, stochastic analysis, and Banach fixed-point lability, this approach ensures the system follows
theorem. Shukla et al. 36 recently used a sequen- a predefined trajectory. The paper extends frac-
tial technique to establish approximate controlla- tional control theory and provides practical ex-
bility of fractional system of order ν ∈ (1, 2] with amples, demonstrating its relevance in systems
indefinite delay. In recent years, a new concept with memory effects and nonlinear dynamics.
of controllability known as trajectory controlla- The following is the outline for this pa-
bility (T-controllability) has developed as a new per. Under certain necessary conditions, the T-
area of research, describing the path along which controllability of the fractional semilinear inte-
our system goes while under control. There is a grodifferential equations having order ν ∈ (0, 1]
scarcity of the literature on T-controllability. We was explored in Section 2. T-controllability of the
strive to discover a control that drives the system fractional semilinear integrodifferential equations
along a predefined trajectory rather than one that having order ν ∈ (1, 2] was explored in Section 3.
steers from a given start state to the final state We employ the instruments of monotone nonlin-
in T-controllability challenges. Depending on the earities along with coercivity property to demon-
course chosen, it may reduce some of the costs strate this. Certain examples are presented to
associated with steering the system. demonstrate the applicability of the abstract the-
ory in the final section.
George introduced the concept of T-
controllability for one-dimensional nonlinear sys-
2. Fractional differential systems of
tems in 1996. For example, when a rocket is
order ν ∈ (0, 1]
launched into orbit, it may be necessary to have
a precise course along with the targeted destina- The primary target of this section is to present the
tion in order to save money and avoid collisions. T-controllability of the fractional semilinear inte-
It could also be used to protect the system. In grodifferential equations having order ν ∈ (0, 1].
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