Optimal control results for Sobolev‐type fractional mixed Volterra–Fredholm type integrodifferential equations of order 1 < r < 2 with sectorial operators

Abstract

The goal of this research is to investigate the issue of existence and optimal control results for fractional mixed Volterra-Fredholm type integrodifferential systems of order 1 < r < 2 with sectorial operators. Initially, we use fractional derivatives, semigroup theory, mild solutions, Sobolev-type, and Schauder's fixed point theorem to examine the existence of results. Then, we will move on to the systems with optimal control results. Finally, we show a theoretical application to establish the validity of the findings.

An analysis on approximate controllability results for impulsive fractional differential equations of order 1 < r < 2 with infinite delay using sequence method

“…In [28], researchers used stochastic systems, the measure of noncompactness, control problems, the fixed point theorem of Mönch, nonlocal circumstances, and sectorial operators to derive Riemann–Liouville fractional derivatives of order $$$ 1&amp;amp;lt;q&amp;amp;lt;2 $$$. Additionally, using sectorial operators, integrodifferential systems, multivalued functions, and various fixed point procedures, the researchers in [38, 49, 50] developed the existence, optimal controls, and approximate controllability results for fractional derivatives of order $$$ r\in \left(1,2\right) $$$.…”

Sobolev‐type existence results for impulsive nonlocal fractional stochastic integrodifferential inclusions of order ϱ∈(1,2) with infinite delay via sectorial operator

“…Applications of these equations can be found in various fields, including physics, biology, and engineering, to capture phenomena with long‐range interactions and memory‐dependent dynamics. Many scholarly works, such as those cited in [7, 24, 40, 41], provide valuable perspectives on the examination of differential systems that encompass the existence results of Volterra–Fredholm equations.…”

“…The existence results of this proposed problem are derived by employing Martelli's fixed point approach. Very recently, Mohan Raja et al [50–52] have developed the existence, optimal controls, and approximate controllability results for fractional derivatives of order $$$ r\in \left(1,2\right) $$$ by utilizing the sectorial operators, integrodifferential systems, multivalued functions, and various fixed point techniques. Johnson et al [53, 54] discussed the effectiveness of optimal controllability in fractional order $$$ r\in \left(1,2\right) $$$ by utilizing sectorial operators, stochastic systems, impulsive conditions, and fixed point approaches.…”

Optimal control for Hilfer fractional neutral integrodifferential evolution equations with infinite delay

“…In recent years, fractional‐order control systems defined by fractional‐order differential equations have attracted a lot of attention; contain a long list of these distributions. See References 9 and 19‐39 for further information. Achieving more precise bounds based on double and triple integral as proposed by generalized proportional fractional operators in the Hilfer sense, see Reference 40.…”

“…For a class of fractional‐order integrodifferential systems with infinite delay in a reflexive Banach spaces, Wang et al 45 found the solvability and optimal controls. In Reference 22, the authors obtained optimal control results for Sobolev‐type fractional mixed Volterra–Fredholm type integrodifferential equations of order $$$ 1<r<2 $$$ with sectorial operators. A general formulation and solution scheme for fractional optimal control problems, optimal control problems for a neutral integrodifferential system with infinite delay have been investigated by Agrawal 46 and Huang and Fu 47 and proved whether there are any optimal controls for systems governed by fractional integrodifferential systems with indfinite delay.…”