New discussion about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2

Abstract

In this manuscript, our main focus is about the approximate controllability of fractional stochastic differential inclusions with order 1 < r < 2. With the utilization of the fractional calculations, the findings and facts associated with multivalued maps, the fixed‐point method, and the principal discussions are demonstrated. First, we focus on the approximate controllability of our system. Lastly, we provide an application for the demonstration of theoretical results.

Optimal control and approximate controllability for fractional integrodifferential evolution equations with infinite delay of order r∈(1,2)

“…For more specifics, refer to books [1][2][3][4][5][6] and the research papers. [7][8][9][10][11][12][13] The numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator is proved in Reference 14.…”

“…In recent year, approximate controllability of systems described from impulsive functional inclusions, integrodifferential equations, semilinear functional equations, neutral functional differential equations and evolution inclusions have been thought in many researchers, one can refer to the researcher's articles. [9][10][11][37][38][39] Furthermore, integrodifferential systems are used in a variety of scientific fields where an aftereffect or delay must be considered, such as biology, control theory, ecology, and medicine. In practice, integrodifferential systems are always used to describe a model that has hereditary features, for more details, refer to articles.…”

“…So, recently, many researchers have done valuable performances in electromagnetic, control theory, signal, porous media, viscoelasticity, biological, engineering problems, image processing, fluid flow, diffusion, theology, and so on. For more specifics, refer to books 1‐6 and the research papers 7‐13 . The numerical approximation of the solutions of a class of abstract parabolic time optimal control problems with unbounded control operator is proved in Reference 14.…”

“…The notation of exact and approximate controllability has performed as an important tool in analysis and design control systems. In recent year, approximate controllability of systems described from impulsive functional inclusions, integrodifferential equations, semilinear functional equations, neutral functional differential equations and evolution inclusions have been thought in many researchers, one can refer to the researcher's articles 9‐11,37‐39 …”

Results on approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1 < r < 2

“…Very recently, He et al 41 presented the way of finding the mild solution for the fractional system of order 1 < 𝛼 < 2 using sine and cosine functions of operators, fractional calculus, and fixed point technique. In previous studies, [42][43][44][45][46][47] the authors used the approach described in He et al 41 to demonstrate the existence and controllability discussion regarding the fractional evolution system of order 𝛼 ∈ (1, 2) with or without delay. Meanwhile, Volterra and Fredholm integro-differential equations are significant in the analysis of parabolic boundary value problems, population dynamics, mathematical modeling of epidemic Spatio-temporal growth, and numerous physical and biological models that can be affirmed.…”