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

Abstract

The main motivation of our conversation is the approximate controllability of fractional stochastic Sobolev‐type Volterra–Fredholm integro‐differential equation of order 1 < r < 2. Using principles and ideas from stochastic analysis, the theory of cosine family, fractional calculus, and Banach fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions and then demonstrate the approximate controllability of the fractional stochastic control equation. We then apply our findings to the theory of nonlocal conditions. At last, an application is established for drawing the theory of the main results.

Exponential Stability of Highly Nonlinear Hybrid Differently Structured Neutral Stochastic Differential Equations with Unbounded Delays

“…r(t) is the same Markov chain as that in Examples 1 and 2. For i ∈ S, z 1 , z 2 , z 3 ∈ R, set F 1 (z 1 , z 2 , z 3 , t, 1) = −2.4z 1 + 5e −0.3t z 2 ; G 1 (z 1 , z 2 , z2, t, 1) = 0.2e −0.3t z 3 ; F 2 (z 1 , z 2 , z 3 , t, 1) = −z 1 + 3.9e −0.3t z 2 ; G 1 (z 1 , z 2 , z, t, 1) = 0.06e −0.3t z 3 ; F 1 (z 1 , z 2 , z 3 , t, 2) = −2.5z 1 + 1.6e −0.3t z 2 − 0.3e −0.3t z 3 ; G 1 (z 1 , z 2 , z, t, 2) = 1.58z 1 + 0.1e −0.3t z 3 ; F 2 (z 1 , z 2 , z 3 , t, 2) = −3z 1 + 0.85e −0.3t z 2 − 0.5e −0.3t z 3 ; G 2 (z 1 , z 2 , z, t, 2) = 2.7z 1 + 0.3e −0.3t z 3 ; F 1 (z 1 , z 2 , z 3 , t, 3) = −5z 3 1 − 6z 1 − 0.6912e −0.6t z 1 z 2 2 − 0.05e −0.9t z 3 2 ; G 1 (z 1 , z 2 , z, t, 3) = 0.5e −0.6t z 2 3 ; F 2 (z 1 , z 2 , z 3 , t, 3) = −1.7z 3 1 − 5z 1 − 0.7e −0.6t z 1 z 2 2 − 0.2e −0.9t z 3 2 ; G 2 (z 1 , z 2 , z, t, 3) = 1.32e −0.6t z 2 3 .…”

“…In the paper [2], the authors dealt with the complete controllability of a semi-linear stochastic system with delay under the assumption that the corresponding linear system is completely controllable. The paper [3] investigated the approximate controllability of fractional stochastic Sobolev-type Volterra-Fredholm integrodifferential equation of order 1 < r < 2. The paper [4] studied the time fractional system in the Caputo sense of fluid-conveying single-walled carbon nanotubes (SWCNT).…”

Some results on the approximate controllability of impulsive stochastic integro-differential equations with nonlocal conditions and state-dependent delay

“…Therefore, it is important to investigate the problem of approximate controllability of nonlinear systems. Approximate controllability of nonlinear stochastic differential and integrodifferential systems with and without delay in infinite-dimensional spaces has been extensively studied (see [1,2,12,17,30,41,47] for example). Recently, Muthukumar and Rajivganthi in [41] proved the approximate controllability of control systems governed by a class of impulsive neutral stochastic functional differential systems with state-dependent delay in Hilbert spaces.…”

Null Controllability of Hilfer Fractional Stochastic Differential Inclusions

“…The controllability of various deterministic and stochastic control systems has been investigated in many works (see [1][2][3][4][5][6][7][8][9]). It should be emphasized that there are many different notions of controllability for fractiona-evolution systems-for example, approximate controllability, complete controllability, null controllability, and so on.…”

“…where D h, 0+ is the Hilfer fractional derivative, 0 ≤ h ≤ 1, 1 2 < < 1, and is the infinitesimal generator of a compact semigroup {ℵ(ς), ς ≥ 0} in Hilbert space Λ, where sup ς∈T ℵ(ς) ≤ Π, Π > 1.…”