Controllability of nonlinear fractional Langevin delay systems

Abstract

In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α, β 1. Necessary and sufficient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufficient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders fixed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory.

Existence and uniqueness of solutions of the fractional integro-differential equations in vector-valued function space

“…R-boundedness-UMD space, L p -multiplier and Riemann-Liouville fractional integral. For results on operator-valued Fourier multipliers and R-boundedness (used in the next section), as well as some applications to evolutionary partial differential equations, we refer to Bourgain [5,6], Clément-de Pagter-Sukochev--Witvliet [12], Weis [28,29], Girardi-Weis [17,18], Kunstmann-Weis [23], Clément-Prüss [13], Arendt [2], Arendt-Bu [3], Ataricio-Keyantuo [1] and Suresh [27].…”

Controllability of nonlinear Hilfer fractional Langevin dynamical system

“…Radhakrishnan et al [22][23][24] discussed the controllability results in infinite dimensional space. Also, by using the Grammian matrix in the finite dimensional spaces Sureshkumar et al [26] investigated…”

Existence and Ulam–Hyers stability results for nonlinear fractional Langevin equation with modified argument

“…Recently, the existence‐uniqueness and stability results for nonlinear fractional Langevin equations are studied. We mention here some papers 20–30 …”

“…We mention here some papers. [20][21][22][23][24][25][26][27][28][29][30] This paper deals with the existence and uniqueness of solutions as well as the Ulam-Hyers stability for the following nonlinear fractional-order Langevin equation with a modified argument:…”