Approximate Controllability for Functional Equations with Riemann-Liouville Derivative by Iterative and Approximate Method

Abstract

We discuss the functional control systems governed by differential equations with Riemann-Liouville fractional derivative in general Banach spaces in the present paper. First we derive the uniqueness and existence of mild solutions for functional differential equations by the approach of fixed point and fractional resolvent under more general settings. Then we present new sufficient conditions for approximate controllability of functional control system by means of the iterative and approximate method. Our results unify and generalize some previous works on this topic.

Non‐instantaneous impulsive Riemann–Liouville fractional differential systems: Existence and controllability analysis

“…Numerous books and articles closely illustrate the findings of existence and controllability for different types of integer and fractional order mathematical models involving Caputo and Riemann–Liouville derivatives. (refer to earlier studies [1–32, 33–45] and references therein). Amidst them, previous studies [12, 37, 40] established exact controllability of fractional differential models by reshaping the controllability problem to the fixed point problem.…”

Approximate controllability of higher order Riemann–Liouville fractional systems via cosine family

“…Under integral contractor assumption of nonlinear operator, authors discussed the existence and controllability of Riemann–Liouville fractional differential equations [35]. In [36], authors proved the controllability of nonlinear Riemann–Luouville fractional equation involving different initial condition by using the concept of $$$ \varsigma $$$‐order resolvent. Mahmudov and McKibben [37] determined the approximate controllability of semilinear systems having generalized Riemann–Liouville derivative.…”

Existence and controllability of higher‐order nonlinear fractional integrodifferential systems via fractional resolvent

“…The existence and controllability results for various types of systems are proved by many researchers 21–52,53–61 . In recent years, the controllability properties of Caputo fractional systems have been extensively studied.…”

“…$$ in Banach spaces using the theory of $$$ {C}_0 $$$‐semigroup together with the probability density function. Ibrahim et al 37 determined the existence and controllability results for the same system with the initial condition $$$ {\lim}_{t\to {0}^{+}}\Gamma \left(\kappa \right){t}^{1-\kappa }z(t)={y}_0 $$$ using the concept of $$$ \kappa $$$‐order resolvent rather than $$$ {C}_0 $$$‐semigroup. Mahmudov and McKibben 38 determined the approximate controllability of fractional systems with generalized Riemann–Liouville derivatives.…”