New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces

Abstract

The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions.

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

“…The control operator is usually assumed to be reversible. Then, the controllability problem is transformed into a fixed point problem [9][10][11][12][13]. Furthermore, an induced inverse of the control operator is not necessarily true in infinite-dimensional space.…”

A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

“…The results are obtained by utilizing Banach contraction mapping theorem due to the Lipschitz conditions of the systems. In addition, some excellent results of exact controllability for various fractional differential equations have also been established recently [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23], but the limitation is also that the functions in the systems are either Lipschitz continuous, compact or satisfy some special growth suppositions. Although the exact controllability studied in [13] does not require the nonlinear term to satify Lipschitz condition, the considered evolution system in [13] have no effects of time delay and impulse.…”

Controllability of nonlinear fractional evolution systems in Banach spaces: A survey

“…After a long and tortuous development, fractional calculus has received considerable attention due mainly to its potential and wide applications in various kinds of scientific fields such as chemical physics, pure mathematics, signal processing, mechanics and engineering, viscoelasticity, biology, neural network model, fractal theory, etc. See [22,55,62,75,96,102,118,120,121] and references therein for further details. Actually, fractional calculus can describe mathematical models involving practical background with less parameters, and present a more vivid and accurate description over things than integral order ones [10,23,38,41,57,91,101,117,130].…”