Compact finite-difference method for 2D time-fractional convection–diffusion equation of groundwater pollution problems

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

“…g is a given nonlinear item satisfying some appropriate hypotheses and The theory of fractional calculus has a long-standing history, and has received considerable attention due mainly to its potential and wide applications in various fields, such as viscoelasticity, signal processing, pure mathematics, control, electromagnetics, etc. (see [1][2][3][4][5][6][7]). In the modeling of many phenomena in various science and technology fields, fractional differential equations, including both ordinary and partial ones, are considered to be more powerful tools than their corresponding integer-order counterparts.…”

Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

“…Fractional calculus has wide applications in many fields including anomalous diffusion processes [1][2][3], control theory [4][5][6][7][8], fractional-order neural networks [9], biomedical applications [10,11], mechatronics [12,13], etc. In the past decades lots of works [14][15][16][17][18][19] are devoted to develop numerical methods or algorithms for fractional differential equations. In recent years optimal control problems governed by different types of fractional differential equations have attracted increasing attentions [20][21][22][23][24][25][26][27][28][29][30].…”

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

“…For the sake of using fixed-point theorem, the controllability problem of nonlinear systems is transformed to a fixed-point problem of corresponding nonlinear operator in a appropriate function space. Frequently used fixed-point theorems include Banach's fixed-point theorem [12], Schauder's fixed-point theorem [21,29,129], Darbo's fixed-point theorem [16,26], Schaefer's fixed-point theorem [16], Krasnoselskii's fixed-point theorem [59,100,116], Sadovskii's fixed-point theorem [40,68,122], Mönch's fixed-point theorem [14,32,54,123,125], etc. It should be particularly noted that the controllability of fractional evolution systems (FESs) is an important issue for lots of practical problems since the fractional calculus can derive better results than integeral order one.…”

“…This section mainly lists some necessary notations, definitions and lemmas, which will be used throughout the present paper. The preliminaries here can be found in, for example, [40,72,73].…”

“…For equations with unbounded operators in infinite-dimensional space, the resolvent operator is more appropriate because it is a direct generalization of C 0semigroups and cosine families. From the viewpoint of dependent parameters, it can be generally classified into two types: univariate resolvent operator [40,68,129] and bivariate resolvent operator [6,12,111]. Definition 6.1.…”