Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Abstract

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

“…The conception of controllability was first initiated and established by Kalman [13] in 1960, and since then it is the matter of prime importance for the researchers worldwide. The results of existence and controllability for various differential systems of integer and fractional order involving Riemann-Liouville and Caputo derivatives have closely been demonstrated in many artefacts (refer [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]and references therein).…”

Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

“…The conception of controllability was first initiated and established by Kalman [13] in 1960, and since then it is the matter of prime importance for the researchers worldwide. The results of existence and controllability for various differential systems of integer and fractional order involving Riemann-Liouville and Caputo derivatives have closely been demonstrated in many artefacts (refer [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]and references therein).…”

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…Compared with some recent results in the literature, such as [6,[8][9][10][11][12][13] and some others, the chief contributions of our study contain at least the following four issues:…”

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

“…We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non‐instantaneous.) It is stated in Agarwal et al 27 that the solution x ( t ) of the fractional differential Equation () satisfies the following fractional integral equation: …”

“…As far as we can ascertain, the first studies of these appear to have been in 2014 23,24 and 2015. 25 Research into non-instantaneous impulsive fractional differential equations (NIFDEs) has continued in the last few years, as can be seen from the papers of Agarwal, Hristova, O'Regan, et al [26][27][28][29][30] and the papers of Zada, Ali, et al [31][32][33] The purpose of this work is to investigate the different approaches that have been used to study NIFDEs in the works cited above. We discover that the ways of rewriting the differential equation as an equivalent integral equation fall into two general approaches, according to the lower bound used in the integral equation, and that these approaches have been applied to two different general types of NIFDEs, according to the lower limit of the fractional derivative.…”

“…where q ∈ (0, 1) is the order of the Caputo fractional derivative and 0 ≤ t 0 < s 0 < t 1 < s 1 < t 2 < s 2 < … are the points forming the subdivision of the interval J = [0, ∞) into subintervals. (We have adjusted notation here from that used in, 27 for consistency within the current work. We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non-instantaneous.)…”

Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

“…The conception of controllability was first initiated and established by Kalman [13] in 1960, and since then it is the matter of prime importance for the researchers worldwide. The results of existence and controllability for various differential systems of integer and fractional order involving Riemann-Liouville and Caputo derivatives have closely been demonstrated in many artefacts (refer [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]and references therein).…”

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…Compared with some recent results in the literature, such as [6,[8][9][10][11][12][13] and some others, the chief contributions of our study contain at least the following four issues:…”

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

“…We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non‐instantaneous.) It is stated in Agarwal et al 27 that the solution x ( t ) of the fractional differential Equation () satisfies the following fractional integral equation: …”

“…As far as we can ascertain, the first studies of these appear to have been in 2014 23,24 and 2015. 25 Research into non-instantaneous impulsive fractional differential equations (NIFDEs) has continued in the last few years, as can be seen from the papers of Agarwal, Hristova, O'Regan, et al [26][27][28][29][30] and the papers of Zada, Ali, et al [31][32][33] The purpose of this work is to investigate the different approaches that have been used to study NIFDEs in the works cited above. We discover that the ways of rewriting the differential equation as an equivalent integral equation fall into two general approaches, according to the lower bound used in the integral equation, and that these approaches have been applied to two different general types of NIFDEs, according to the lower limit of the fractional derivative.…”

“…where q ∈ (0, 1) is the order of the Caputo fractional derivative and 0 ≤ t 0 < s 0 < t 1 < s 1 < t 2 < s 2 < … are the points forming the subdivision of the interval J = [0, ∞) into subintervals. (We have adjusted notation here from that used in, 27 for consistency within the current work. We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non-instantaneous.)…”

Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

“…The conception of controllability was first initiated and established by Kalman [13] in 1960, and since then it is the matter of prime importance for the researchers worldwide. The results of existence and controllability for various differential systems of integer and fractional order involving Riemann-Liouville and Caputo derivatives have closely been demonstrated in many artefacts (refer [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]and references therein).…”

Existence and finite-time stability of solutions for a class of nonlinear fractional differential equations with time-varying delays and non-instantaneous impulses

“…Compared with some recent results in the literature, such as [6,[8][9][10][11][12][13] and some others, the chief contributions of our study contain at least the following four issues:…”

On non‐instantaneous impulsive fractional differential equations and their equivalent integral equations

“…We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non‐instantaneous.) It is stated in Agarwal et al 27 that the solution x ( t ) of the fractional differential Equation () satisfies the following fractional integral equation: …”

“…As far as we can ascertain, the first studies of these appear to have been in 2014 23,24 and 2015. 25 Research into non-instantaneous impulsive fractional differential equations (NIFDEs) has continued in the last few years, as can be seen from the papers of Agarwal, Hristova, O'Regan, et al [26][27][28][29][30] and the papers of Zada, Ali, et al [31][32][33] The purpose of this work is to investigate the different approaches that have been used to study NIFDEs in the works cited above. We discover that the ways of rewriting the differential equation as an equivalent integral equation fall into two general approaches, according to the lower bound used in the integral equation, and that these approaches have been applied to two different general types of NIFDEs, according to the lower limit of the fractional derivative.…”

“…where q ∈ (0, 1) is the order of the Caputo fractional derivative and 0 ≤ t 0 < s 0 < t 1 < s 1 < t 2 < s 2 < … are the points forming the subdivision of the interval J = [0, ∞) into subintervals. (We have adjusted notation here from that used in, 27 for consistency within the current work. We also assume that s i − 1 < t i instead of s i − 1 ≤ t i so that this impulsive problem is definitely non-instantaneous.)…”