Snezhana Hristova

Snezhana Hristova
Plovdiv University "Paisii Hilendarski" · Department of Applied Mathematics and Modeling

PhD, DSci, Professor
impulsive differential equations, fractional differential equations, delay, non-instantaneous impulses, stability

About

215
Publications
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1,730
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Additional affiliations
August 2014 - May 2015
Denison University
Position
  • viitin Assoc. Professor

Publications

Publications (215)
Article
Full-text available
The synchronization problem for impulsive fractional-order Cohen–Grossberg neural networks with generalized proportional Caputo fractional derivatives with changeable lower limit at any point of impulse is studied. We consider the cases when the control input is acting continuously as well as when it is acting instantaneously at the impulsive times...
Article
Full-text available
The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differentia...
Article
Full-text available
This paper deals with multi-agent systems that, due to using the generalized proportional Caputo fractional derivative, possess memories. The information exchange between agents does not occur continuously but only at fixed given update times, and the lower limit of the fractional derivative changes according to the update times. Two types of multi...
Article
Full-text available
Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on...
Conference Paper
Full-text available
This paper is dedicated to studying the existence and uniqueness of solutions to a system of fractional differential equations with anti-periodic fractional integral boundary conditions in the frame of the generalized proportional fractional derivatives of Riemann-Liouville type. With the aid of the Schaefer fixed point theorem and the Banach fixed...
Article
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A scalar nonlinear integro-differential equation with time-variable and bounded delays and generalized Caputo proportional fractional derivative is considered. The main goal of this paper is to study the stability properties of the zero solution. Results are given concerning stability, exponential stability, asymptotic stability, and boundedness of...
Article
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A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functi...
Article
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In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some compari...
Article
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Caputo fractional differential equations with impulses are a very useful apparatus for adequate modeling of the dynamics of many real world problems. It requires developments of good and consistent theoretical proofs and the results for various problems. In this note we point out and correct the statement of the boundary value problem with Riemann–...
Article
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A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordin...
Article
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Difference equations with a special type of delay is studied. This type of delay is characterized by the maximum value of the unknown function on the past time interval with a constant length. It is studied the exponential stability. Some sufficient conditions are theoretically proved and computer simulated on several examples. The influence of the...
Article
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Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riem...
Article
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A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alte...
Article
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Initial and impulsive conditions for initial value problems of systems of nonlinear impulsive Riemann-Liouville fractional differential equations are introduced. The case when the lower limit of the fractional derivative is changed at each time point of the impulses is studied. In the case studied, the solution has a singu-larity at the initial tim...
Article
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The present paper is concerned with the existence of solutions of a new class of nonlinear generalized proportional fractional differential inclusions with the right-hand side contains a Carathèodory-type multi-valued nonlinearity on infinite intervals. The investigation of the proposed inclusion problem relies on the multi-valued form of Leray-Sc...
Article
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The main aim of the paper is to present an algorithm to solve approximately initialvalue problems for a scalar non-linear fractional differential equation with generalized proportionalfractional derivative on a finite interval. The main condition is connected with the one sidedLipschitz condition of the right hand side part of the given equation. A...
Article
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The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of sol...
Article
Full-text available
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied where initial conditions and impulsive conditions are set up in appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equat...
Article
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Nonlinear delay Caputo fractional differential equations with non-instantaneous impulses are studied and we consider the general case of delay, depending on both the time and the state variable. The case when the lower limit of the Caputo fractional derivative is fixed at the initial time, and the case when the lower limit of the fractional derivat...
Article
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A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value pr...
Article
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In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several...
Article
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The synchronization is one of the main problems connected with neural networks. We examine a model of a neural network with time varying delay, and also the case when the connection weights (the influential strength of the j-th neuron to the i-th neuron) are variable in time and unbounded. The rate of change of the dynamic of all neurons is describ...
Article
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A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapun...
Article
Full-text available
In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time po...
Conference Paper
Ulam type stability concepts for non-instantaneous impulsive difference equations are introduced and studied. Sufficient conditions for Ulam-Hyers stability, generalized Ulam-Hyers-Rassias stability and Ulam- Hyers-Rassias stability are obtained. Also some conditions for instability are provided. Some of the obtained results are illustrated on exam...
Conference Paper
The stability in terms of two different measures for differential equations with supremum and non-instantaneous impulses is studied using Lyapunov like functions and Razumikhin technique. In these differential equation we have impulses, which start abruptly at some points and their action continue on given finite intervals. An example is given to i...
Conference Paper
Practical stability for nonlinear differential equations with non-instantaneous impulses and supremum is studied. The impulses start abruptly at some points and their actions continue on given finite intervals. Some sufficient conditions for practical stability and strong practical stability are obtained. An example is given to illustrate some of t...
Conference Paper
Linear Riemann-Liouville fractional differential equations with a constant delay and non-instantaneous impulses are studied. Both cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point of i...
Conference Paper
Full-text available
In this paper, we study a class of nonlocal functional evolution inclusions in the form of multivalued perturbations of m–dissipative operators. We prove some new existence results in case of dissipative type conditions on the right-hand side. Illustrative example is provided.
Conference Paper
Lipschitz stability for nonlinear differential equations with non-instantaneous impulses and supremum of the unknown function over a past time interval is studied. The impulses start abruptly at some points and their action continue on given finite intervals. Some sufficient conditions for Lipschitz stability are obtained. Modified Razumikhin metho...
Conference Paper
In this paper we study nonlocal semilinear fractional evolution inclusions involving Riemann-Liouville derivative with right-hand side depending on casual operator. The existence of solutions is proved under compactness type conditions on the multivalued term.
Conference Paper
Caputo fractional differential equations of order q ∈ (1, 2) with impulses at random moments are set up and investigated in this paper. The main characteristic of the studied equations is that the impulses occur at random moments which are exponentially distributed random variables. It leads to a total change of the behavior of the solution which i...
Conference Paper
A system of nonlinear Riemann-Liouville fractional differential equations with a constant delay is studied. The initial condition is set up similarly to the case of ordinary derivative. Sufficient conditions for finite time stability are obtained.
Conference Paper
In this paper, the direct Lyapunov method is extended to fractional case of Volterra integro-Caputo fractional differential equations with delays. Two types of homogeneous scalar Voterra integro-Caputo fractional differential equations with delays are studied. Three types of fractional derivatives of Lyapunov functions known in the literature are g...
Article
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First, we set up in an appropriate way the initial value problem for nonlinear delay differential equations with a Riemann-Liouville (RL) fractional derivative. We define stability in time and generalize Mittag-Leffler stability for RL fractional differential equations and we study stability properties by an appropriate modification of the Razumikh...
Article
Full-text available
In this paper a model of Hopfield’s graded response neural network is investigated. A network whose neurons are subject to a certain impulsive state displacement at random times is considered. The model is set up and studied. The presence of random moments of impulses in the model leads to a change of the solutions to stochastic processes. Also, we...
Article
A nonlinear Riemann–Liouville fractional differential equation with constant delay is studied. Initially, some existence results are proved. Three Ulam type stability concepts are defined and studied. Several sufficient conditions are obtained. Some of the obtained results are illustrated on fractional biological models.
Article
Full-text available
The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time‐varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, expli...
Article
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The bounded input bounded output (BIBO) stability for a nonlinear Caputo fractional system with time-varying bounded delay and nonlinear output is studied. Utilizing the Razumikhin method, Lyapunov functions and appropriate fractional derivatives of Lyapunov functions some new bounded input bounded output stability criteria are derived. Also, expli...
Article
Full-text available
Fractional differential equations with impulses arise in modeling real world phenomena where the state changes instantaneously at some moments. Often, these instantaneous changes occur at random moments. In this situation the theory of Differential equations has to be combined with Probability theory to set up the problem correctly and to study the...
Article
Full-text available
The initial value problem for a special type of scalar nonlinear fractional differential equation with a Riemann–Liouville fractional derivative is studied. The main characteristic of the equation is the presence of the supremum of the unknown function over a previous time interval. This type of equation is difficult to be solved explicitly and we...
Article
Full-text available
The main objective of this paper is to introduce the (α,β)-type ϑ-contraction, (α,β)-type rational ϑ-contraction, and cyclic (α-ϑ) contraction. Based on these definitions we prove fixed point theorems in the complete metric spaces. These results extend and improve some known results in the literature. As an application of the proved fixed point The...
Article
Full-text available
Riemann-Liouville fractional differential equations with a constant delay and impulses are studied in this article. The following two cases are considered: the case when the lower limit of the fractional derivative is fixed on the whole interval of consideration and the case when the lower limit of the fractional derivative is changed at any point...
Article
Full-text available
This paper studies the leader-following consensus problem in continuous-time multi-agent networks with communications/updates occurring only at random times. The time between two consecutive controller updates is exponentially distributed. Some sufficient conditions are derived to design the control law that ensures the leader-following consensus i...
Article
Full-text available
Abstract A system of linear Riemann–Liouville fractional differential equations with constant delay is studied. The initial condition is set up similar to the case of the ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.
Article
Full-text available
Non-linear scalar Riemann-Liouville fractional differential equation with a constant delay is studied on a finite interval. An initial value problem is set up in appropriate way combining the idea of the initial time interval in ordinary differential equations with delays and the properties of Riemann- Liouville fractional derivatives. The mild sol...
Article
Full-text available
Four Ulam type stability concepts for non-instantaneous impulsive fractional differential equations with state dependent delay are introduced. Two different approaches to the interpretation of solutions are investigated. We study the case of an unchangeable lower bound of the Caputo fractional derivative and the case of a lower bound coinciding wit...
Article
Full-text available
Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the...
Article
Full-text available
The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is...
Article
Full-text available
The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of t...
Article
Full-text available
A discrete Hopfield-type neural network with constant delays, instantaneous switching topologies at certain times and time variable connection weights is studied. Some criteria for exponential stability are derived. The obtained results are illustrated on an example with different activation functions such as tanh, Swish, and the error function. Th...
Article
Full-text available
The main aim of this paper is to suggest some algorithms and to use them in an appropriate computer environment to solve approximately the initial value problem for scalar nonlinear Riemann–Liouville fractional differential equations on a finite interval. The iterative schemes are based on appropriately defined lower and upper solutions to the give...
Article
Full-text available
In this paper we consider systems of functional differential equations whose dynamics depends on the maximum value of solution over a prehistory time interval. Such kind of systems are infinite-dimensional and nonlinear. We consider controlled systems with maximum and study their input-to-state stability property. As well we compare stability prope...
Article
Full-text available
In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.
Conference Paper
The paper deals with some stability properties of the solutions of impulsive differential equations with supremum. The main characteristic of the impulses in the system is their duration- the impulsive action starts at an arbitrary fixed point and remains active on a finite time interval. Note the impulsive differential systems originate from the r...
Conference Paper
Full-text available
We consider the Hopfield’s graded response neural network in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (they are known as non-instantaneous impulses). We examine the case when the present state of any neuron depends on its maximum va...
Article
Full-text available
In this paper some practical stability results for nonlinear differential equations with non-instantaneous impulses and state dependent delays are presented. The impulses start abruptly at some points and their action continue on given finite intervals. The delay depends on both the time and the state variable which is a generalization of time vari...
Conference Paper
Recent modeling of real world phenomena give rise to fractional differential equations with non-instantaneous impulses. The main goal of this paper is to provide an existence and uniqueness results for Riemann-Liouville fractional differential equation with non-instantaneous impulses. It is studied the case when the lower bound of the Riemann-Liouv...
Conference Paper
Recent modeling of real world phenomena give rise to fractional differential equations with non-instantaneous impulses. The main goal of the paper is to highlight basic points in introducing non-instantaneous impulses in Riemann-Liouville fractional differential equations. The case when the lower limit of the fractional derivative is changed at any...
Article
Full-text available
A leader-following consensus of discrete-time multi-agent systems with nonlinear intrinsic dynamics and impulses is investigated. We propose and prove conditions ensuring a leader-following consensus. Numerical examples are given to illustrate effectiveness of the obtained results. Also, the necessity and sufficiency of the obtained conditions are...
Article
Full-text available
A leader-following consensus for Caputo fractional multi-agent systems with nonlinear intrinsic dynamics is investigated. The second Lyapunov method is used to design a control protocol ensuring consensus for two types of multi-agent systems. Contrary to the previous studies on leader-following consensus , the investigation covers systems with boun...
Article
Full-text available
Practical stability properties of Caputo fractional delay differential equations is studied and, in particular, the case with state dependent delays is considered. These type of delays is a generalization of several types of delays such as constant delays, time variable delays, or distributed delays. In connection with the presence of a delay in a...
Article
Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions an...
Article
Full-text available
The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is bas...
Article
Full-text available
In this paper a nonlinear system of Riemann–Liouville (RL) fractional differential equationswith non-instantaneous impulses is studied. The presence of non-instantaneous impulses requireappropriate definitions of impulsive conditions and initial conditions. In the paper several types ofinitial value problems are considered and their mild solutions...
Preprint
A leader-following consensus for Caputo fractional multi-agent systems with nonlinear 1 intrinsic dynamics is investigated. The second Lyapunov method is used to design a control 2 protocol ensuring consensus for two types of multi-agent systems. Contrary to the previous 3 studies on leader-following consensus, investigation covers systems with bou...
Article
Full-text available
We use Lyapunov functions to study stability of the first-order Volterra integro-differential equation with Caputo fractional derivative C t 0 D q t x(t) = −a(t)f (x(t)) + Z t t−r B(t, s)g(s, x(s))ds + h(t, x(t), x(t − τ (t))). For the Lyapunov functions, we consider three types of fractional derivatives. By means of these derivatives, we obtain ne...
Article
In this paper, we consider neural networks in the case when the neurons are subject to a certain impulsive state displacement at fixed moments and the duration of this displacement is not negligible small (these are known as non-instantaneous impulses). We examine some stability properties of the equilibrium of the model. Several sufficient conditi...
Article
Full-text available
The Lipschitz stability for nonlinear differential equations with non-instantaneous impulses and variable delays is studied. The impulses start abruptly at some points and their action continue on given finite intervals. The delay is time variable. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are...
Article
Full-text available
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses...
Conference Paper
Full-text available
The stability in terms of two different measures for delay differential equations with non-instantaneous impulses is studied using Lyapunov like functions and Razumikhin technique. In these differential equation we have impulses, which start abruptly at some points and their action continue on given finite intervals. Examples are given to illustrat...
Conference Paper
Nonlinear differential equations with non-instantaneous impulses are studied. In these differential equations we have impulses, which start abruptly at some points and their action continue on given finite intervals. The stability with respect to part of variables of a nonlinear differential equation with non-instantaneous impulses is studied using...
Article
Full-text available
One approach to study various stability properties of solutions of nonlinear Caputo fractional differential equations is based on using Lyapunov like functions. A basic question which arises is the definition of the derivative of the Lyapunov like function along the given fractional equation. In this paper, several definitions known in the literatu...