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Elena Braverman

Publications and source records attributed to Elena Braverman.

At least 37 records · Page 2Linked to original sources
arXiv2021

Exponential stability of systems of vector delay differential equations with applications to second order equations

Various results and techniques, such as Bohl-Perron theorem, a priori solution estimates, M-matrices and the matrix measure, are applied to obtain new explicit exponential stability conditions for the system of vector functional differential equations $$ \dot{x_i}(t)=A_i(t)x_i(h_i(t)) +\sum_{j=1}^n \sum_{k=1}^{m_{ij}} B_{ij}^k(t)x_j(h_{ij}^k(t)) + \sum_{j=1}^n\int\limits_{g_{ij}(t)}^t K_{ij}(t,s)x_j(s)ds,~i=1,\dots,n. $$ Here $x_i$ are unknown vector functions, $A_i, B_{ij}^k, K_{ij}$ are matrix functions, $h_i,h_{ij}^k, g_{ij}$ are delayed arguments. Using these results, we deduce explicit exponential stability tests for second order vector delay differential equations.

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arXiv2020

Asymptotic properties of neutral type linear systems

Exponential stability and solution estimates are investigated for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$ are delayed arguments. Stability tests are applicable to a wide class of linear neutral systems with time-varying coefficients and delays. In addition, explicit exponential estimates for solutions of both homogeneous and non-homogeneous neutral systems are obtained for the first time. These inequalities are not just asymptotic estimates, they are valid on every finite segment and evaluate both short- and long-term behaviour of solutions.

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arXiv2020

Solution estimates and stability tests for linear neutral differential equations

Explicit exponential stability tests are obtained for the scalar neutral differential equation $$ \dot{x}(t)-a(t)\dot{x}(g(t))=-\sum_{k=1}^m b_k(t)x(h_k(t)), $$ together with exponential estimates for its solutions. Estimates for solutions of a non-homogeneous neutral equation are also obtained, they are valid on every finite segment, thus describing both asymptotic and transient behavior. For neutral differential equations, exponential estimates are obtained here for the first time. Both the coefficients and the delays are assumed to be measurable, not necessarily continuous functions.

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arXiv2020

Stabilization of cycles for difference equations with a noisy PF control

Difference equations, such as a Ricker map, for an increased value of the parameter, experience instability of the positive equilibrium and transition to deterministic chaos. To achieve stabilization, various methods can be applied. Proportional Feedback control suggests a proportional reduction of the state variable at every $k$th step. First, if $k \neq 1$, a cycle is stabilized rather than an equilibrium. Second, the equation can incorporate an additive noise term, describing the variability of the environment, as well as multiplicative noise corresponding to possible deviations in the control intensity. The present paper deals with both issues, it justifies a possibility of getting a stable blurred $k$-cycle. Presented examples include the Ricker model, as well as equations with unbounded $f$, such as the bobwhite quail population models. Though the theoretical results justify stabilization for either multiplicative or additive noise only, numerical simulations illustrate that a blurred cycle can be stabilized when both multiplicative and additive noises are involved.

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arXiv2020

Stabilization of cycles with stochastic prediction-based and target-oriented control

We stabilize a prescribed cycle or an equilibrium of the difference equation using pulsed stochastic control. Our technique, inspired by the Kolmogorov's Law of Large Numbers, activates a stabilizing effect of stochastic perturbation and allows for stabilization using a much wider range for the control parameter than would be possible in the absence of noise. Our main general result applies to both Prediction-Based and Target-Oriented Controls. This analysis is the first to make use of the stabilizing effects of noise for Prediction-Based Control; the stochastic version has been examined in the literature, but only the destabilizing effect of noise was demonstrated. A stochastic variant of Target-Oriented Control has never been considered, to the best of our knowledge, and we propose a specific form that uses a point equilibrium or one point on a cycle as a target. We demonstrate our results numerically on the logistic, Ricker and Maynard Smith models from population biology.

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arXiv2020

Stability and oscillation of linear delay differential equations

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation $$ x^{\prime}(t)+c(t)x(τ(t))=0,~~t\geq 0, $$ where $c$ is locally integrable of any sign, $τ(t)\leq t$ is Lebesgue measurable, $\lim_{t\rightarrow\infty}τ(t)=\infty$, we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the $3/2-$stability criterion to the case of measurable parameters, improving $1+1/e$ to the sharp $3/2$ constant.

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arXiv2020

Global stabilization and destabilization by the state dependent noise with particular distributions

Under natural assumptions, an unstable equilibrium of a difference equation can be stabilized by a bounded multiplicative noise, identically distributed at each step. This includes stabilization of an otherwise unstable positive equilibrium of Ricker, logistic, and Beverton-Holt maps. Introduction of a multiplicative noise also allows to destabilize a stable equilibrium in a sense that all solutions stay away from this point, almost surely. In our examples a noise has symmetric, discrete or continuous, distribution with support $[-1,1]$, including Bernoulli and uniform continuous distribution. We obtain conditions on the noise amplitudes in each case that allow to either stabilize or destabilize an equilibrium. Computer simulations illustrate our results.

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arXiv2020

Solution estimates for linear differential equations with delay

In this paper, we give explicit exponential estimates $\displaystyle |x(t)|\leq M e^{ -γ(t-t_0) }$, where $t\geq t_0$, $M>0$, for solutions of a linear scalar delay differential equation $$ \dot{x}(t)+\sum_{k=1}^m b_k(t)x(h_k(t))=f(t),~~ t\geq t_0,~ x(t)=ϕ(t),~t\leq t_0. $$ We consider two different cases: when $γ>0$ (corresponding to exponential stability) and the case of $γ<0$ when the solution is, generally, growing. In the first case, together with the exponential estimate, we also obtain an exponential stability test, in the second case we get estimation for solution growth. Here both the coefficients and the delays are measurable, not necessarily continuous.

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arXiv2020

On the global attractivity of non-autonomous neural networks with a distributed delay

We consider a system of several nonlinear equations with a distributed delay and obtain absolute asymptotic stability conditions, independent of the delay. The ideas of the proofs are based on the notion of a strong attractor. The results are applied to Hopfield neural networks, Nicholson's blowflies type system, and compartment models of population dynamics.

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arXiv2019

On stability of delay equations with positive and negative coefficients with applications

We obtain new explicit exponential stability conditions for linear scalar equations with positive and negative delayed terms $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))- \sum_{k=1}^l b_k(t)x(g_k(t))=0 $$ and its modifications, and apply them to investigate local stability of Mackey--Glass type models $$\dot{x}(t)=r(t)\left[β\frac{x(g(t))}{1+x^n(g(t))}-γx(h(t))\right]$$ and $$\dot{x}(t)=r(t)\left[β\frac{x(g(t))}{1+x^n(h(t))}-γx(t)\right].$$

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arXiv2019

Explicit stability tests for linear neutral delay equations using infinite series

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ (x(t)-a(t)x(g(t)))'+b(t)x(h(t))=0, $ where $|a(t)| \leq A_0 < 1$, $0<b_0\leq b(t)\leq B_0$, assuming that all parameters of the equation are measurable functions. To analyze exponential stability, we apply the Bohl-Perron theorem and a reduction of a neutral equation to an equation with an infinite number of non-neutral delay terms. This method has never been used before for this neutral equation; its application allowed to omit a usual restriction $|a(t)|<\frac{1}{2}$ in known asymptotic stability tests and consider variable delays.

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arXiv2019

On oscillation of difference equations with continuous time and variable delays

We consider existence of positive solutions for a difference equation with continuous time, variable coefficients and delays $$ x(t+1)-x(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))=0, \quad a_k(t) \geq 0, ~~h_k(t) \leq t, \quad t \geq 0, \quad k=1, \dots, m. $$ We prove that for a fixed $h(t)\not\equiv t$, a positive solution may exist for $a_k$ exceeding any prescribed $M>0$, as well as for constant positive $a_k$ with $h_k(t) \leq t-n$, where $n \in {\mathbb N}$ is arbitrary and fixed. The point is that for equations with continuous time, non-existence of positive solutions with $\inf x(t)>0$ on any bounded interval should be considered rather than oscillation. Sufficient conditions when such solutions exist or do not exist are obtained. We also present an analogue of the Grönwall-Bellman inequality for equations with continuous time, and examine the question when the equation has no positive non-increasing solutions. Counterexamples illustrate the role of variable delays.

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arXiv2019

On the interplay of harvesting and various diffusion strategies for spatially heterogeneous populations

The paper explores the influence of harvesting (or culling) on the outcome of the competition of two species in a spatially heterogeneous environment. The harvesting effort is assumed to be proportional to the space-dependent intrinsic growth rate. The differences between the two populations are the diffusion strategy and the harvesting intensity. In the absence of harvesting, competing populations may either coexist, or one of them may bring the other to extinction. If the latter is the case, introduction of any level of harvesting to the successful species guarantees survival to its non-harvested competitor. In the former case, there is a strip of "close enough" to each other harvesting rates leading to preservation of the original coexistence. Some estimates are obtained for the relation of the harvesting levels providing either coexistence or competitive exclusion.

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arXiv2019

On stability of linear neutral differential equations with variable delays

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $$ where $$ |a(t)|<1,~ b(t)\geq 0, ~h(t)\leq t, ~g(t)\leq t, $$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

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arXiv2019

On stability of linear neutral differential equations in the Hale form

We present new explicit exponential stability conditions for the linear scalar neutral equation with two variable coefficients and delays $$ (x(t)-a(t)x(g(t)))'=-b(t)x(h(t)), $$ where $|a(t)|<1$, $b(t)\geq 0$, $h(t)\leq t$, $g(t)\leq t$, in the case when the delays $t-h(t)$, $t-g(t)$ are bounded, as well as an asymptotic stability condition, if the delays can be unbounded.

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arXiv2019

A new stability test for linear neutral differential equations

We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $ \dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $ where $ 0\leq a(t)\leq A_0<1$, $0<b_0\leq b(t)\leq B$, using the Bohl-Perron theorem and a transformation of the neutral equation into a differential equation with an infinite number of delays. The results are applied to the neutral logistic equation.

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arXiv2019

Stochastic control stabilizing unstable or chaotic maps

The paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value.

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