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Alexander Domoshnitsky

Publications and source records attributed to Alexander Domoshnitsky.

8 recordsLinked to original sources
arXiv2026

Integro-differential equations in angular stabilization of drone motion by distributed feedback control

In this paper, we propose angular stabilization of drone motion using distributed feedback control in the form of an integral operator. It should be stressed that the memory of this integral operator could be unbounded. It is intuitively clear that large length of the observation time open new possibilities to construct better control based on previous states of the control object. Unbounded memory in control requires the creation of a certain approach different from standard ones to the study of integro-differential equations. One of the goals of this article is to propose a certain universal approach that allows us to study the stability of integro-differential equations in the case of unbounded memory in the integral operator specifying the feedback control in stabilization. The approach we propose allows us to reduce the study of integro-differential equations to the analysis of systems of ordinary differential equations. In general, such systems can consist of an infinite number of equations. In relation to the so-called linear approximation in the problem of angle stabilization manages to limit itself to relatively simple exponential kernels in the integral control and arrive at a system with a finite number of equations. The examples explain that more complex kernels, for example, linear combinations of the exponential kernels, can enhance the stabilization capabilities. We obtain new unexpectable results on the exponential stability of integro-differential equations. Then we apply them to stabilization of drone flight.

cs.AI↗
arXiv2026

On exponential stability of linear and nonlinear delay differential equations: a review and new results

An extensive overview of existing criteria, as well as some new uniform exponential stability tests are included for a scalar delay equation $$ \dot{x}(t)+ \sum_{j=1}^n a_j(t)x(h_j(t))=0. $$ Both cases of continuous and measurable parameters $h_j$, $a_j$ are explored. We apply the global linearisation approach and employ linear results to explore global exponential stability for nonlinear models of the form $$ \dot{x}(t)+\sum_{j=1}^n f_j\left( t,x(h_j(t)) \right) =0. $$ The proofs are based on solution estimations. Further, the Bohl-Perron theorem on exponential dichotomy is instrumental for establishing global exponential stability for nonlinear models. Conclusions are illustrated with numerical examples.

math.DS↗
arXiv2026

Exponential stability of second order delay differential equations through Floquet theory

In this paper, we obtain results on exponential stability of second order delay differential equations, which are based on a version of the Floquet theory for delay differential equations of the second order we proposed. Our version allows researchers to preserve the order of equation and to obtain analogues of the classical results of the Floquet theory known for ordinary differential equations. On the basis of our version of the Floquet theory, new original unexpected results on the exponential stability are proposed. We demonstrate that choosing period of coefficients and delays of the gain in corresponding intervals allows to achieve the exponential stabilization in the cases considered as impossible when the standard technique was applied.

math.DS↗
arXiv2025

Existence of periodic solution of a non-autonomous allelopathic phytoplankton model with fear effect

In this paper, we consider a non-autonomous allelopathic phytoplankton competition ODE model, incorporating the influence of fear effects observed in natural biological phenomena. Based on Mawhin's coincidence degree theory some sufficient conditions for existence of periodic solutions are obtained. We validate our findings through an illustrative example and numerical simulations, showing that constant coefficients lead to steady-state dynamics, while periodic variations induce oscillatory behavior.

math.DS↗
arXiv2023

Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-τ(t))=0,~~t\geq s\text{ }\ \end{equation*} where $τ:\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}% \rightarrow \mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$

math.DS↗
arXiv2020

Marchuk's models of infection diseases: new developments

We consider mathematical models of infection diseases built by G.I. Marchuk in his well known book on immunology. These models are in the form of systems of ordinary delay differential equations. We add a distributed control in one of the equations describing the dynamics of the antibody concentration rate. Distributed control looks here naturally since the change of this concentration rather depends on the corresponding average value of the difference of the current and normal antibody concentrations on the time interval than on their difference at the point t only.

q-bio.CB↗
arXiv2008

Nonoscillation and Stability of the Second Order Ordinary Differential Equations with a Damping Term

In this paper we consider the linear ordinary equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[α,β]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), unique solvability of the interpolation problems, kernels oscillation, separation of zeros, zones of Lyapunov's stability and some others have a certain common basis - nonoscillation. Presumably Sturm was the first to consider the two problems which naturally appear here: to develop corollaries of nonoscillation and to find methods to check nonoscillation. In this paper we obtain several tests for nonoscillation on the semiaxis and apply them to propose new results on asymptotic properties and the exponential stability of the second order equation (2). Using the Floquet representations and upper and lower estimates of nonoscillation intervals of oscillatory solutions we deduce results on the exponential and Lyapunov's stability and instability of equation (2).

math.DS↗