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Natalya Sedova

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My research interests are around stability, stabilization, and control theory of dynamical systems. Applications under consideration involve Takagi-Sugeno systems, mechanical systems, communication, population dynamics, economic systems, etc. I have more than 20 years of teaching experience, mathematical physics, mathematical modeling, differential equations. I have more than 60 papers including papers published in international journals, proceedings of international conferences.

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University of Vienna

University of Exeter

Carnegie Mellon University

KU Leuven

Thomas Drakengren

Laurence Jacquet

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Papers by Natalya Sedova

О развитии прямого метода Ляпунова для функционально-дифференциальных уравнений с бесконечным запаздыванием

Математические заметки, 2008

МЕТОД ФУНКЦИЙ ЛЯПУНОВА-РАЗУМИХИНА В ЗАДАЧЕ ОБ УСТОЙЧИВОСТИ СИСТЕМ С ЗАПАЗДЫВАНИЕМ

Автоматика и телемеханика, 2019

Рассмотрена задача стабилизации управляемых систем, описываемых нелинейными функционально-диффере... more Рассмотрена задача стабилизации управляемых систем, описываемых нелинейными функционально-дифференциальными уравнениями запаздывающего типа. Для построения стабилизирующего управления применяются так называемые контролирующие вспомогательные функционалы, свойства которых позволяют установить возможность стабилизации и в некоторых случаях построить стабилизирующие законы управления в явном виде. Обсуждены «обратная оптимальность» и робастность построенных управлений.

Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique

Mathematics

In this article, some new sufficient conditions for the exponential stability of nonlinear time-v... more In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples.

A Remark on the Lyapunov–Razumikhin Method for Equations with Infinite Delay

Differential Equations - DIFFERENTIAL EQUAT-ENGL TR, 2002

The choice of an appropriate phase space plays an important role in the construction of the theor... more The choice of an appropriate phase space plays an important role in the construction of the theory of equations with unbounded (infinite) delay, since it affects even the results concerning the existence and uniqueness of solutions (see the survey [1]). An axiomatic approach to this problem was developed in [2], where a number of axioms dealing with the phase space and the right-hand side of the equation were introduced. Any space and functional satisfying these axioms necessarily have specific properties; in particular, they satisfy conditions guaranteeing the existence and uniqueness of solutions. This approach (later used in numerous papers related to the investigation of various properties of such equations, e.g., see [3-7]) permits one to define conditions under which it is possible to construct limit equations with properties similar to those obtained for ordinary equations in [8-10] and for functional-differential equations with finite delay in [11, 12]. The method of limit equations allows one to derive sufficient conditions for Lyapunov asymptotic stability via finite-dimensional functions; for some class of nonautonomous equations, these conditions are more general than the well-known Lyapunov-Razumikhin results for equations with unbounded delay [13-15]. 1. MAIN DEFINITIONS, ASSUMPTIONS, AND AUXILIARY ASSERTIONS Let B be a real vector space of one of the following types: (i) a space of continuous functions on (−∞, 0] ranging in R n ; two functions ϕ, ψ ∈ B are equal (ϕ = ψ) if ϕ(s) = ψ(s) for all s ∈ (−∞, 0]; (ii) a space of measurable functions on (−∞, 0] ranging in R n ; two functions ϕ, ψ ∈ B are equal (ϕ = ψ) if ϕ(s) = ψ(s) for almost all s ∈ (−∞, 0] and ϕ(0) = ψ(0). We denote the norm in R n by | • |. We also assume that B is equipped with a norm | • | B such that (B, | • | B) is a Banach space. For a function x : (−∞, A) → R n , 0 < A ≤ +∞, we introduce a function x t : (−∞, 0] → R n by setting x t (s) = x(t + s), s ≤ 0, for each t ∈ [0, A). Definition 1 [13]. A space B is said to be admissible if there exist constants K, J > 0 and a continuous function M : [0, ∞) → [0, ∞) such that the following conditions are satisfied. Let 0 ≤ a < A ≤ ∞. If x : (−∞, A) → R n is continuous on [a, A) and x a ∈ B, then, for all t ∈ [a, A), (B1) x t ∈ B, and x t is continuous with respect to t in | • | B ; (B2) |x t | B ≤ K max a≤s≤t |x(s)| + M (t − a) |x a | B ; (B3) |ϕ(0)| ≤ J|ϕ| B for all ϕ ∈ B; (B4) M (t) → 0 as t → ∞. Let B be an admissible separable space, and let B H = {ϕ ∈ B : |ϕ| B < H}. Suppose that if ϕ is bounded and continuous on (−∞, 0], then ϕ ∈ B and all functions ϕ −t , t ≥ 0, are bounded in the norm of B, i.e., |ϕ −t | B ≤ L for some L > 0 [3] (here ϕ −t (s) = ϕ(−t + s), s ∈ (−∞, 0]). Consider the system of functional-differential equationṡ x(t) = X (t, x t) , (SI) where X : R + × B H → R n is a continuous mapping bounded on each set R + × D , that is, |X(t, ϕ)| ≤ m (D) for all (t, ϕ) ∈ R + × D , whenever D is a closed bounded subset of B H. Then

The global asymptotic stability and stabilization in nonlinear cascade systems with delay

Russian Mathematics, 2008

We study certain sufficient conditions for the local and global uniform asymptotic stability, as ... more We study certain sufficient conditions for the local and global uniform asymptotic stability, as well as the stabilizability of the equilibrium in cascade systems of delay differential equations. As distinct from the known results, the assertions presented in this paper are also valid for the cases, when the right-hand sides of equations are nonlinear and depend on time or arbitrarily depend on the historical data of the system. We prove that the use of auxiliary constant-sign functionals and functions with constant-sign derivatives essentially simplifies the statement of sufficient conditions for the asymptotic stability of a cascade. We adduce an example which illustrates the use of the obtained results. It demonstrates that the proposed procedure makes the study of the asymptotic stability and the construction of a stabilizing control easier in comparison with the traditional methods.

Degenerate Functions in the Analysis of Asymptotic Stability of Solutions of Functional Differential Equations

Mathematical Notes, 2005

Sufficient stability conditions and stabilizing control design for delay differential systems of a special type

Journal of Applied and Industrial Mathematics, 2012

This paper introduces some sufficient conditions for uniform and asymptotic global stability as w... more This paper introduces some sufficient conditions for uniform and asymptotic global stability as well as the algorithms for design of stabilizing control for special systems like cascaded (triangular) systems and integrator chains. The results are presented in terms of semidefinite Lyapunov functions, and they hold for nonlinear nonautonomous systems. Application of the results proposed is illustrated by some classical examples.

Razumikhin-Type Theorems in the Problem on Instability of Nonautonomous Equations with Finite Delay

Funkcialaj Ekvacioj, 2004

The paper provides some theorems on complete instability of zero solution relative to a set for n... more The paper provides some theorems on complete instability of zero solution relative to a set for nonautonomous nonlinear equations with delay. The right-hand side of the equation is assumed to satisfy conditions, which provide standard existenceuniqueness-continuous dependence-continuation theory for the equation, as well as precompactness of the collection of translations in time of the right-hand side in a functional space with a metrizable compact open topology. These assumptions allow constructing limiting equations. Using conceptions of Lyapunov-Razumikhin functions and limiting equations, new instability results are obtained, which are applicable, in particular, to autonomous and periodic delayed di¤erential equations and generalize some known results.

Development of the direct Lyapunov method for functional-differential equations with infinite delay

Mathematical Notes, 2008

We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of... more We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of a retarded functional-differential equation with unbounded (infinite) delay. The equation can be nonlinear and nonautonomous. The conditions are formulated in terms of Razumikhin-type functions, and in this case, a function is coupled with a functional related to this function by a certain dependence. In the results presented here, because of additional restrictions imposed on the right-hand side of the equation and the use of the limiting equation techniques, the classical requirements stating that the function and its derivative must be of fixed sign along the solution are weakened to the requirements that the function and its derivative must be of constant signs.

On the problem of tracking for the nonholonomic systems with provision for the feedback delay

Automation and Remote Control, 2013

ABSTRACT For the nonholonomic system of in the form of an arbitrary chain, consideration was give... more ABSTRACT For the nonholonomic system of in the form of an arbitrary chain, consideration was given to the problem of tracking. Proposed were two classes of two-dimensional controls in the form of linear feedback with provision for delay that maintain convergence of all trajectories of the controlled system to the desired trajectory.

On the principle of reduction for the nonlinear delay systems

Automation and Remote Control, 2011

Results were presented enabling one to reduce the problem equilibrium stability for multidimensio... more Results were presented enabling one to reduce the problem equilibrium stability for multidimensional delay systems to a similar problem for more than one system of lower dimensionality. The results were established using fixed-sign (degenerate) Lyapunov functions and the limiting systems. The latter were constructed under more general assumptions about the right-hand side of the system than those used traditionally.

Stability in systems with unbounded aftereffect

Automation and Remote Control, 2009

The paper considers the stability problem for systems described by differential equations with de... more

Control functionals in stabilization problem systems with time delay

Automation and Remote Control, 2010

The stabilization problem is considered for control systems described by nonlinear functional-dif... more The stabilization problem is considered for control systems described by nonlinear functional-differential equations of delayed type. Stabilizing control is constructed using the so-called control auxiliary functional, properties of which make it possible to find possibility of stabilization and, in some cases, to construct stabilizing control laws in an explicit form. "Inverse optimality" and robustness of the constructed controls are also discussed.

Local and semiglobal stabilization in a cascade with delay

Automation and Remote Control, 2008

Consideration is given to the problem of stability and stabilization for a system of differential... more

Lyapunov–Razumikhin pairs in the instability problem for infinite delay equations

Nonlinear Analysis: Theory, Methods & Applications, 2010

ABSTRACT Some theorems on complete instability of the zero solution relative to a set for nonauto... more ABSTRACT Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov–Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in t delay differential equations.

Optimization Problems in Tracking Control Design for an Underactuated Ship with Feedback Delay, State and Control Constraints

Semidefinite Programming Methods of Ascertaining Stability for a Nonlinear Delay System Subject to Constraints

This paper deals with stability problem for nonlinear delay systems. The proposed approach is bas... more This paper deals with stability problem for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi– Sugeno system with delay. The aim of the study is to present sufficient conditions of stability for the system subject to the constraints defining the subset under consideration. Because of the constraints, another important issue is how to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin conditions, we obtain stability conditions and an estimation of the domain of attraction in terms of linear matrix inequalities with scalar parameters. The obtained problems can be formulated by semidefinite programming and solved numerically.

LMI-based Method to Estimate the Domain of Attraction for Nonlinear Cascaded Systems with Delay

WSEAS TRANSACTIONS on SYSTEMS archive, 2018

This paper deals with stability analysis for nonlinear systems with time delay. The proposed appr... more This paper deals with stability analysis for nonlinear systems with time delay. The proposed approach is based on the assumption that on a subset of the state space the system is represented by a continuous-time Takagi– Sugeno system with delay and cascaded structure. The first aim is to present linear matrix inequality conditions to assess non-local stability properties of the system. The second relevant contribution is to present linear matrix inequalities that allow to find an inner estimate of the domain of attraction for the system subject to constraints defining the subset under consideration. The proposed approach is based on common quadratic Lyapunov functions and the Razumikhin technique.

Razumikhin conditions in partial stability problem for delay systems

In this paper nonlinear time-varying delay differential systems are considered and the usage of L... more

О развитии прямого метода Ляпунова для функционально-дифференциальных уравнений с бесконечным запаздыванием

Математические заметки, 2008

О развитии прямого метода Ляпунова для функционально-дифференциальных уравнений с бесконечным запаздыванием

Математические заметки, 2008

МЕТОД ФУНКЦИЙ ЛЯПУНОВА-РАЗУМИХИНА В ЗАДАЧЕ ОБ УСТОЙЧИВОСТИ СИСТЕМ С ЗАПАЗДЫВАНИЕМ

Автоматика и телемеханика, 2019

Рассмотрена задача стабилизации управляемых систем, описываемых нелинейными функционально-диффере... more Рассмотрена задача стабилизации управляемых систем, описываемых нелинейными функционально-дифференциальными уравнениями запаздывающего типа. Для построения стабилизирующего управления применяются так называемые контролирующие вспомогательные функционалы, свойства которых позволяют установить возможность стабилизации и в некоторых случаях построить стабилизирующие законы управления в явном виде. Обсуждены «обратная оптимальность» и робастность построенных управлений.

Exponential Stability of Nonlinear Time-Varying Delay Differential Equations via Lyapunov–Razumikhin Technique

Mathematics

In this article, some new sufficient conditions for the exponential stability of nonlinear time-v... more In this article, some new sufficient conditions for the exponential stability of nonlinear time-varying delay differential equations are given. An extension of the classical asymptotical stability theorem in terms of a Lyapunov–Razumikhin function is obtained. The condition of non-positivity of the time derivative of a Razumikhin function is weakened. Additionally, the resulting sufficient asymptotic stability conditions allow us to guarantee uniform exponential stability and evaluate the exponential convergence rate of the system solutions. The effectiveness of the results is demonstrated by some examples.

A Remark on the Lyapunov–Razumikhin Method for Equations with Infinite Delay

Differential Equations - DIFFERENTIAL EQUAT-ENGL TR, 2002

The choice of an appropriate phase space plays an important role in the construction of the theor... more The choice of an appropriate phase space plays an important role in the construction of the theory of equations with unbounded (infinite) delay, since it affects even the results concerning the existence and uniqueness of solutions (see the survey [1]). An axiomatic approach to this problem was developed in [2], where a number of axioms dealing with the phase space and the right-hand side of the equation were introduced. Any space and functional satisfying these axioms necessarily have specific properties; in particular, they satisfy conditions guaranteeing the existence and uniqueness of solutions. This approach (later used in numerous papers related to the investigation of various properties of such equations, e.g., see [3-7]) permits one to define conditions under which it is possible to construct limit equations with properties similar to those obtained for ordinary equations in [8-10] and for functional-differential equations with finite delay in [11, 12]. The method of limit equations allows one to derive sufficient conditions for Lyapunov asymptotic stability via finite-dimensional functions; for some class of nonautonomous equations, these conditions are more general than the well-known Lyapunov-Razumikhin results for equations with unbounded delay [13-15]. 1. MAIN DEFINITIONS, ASSUMPTIONS, AND AUXILIARY ASSERTIONS Let B be a real vector space of one of the following types: (i) a space of continuous functions on (−∞, 0] ranging in R n ; two functions ϕ, ψ ∈ B are equal (ϕ = ψ) if ϕ(s) = ψ(s) for all s ∈ (−∞, 0]; (ii) a space of measurable functions on (−∞, 0] ranging in R n ; two functions ϕ, ψ ∈ B are equal (ϕ = ψ) if ϕ(s) = ψ(s) for almost all s ∈ (−∞, 0] and ϕ(0) = ψ(0). We denote the norm in R n by | • |. We also assume that B is equipped with a norm | • | B such that (B, | • | B) is a Banach space. For a function x : (−∞, A) → R n , 0 < A ≤ +∞, we introduce a function x t : (−∞, 0] → R n by setting x t (s) = x(t + s), s ≤ 0, for each t ∈ [0, A). Definition 1 [13]. A space B is said to be admissible if there exist constants K, J > 0 and a continuous function M : [0, ∞) → [0, ∞) such that the following conditions are satisfied. Let 0 ≤ a < A ≤ ∞. If x : (−∞, A) → R n is continuous on [a, A) and x a ∈ B, then, for all t ∈ [a, A), (B1) x t ∈ B, and x t is continuous with respect to t in | • | B ; (B2) |x t | B ≤ K max a≤s≤t |x(s)| + M (t − a) |x a | B ; (B3) |ϕ(0)| ≤ J|ϕ| B for all ϕ ∈ B; (B4) M (t) → 0 as t → ∞. Let B be an admissible separable space, and let B H = {ϕ ∈ B : |ϕ| B < H}. Suppose that if ϕ is bounded and continuous on (−∞, 0], then ϕ ∈ B and all functions ϕ −t , t ≥ 0, are bounded in the norm of B, i.e., |ϕ −t | B ≤ L for some L > 0 [3] (here ϕ −t (s) = ϕ(−t + s), s ∈ (−∞, 0]). Consider the system of functional-differential equationṡ x(t) = X (t, x t) , (SI) where X : R + × B H → R n is a continuous mapping bounded on each set R + × D , that is, |X(t, ϕ)| ≤ m (D) for all (t, ϕ) ∈ R + × D , whenever D is a closed bounded subset of B H. Then

The global asymptotic stability and stabilization in nonlinear cascade systems with delay

Russian Mathematics, 2008

We study certain sufficient conditions for the local and global uniform asymptotic stability, as ... more We study certain sufficient conditions for the local and global uniform asymptotic stability, as well as the stabilizability of the equilibrium in cascade systems of delay differential equations. As distinct from the known results, the assertions presented in this paper are also valid for the cases, when the right-hand sides of equations are nonlinear and depend on time or arbitrarily depend on the historical data of the system. We prove that the use of auxiliary constant-sign functionals and functions with constant-sign derivatives essentially simplifies the statement of sufficient conditions for the asymptotic stability of a cascade. We adduce an example which illustrates the use of the obtained results. It demonstrates that the proposed procedure makes the study of the asymptotic stability and the construction of a stabilizing control easier in comparison with the traditional methods.

Degenerate Functions in the Analysis of Asymptotic Stability of Solutions of Functional Differential Equations

Mathematical Notes, 2005

Sufficient stability conditions and stabilizing control design for delay differential systems of a special type

Journal of Applied and Industrial Mathematics, 2012

This paper introduces some sufficient conditions for uniform and asymptotic global stability as w... more This paper introduces some sufficient conditions for uniform and asymptotic global stability as well as the algorithms for design of stabilizing control for special systems like cascaded (triangular) systems and integrator chains. The results are presented in terms of semidefinite Lyapunov functions, and they hold for nonlinear nonautonomous systems. Application of the results proposed is illustrated by some classical examples.

Razumikhin-Type Theorems in the Problem on Instability of Nonautonomous Equations with Finite Delay

Funkcialaj Ekvacioj, 2004

The paper provides some theorems on complete instability of zero solution relative to a set for n... more The paper provides some theorems on complete instability of zero solution relative to a set for nonautonomous nonlinear equations with delay. The right-hand side of the equation is assumed to satisfy conditions, which provide standard existenceuniqueness-continuous dependence-continuation theory for the equation, as well as precompactness of the collection of translations in time of the right-hand side in a functional space with a metrizable compact open topology. These assumptions allow constructing limiting equations. Using conceptions of Lyapunov-Razumikhin functions and limiting equations, new instability results are obtained, which are applicable, in particular, to autonomous and periodic delayed di¤erential equations and generalize some known results.

Development of the direct Lyapunov method for functional-differential equations with infinite delay

Mathematical Notes, 2008

We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of... more We propose new sufficient conditions for the uniform asymptotic stability of the zero solution of a retarded functional-differential equation with unbounded (infinite) delay. The equation can be nonlinear and nonautonomous. The conditions are formulated in terms of Razumikhin-type functions, and in this case, a function is coupled with a functional related to this function by a certain dependence. In the results presented here, because of additional restrictions imposed on the right-hand side of the equation and the use of the limiting equation techniques, the classical requirements stating that the function and its derivative must be of fixed sign along the solution are weakened to the requirements that the function and its derivative must be of constant signs.

On the problem of tracking for the nonholonomic systems with provision for the feedback delay

Automation and Remote Control, 2013

ABSTRACT For the nonholonomic system of in the form of an arbitrary chain, consideration was give... more ABSTRACT For the nonholonomic system of in the form of an arbitrary chain, consideration was given to the problem of tracking. Proposed were two classes of two-dimensional controls in the form of linear feedback with provision for delay that maintain convergence of all trajectories of the controlled system to the desired trajectory.

On the principle of reduction for the nonlinear delay systems

Automation and Remote Control, 2011

Results were presented enabling one to reduce the problem equilibrium stability for multidimensio... more Results were presented enabling one to reduce the problem equilibrium stability for multidimensional delay systems to a similar problem for more than one system of lower dimensionality. The results were established using fixed-sign (degenerate) Lyapunov functions and the limiting systems. The latter were constructed under more general assumptions about the right-hand side of the system than those used traditionally.

Stability in systems with unbounded aftereffect

Automation and Remote Control, 2009

The paper considers the stability problem for systems described by differential equations with de... more

Control functionals in stabilization problem systems with time delay

Automation and Remote Control, 2010

The stabilization problem is considered for control systems described by nonlinear functional-dif... more The stabilization problem is considered for control systems described by nonlinear functional-differential equations of delayed type. Stabilizing control is constructed using the so-called control auxiliary functional, properties of which make it possible to find possibility of stabilization and, in some cases, to construct stabilizing control laws in an explicit form. "Inverse optimality" and robustness of the constructed controls are also discussed.

Local and semiglobal stabilization in a cascade with delay

Automation and Remote Control, 2008

Consideration is given to the problem of stability and stabilization for a system of differential... more

Lyapunov–Razumikhin pairs in the instability problem for infinite delay equations

Nonlinear Analysis: Theory, Methods & Applications, 2010

ABSTRACT Some theorems on complete instability of the zero solution relative to a set for nonauto... more ABSTRACT Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov–Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in t delay differential equations.

Optimization Problems in Tracking Control Design for an Underactuated Ship with Feedback Delay, State and Control Constraints

Semidefinite Programming Methods of Ascertaining Stability for a Nonlinear Delay System Subject to Constraints

This paper deals with stability problem for nonlinear delay systems. The proposed approach is bas... more This paper deals with stability problem for nonlinear delay systems. The proposed approach is based on the assumption that on a subset of instant states the system is represented by a continuous-time Takagi– Sugeno system with delay. The aim of the study is to present sufficient conditions of stability for the system subject to the constraints defining the subset under consideration. Because of the constraints, another important issue is how to estimate the set of initial points such that the trajectories starting at those points remain to satisfy the constraints. Using Lyapunov functions with the Razumikhin conditions, we obtain stability conditions and an estimation of the domain of attraction in terms of linear matrix inequalities with scalar parameters. The obtained problems can be formulated by semidefinite programming and solved numerically.

LMI-based Method to Estimate the Domain of Attraction for Nonlinear Cascaded Systems with Delay

WSEAS TRANSACTIONS on SYSTEMS archive, 2018

This paper deals with stability analysis for nonlinear systems with time delay. The proposed appr... more This paper deals with stability analysis for nonlinear systems with time delay. The proposed approach is based on the assumption that on a subset of the state space the system is represented by a continuous-time Takagi– Sugeno system with delay and cascaded structure. The first aim is to present linear matrix inequality conditions to assess non-local stability properties of the system. The second relevant contribution is to present linear matrix inequalities that allow to find an inner estimate of the domain of attraction for the system subject to constraints defining the subset under consideration. The proposed approach is based on common quadratic Lyapunov functions and the Razumikhin technique.

Razumikhin conditions in partial stability problem for delay systems

In this paper nonlinear time-varying delay differential systems are considered and the usage of L... more

О развитии прямого метода Ляпунова для функционально-дифференциальных уравнений с бесконечным запаздыванием

Математические заметки, 2008