# Martynyuk A. A.

### Analysis of the Set of Trajectories of Fuzzy Equations of Perturbed Motion

Martynyuk A. A., Martynyuk-Chernienko Yu. A.

Ukr. Mat. Zh. - 2014. - 66, № 11. - pp. 1512–1527

The paper presents a new approach to the investigation of the first-order fuzzy initial-value problems. We use different versions of the comparison principle to establish conditions for the existence of solutions of a set of differential equations.

### Existence, uniqueness, and estimation of solutions for a set of equations of perturbed motion

Martynyuk A. A., Martynyuk-Chernienko Yu. A.

Ukr. Mat. Zh. - 2013. - 65, № 2. - pp. 273-295

We propose a regularization procedure for a set of equations of perturbed motion with uncertain values of parameters. Using the comparison principle, we establish conditions for the existence of solutions of the original system and the regularized system.

### Stability of motion of nonlinear systems with fuzzy characteristics of parameters

Martynyuk A. A., Martynyuk-Chernienko Yu. A.

Ukr. Mat. Zh. - 2012. - 64, № 1. - pp. 50-70

We investigate the stability of a stationary solution of a fuzzy dynamical system by a generalized Lyapunov direct method.

### Integral inequalities and stability of an equilibrium state on a time scale

Luk’yanova T. A., Martynyuk A. A.

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1490–1499

We present some integral inequalities on a time scale and establish sufficient conditions for the uniform stability of an equilibrium state of a nonlinear system on a time scale.

### On the mappings preserving the Lyapunov stability of Takagi–Sugeno fuzzy systems

Denisenko V. S., Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 641-649

We propose a general principle of comparison for stability-preserving mappings and establish sufficient conditions of stability for the Takagi – Sugeno continuous fuzzy systems.

### On the theory of stability of matrix differential equations

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 464-471

We establish the conditions of asymptotic stability of a linear system of matrix differential equations with quasiperiodic coefficients on the basis of constructive application of the principle of comparison with a Lyapunov matrix-valued function.

### On stability of linear hybrid mechanical systems with distributed components

Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 204–216

We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional.

### Aleksandr Mikhailovich Lyapunov (the 150th anniversary of his birth)

Martynyuk A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 996-1000

### Stability analysis of large-scale functional differential systems

Ukr. Mat. Zh. - 2007. - 59, № 3. - pp. 382–394

The present paper is focused on a new method for analysis of stability of solutions of a large-scale functional differential system via matrix-valued Lyapunov-Krasovskii functionals. The stability conditions are based on information about the dynamical behavior of subsystems of the general system and properties of the functions of interconnection between them.

### Stability analysis with respect to two measures of impulsive systems under structural perturbations

Martynyuk A. A., Stavroulakis I. P.

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1476–1484

The asymptotic stability with respect to two measures of impulsive systems under structural perturbations is investigated. Conditions of asymptotic (ρ_{0}, ρ)-stability of the system in terms of the fixed signs of some special matrices are established.

### Stability analysis of linear impulsive differential systems under structural perturbation

Martynyuk A. A., Stavroulakis I. P.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 784–795

The stability and asymptotic stability of solutions of large-scale linear impulsive systems under structural perturbations are investigated. Sufficient conditions for stability and instability are formulated in terms of the fixed signs of special matrices.

### On the practical $μ$-stability of solutions of standard systems with delay

Martynyuk A. A., Sun' Chzhen-tsi

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 204–213

We study the problem of $μ$-stability of a dynamical system with delay. Conditions of the practical $μ$-stability are established for the general case and for a quasilinear system. The conditions suggested are illustrated by an example.

### Analytical construction of the hierarchical matrix Lyapunov function for impulse systems

Begmuratov K. A., Martynyuk A. A.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 548–557

For impulse systems, we develop a method for the construction of the hierarchical matrix Lyapunov ffunction.

### Stability of large-scale discrete systems under structural perturbations

Martynyuk A. A., Miladzhanov V. G., Muminov M. M.

Ukr. Mat. Zh. - 1996. - 48, № 10. - pp. 1352-1362

By using the matrix Lyapunov function, we establish conditions of (uniform) stability and (uniform) asymptotic stability of a large-scale discrete system under structural perturbations.

### Exponential polystability of separable motions

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 642-649

We establish conditions of exponential*x* _{1}-stability and polystability for systems with separable motions. Stability conditions of these types are obtained by using the Lyapunov functions (scalar and matrix).

### Aleksandr Mikhailovich Lyapunov (the 125th anniversary of his birth)

Martynyuk A. A., Mitropolskiy Yu. A., Zubov V. I.

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 536—537

### Comparison principle for systems of differential equations with rapidly rotating phase

Martynyuk A. A., Matviichuk K. S.

Ukr. Mat. Zh. - 1979. - 31, № 5. - pp. 498–503

### Some problems in the comparison method in nonlinear mechanics

Martynyuk A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1978. - 30, № 6. - pp. 823–829

### A theorem of lyapunov type concerning the stability of a multidimensional system

Ukr. Mat. Zh. - 1972. - 24, № 4. - pp. 532–537

### An iteration formula for Lyapunov-function construction

Ukr. Mat. Zh. - 1972. - 24, № 2. - pp. 253—258

### Sufficient conditions of stability of systems with steadily acting perturbations

Kozubovskaya I. G., Martynyuk A. A.

Ukr. Mat. Zh. - 1971. - 23, № 3. - pp. 405–410

### Stability criterion for solutions of sets of nonlinear differential equations

Ukr. Mat. Zh. - 1971. - 23, № 2. - pp. 253–257

### On a rapidly converging iterational process for solving diffential equations and some of its applications

Ukr. Mat. Zh. - 1970. - 22, № 6. - pp. 734—748

### Polynomial approximations of the solutions of nonlinear equations

Ukr. Mat. Zh. - 1970. - 22, № 4. - pp. 557–563

### Construction of solutions of systems of differential equations in the region of asymptotic stability

Ukr. Mat. Zh. - 1970. - 22, № 3. - pp. 403–412

### Construction of approximate solutions of denumerable systems and their stability

Martynyuk A. A., Sukennik A. A.

Ukr. Mat. Zh. - 1969. - 21, № 5. - pp. 706–711

### Sufficient conditions for stability in a finite system with delay

Kozubovskaya I. G., Martynyuk A. A.

Ukr. Mat. Zh. - 1969. - 21, № 3. - pp. 399–406