Finding of bounded solutions to linear impulsive systems

Asrorov, Farkhod; Sobchuk, Valentyn; Kurylko, Оlexandr

doi:10.15587/1729-4061.2019.178635

Cited by 13 publications

(11 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Paper [1] established the conditions that warrant the hyperbolicity of systems of differential equations with pulse action. The resulting hyperbolicity conditions make it possible to investigate the existence of bounded solutions to the heterogeneous multidimensional systems of differential equations with pulse perturbation.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Studies [1][2][3][4][5][6][7][8][9][10][11] lay out the basics of the qualitative theory of differential equations with pulse action. In fact, the basics of the high-quality theory of pulse systems, based on the qualitative theory of differential equations, methods of asymptotic integration of such equations, the theory of difference equations and generalized functions were laid.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…According to Theorem 1 [1], for the arbitrary R-bounded functions f(t) and the sequence {a i }, the system of equations has a single solution bounded along the entire axis.…”

Section: T I T T Tmentioning

confidence: 99%

“…Let A be the manifold of the initial values of solutions to equations (5) bounded along the entire axis and let P : R n →A be the projector that maps R n onto A. According to theorem 1 [1], with any limited piecewise-continuous function f(t) and a constrained sequence {a i }, the system of equations ( 6) has a single solution bounded along the entire axis x = f(t) that satisfies the condition Pf(0) = 0 and the following inequality: sup m ax sup ,sup .…”

Section: T I T T Tmentioning

confidence: 99%

“…, that satisfies the condition Pf(0) = 0. Considering Theorem 1 [1] and conditions (3), we obtain an estimate: sup max sup , , ,…”

Section: The Existence Of Singleness Of the Bounded Solution To The Weakly Nonlinear Pulse Systemsmentioning

confidence: 99%

See 4 more Smart Citations

Establishing conditions for the existence of bounded solutions to the weakly nonlinear pulse systems

Asrorov

Perehuda

Sobchuk

et al. 2021

EEJET

Self Cite

View full text Add to dashboard Cite

Processes that involve jump-like changes are observed in mechanics (the movement of a spring under an impact; clockwork), in radio engineering (pulse generation), in biology (heart function, cell division). Therefore, high-quality research of pulse systems is a relevant task in the modern theory of mathematical modeling. This paper considers the issue related to the existence of bounded solutions along the entire real axis (semi-axis) of the weakly nonlinear systems of differential equations with pulse perturbation at fixed time moments. A concept of the regular and weakly regular system of equations for the class of the weakly nonlinear pulse systems of differential equations has been introduced. Sufficient conditions for the existence of a bounded solution to the heterogeneous system of differential equations have been established for the case of poorly regularity of the corresponding homogeneous system of equations. The conditions for the existence of singleness of the bounded solution along the entire axis have been defined for the weakly nonlinear pulse systems. The results were applied to study bounded solutions to the systems with pulse action of a more general form. The established conditions make it possible to use the classical methods of differential equations to obtain statements about solvability and the continuous dependence of solutions on the parameters of a pulse system. It has been shown that classical qualitative methods for studying differential equations are mainly naturally transferred to dynamic systems with discontinuous trajectories. However, the presence of a pulse action gives rise to a series of new specific problems. The theory of systems with pulse influence has a wide range of applications. Such systems arise when studying pulsed automatic control systems, in the mathematical modeling of various mechanical, physical, biological, and other processes.

show abstract

Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…According to Theorem 1 [1], for the arbitrary R-bounded functions f(t) and the sequence {a i }, the system of equations has a single solution bounded along the entire axis.…”

Section: T I T T Tmentioning

confidence: 99%

Section: T I T T Tmentioning

confidence: 99%

“…, that satisfies the condition Pf(0) = 0. Considering Theorem 1 [1] and conditions (3), we obtain an estimate: sup max sup , , ,…”

Section: The Existence Of Singleness Of the Bounded Solution To The Weakly Nonlinear Pulse Systemsmentioning

confidence: 99%

See 3 more Smart Citations