Necessary conditions for the asymptotic stability of systems of differential equations with pulse action on surfaces

2011 ◽  
Vol 14 (1) ◽  
pp. 21-31
Author(s):  
R. I. Gladilina
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Necessary Conditions ◽  
Pulse Action ◽  
Systems Of Differential Equations

scholarly journals CONSTRUCTION OF STABILITY AREAS FOR CONTROLLED SYSTEMS WITH PARAMETRIC AND DYNAMIC UNCERTAINTY

2021 ◽  
pp. 117-122
Author(s):  
Alla Savranska ◽  
Oleksandr Denisenko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Small Parameter ◽  
Singularly Perturbed ◽  
Singularly Perturbed Systems ◽  
Systems Of Differential Equations ◽  
Right Hand ◽  
Perturbed Systems ◽  
Small Factor ◽  
The Right

The subject of research in the article is sigularly perturbed controllable systems of differential equations containing terms with a small parameters on the right-hand side, which are not completely known, but only satisfy some constraints. The aim of the work is to expand the study of the behavior of solutions of singularly perturbed systems of differential equations to the case when the system is influenced not only by dynamic (small factor at the derivative) but also parametric (small factor at the right side of equations) uncertainties and to determine conditions under which such systems will be asymptotically resistant to any perturbations, estimate the upper limit of the small parameter, so that for all values of this parameter less than the obtained estimate, the undisturbed solution of the system was asymptotically stable. The following problems are solved in the article: singularly perturbed systems of differential equations with regular perturbations in the form of terms with a small parameter in the right-hand sides, which are not fully known, are investigated; an estimate is made of the areas of asymptotic stability of the unperturbed solution of such systems, that is, the class of systems that can be investigated for stability is expanded, the formulas obtained that allow one to analyze the asymptotic stability of solutions to systems even under conditions of incomplete information about the perturbations acting on them. The following methods are used: mathematical modeling of complex control systems; vector Lyapunov functions investigation of asymptotic stability of solutions of systems of differential equations. The following results were obtained: an estimate was made for the upper bound of a small parameter for sigularly perturbed systems of differential equations with fully known parametric (fully known) and dynamic uncertainties, such that for all values of this parameter less than the obtained estimate, such an unperturbed solution is asymptotically stable; a theorem is proved in which sufficient conditions for the uniform asymptotic stability of such a system are formulated. Conclusions: the method of vector Lyapunov functions extends to the class of singularly perturbed systems of differential equations with a small factor in the right-hand sides, which are not completely known, but only satisfy certain constraints.

Stability of Linear Systems With Multitask Right-Hand Member

2018 ◽  
pp. 74-112 ◽  
Author(s):  
G. V. Alferov ◽  
G. G. Ivanov ◽  
P. A. Efimova ◽  
A. S. Sharlay
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Computational Models ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Point Of View ◽  
Control Laws ◽  
Systems Of Differential Equations ◽  
The Stability

To study the dynamics of mechanical systems and to define the construction parameters and control laws, it is necessary to have computational models accurately describing properties of real mechanisms. From a mathematical point of view, the computational models of mechanical systems are actually the systems of differential equations. These models can contain equations that also describe non-mechanical phenomena. In this chapter, the problems of stability and asymptotic stability conditions for the motion of mechanical systems with holonomic and non-holonomic constraints are under consideration. Stability analysis for the systems of differential equations is given in term of the second Lyapunov's method. With the use of the set-theoretic approach, the necessary and sufficient conditions for stability and asymptotic stability of zero solution of the considered system are formulated. The proposed approaches can be used to study the stability of the motion for robot manipulators, transport, space, and socio-economic systems.

scholarly journals Class of controllable systems of differential equations for infinite time

2018 ◽  
Vol 20 (4) ◽  
pp. 439-447
Author(s):  
A. Yu. Pavlov
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Nonlinear Differential Equations ◽  
Arbitrary Point ◽  
Infinite Time ◽  
Left Hand ◽  
Systems Of Differential Equations ◽  
Controllability Of Systems ◽  
Controllable Systems ◽  
Asymptotic Equilibrium

In the article necessary conditions for a controllability of systems of nonlinear differential equations in an infinite time are obtained without assuming the existence of an asymptotic equilibrium for the system of linear approximation. Thus, a new class of controlled systems of differential equations is presented. The problem of controllability for an infinite time (i.e. the transfer of an arbitrary point into an arbitrary small domain of another point) comes down to choosing an operator depending on the selected control, which in turn depends on the point being transferred. Then one is to prove the existence of a fixed point for this operator. It is known that the theorems on controllability require existence of an asymptotic equilibrium for system of the first approximation. It is shown in the paper that in general case the condition of asymptotic equilibrium’s existence is not necessary for controllability of systems in an infinite time. An example on the theorem on controllability for an infinite time is given. The theorem generalizing Vazhevsky inequality is proved by implementation of Cauchy-Bunyakovsky inequality. A remark is made about the theorem’s validity for the case when the matrix and vector from the right-hand side of nonlinear differential equation are complex and x is vector with complex components. Basing on the left-hand side of the inequality in the theorem generalizing Vazhevsky inequality, the necessary conditions for controllability in an infinite time are obtained. These conditions are verified on the same example of a scalar equation that was mentioned before.

General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation

1988 ◽  
Vol 108 (1-2) ◽  
pp. 45-56 ◽  
Author(s):  
Gerhard Freiling
Keyword(s):  
Differential Equations ◽  
Characteristic Equation ◽  
Eigenvalue Problems ◽  
Necessary Conditions ◽  
General Boundary ◽  
Multiple Roots ◽  
Systems Of Differential Equations ◽  
Associated Functions ◽  
Boundary Eigenvalue

SynopsisWe consider a class of non-self adjoint multipoint eigenvalue problems. Using necessary conditions for the regularity of these problems, we obtain a theorem on the expansion of certain functions into a series of eigen- and associated functions.

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