Ordinary differential equations with singularities on the right-hand side

Филиппов, В. В.

doi:10.1007/bf01157013

Cited by 10 publications

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“…The aim of the present paper is to obtain a conveniently verifiable condition for the equicontinuity of a sequence of solution spaces, which is one of the two key points in the convergence proof. The place of our result in the theory is similar to that of Theorem 4 in [3] and Theorem 1 in [4].We use some notation from [1-9] without additional explanations. Let U be an open subset of the product ]~ x R = .…”

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confidence: 88%

“…The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition.KEY WORDS: Cauchy problem for equations with discontinuous right-hand side, generalized differential equations, convergence of solutions.In [1][2][3], a new approach to the theory of ordinary differential equations was initiated. In particular, this approach is aimed at studying differential equations with discontinuous right-hand side and the corresponding generalized differential equations and is based on systematic use of the topological structures introduced in [1][2][3].…”

mentioning

confidence: 99%

“…In particular, this approach is aimed at studying differential equations with discontinuous right-hand side and the corresponding generalized differential equations and is based on systematic use of the topological structures introduced in [1][2][3]. This approach was further developed in [4][5][6][7][8][9].…”

mentioning

confidence: 99%

“…Thus, it is important to study sufficient conditions for convergence. The general outline of the convergence proof is discussed in [3,4,[7][8][9]. The aim of the present paper is to obtain a conveniently verifiable condition for the equicontinuity of a sequence of solution spaces, which is one of the two key points in the convergence proof.…”

mentioning

confidence: 99%

“…We use some notation from [1][2][3][4][5][6][7][8][9] without additional explanations. Let U be an open subset of the product ]~ x R = .…”

mentioning

confidence: 99%

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The equicontinuity condition for sequences of solution spaces

Филиппов

1997

Math Notes

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ABSTRACT. The verification of an appropriately stated equicontinuity condition for a sequence of solution spaces is one of the two key points in the theory of the Cauchy problem for equations with singular right-hand sides. We obtain a related sufficient condition.KEY WORDS: Cauchy problem for equations with discontinuous right-hand side, generalized differential equations, convergence of solutions.In [1][2][3], a new approach to the theory of ordinary differential equations was initiated. In particular, this approach is aimed at studying differential equations with discontinuous right-hand side and the corresponding generalized differential equations and is based on systematic use of the topological structures introduced in [1][2][3]. This approach was further developed in [4][5][6][7][8][9]. It turns out that the central notion of the theory is that of convergence of sequences of solution spaces, which is adequate to the idea of continuous dependence of solutions on the parameters occurring in the equations. Thus, it is important to study sufficient conditions for convergence. The general outline of the convergence proof is discussed in [3,4,[7][8][9]. The aim of the present paper is to obtain a conveniently verifiable condition for the equicontinuity of a sequence of solution spaces, which is one of the two key points in the convergence proof. The place of our result in the theory is similar to that of Theorem 4 in [3] and Theorem 1 in [4].We use some notation from [1-9] without additional explanations. Let U be an open subset of the product ]~ x R = . We consider the class of set-valued mappings F: U -* R n satisfying the following condition:(1) the values of F are nonempty convex (one-sided) cones with vertex at the origin (note that {0} is an admissible value).Before proceeding to the main part of our exposition, we choose (2) set-valued mappings F, Fi: U ~ R '~, i = 1, 2,..., each of which satisfies condition (1), and moreover, (3) for any point x E U one has F(z) ~_ rq{cc (U{F,(Ox) : i= k, lc + 1, ... } U F(Oz) (5) Gi(t, y) = Fi(t, y) + Si(t) = {u + v: u E Fi(t, y), v E Si(t) } for (t, y) E U, i = 1,2, ....Let V be a cone with vertex at the origin, and let e > 0. Then by Pc we denote the e-neighborhood of the set {u: u E Y, IluH = 1} on the unit sphere S, and by V~ we denote the cone

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confidence: 99%

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“…We use some notation from [1][2][3][4][5][6][7][8][9] without additional explanations. Let U be an open subset of the product ]~ x R = .…”

mentioning

confidence: 99%

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The equicontinuity condition for sequences of solution spaces

Филиппов

1997

Math Notes

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Differential Games with Impulse Control

Чикрий

Matychyn

Chikrii

Annals of the International Society of Dynamic Games

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Theory of Differential Inclusions and Its Application in Mechanics

Kiseleva

Kuznetsov

Леонов

2017

New Perspectives and Applications of Modern Control Theory

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The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load chance is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems.

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Ordinary differential equations with singularities on the right-hand side

Cited by 10 publications

References 1 publication

The equicontinuity condition for sequences of solution spaces

The equicontinuity condition for sequences of solution spaces

Differential Games with Impulse Control

Theory of Differential Inclusions and Its Application in Mechanics

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