Integral manifolds and decomposition of singularly perturbed systems

Sobolev, Vladimir

doi:10.1016/s0167-6911(84)80099-7

Cited by 152 publications

(40 citation statements)

References 12 publications

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“…Then the following result is valid (see e.g. [8,9]): Proposition 1. Under the assumptions (A 1 ) − (A 3 ) there is a sufficiently small positive ε 1 , ε 1 ≤ ε 0 , such that for ε ∈ I 1 system (1) has a smooth integral manifold M ε ( slow integral manifold) with the representation…”

Section: Mathematical Modellingmentioning

confidence: 99%

Critical Cases in Slow/Fast Control Problems

Sobolev

2016

Collection of Selected Papers of the II International Conference on Information Technology and Nanotechnology

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Abstract. The aim of the paper is to describe the main critical cases in the theory of singularly perturbed optimal control problems and to give examples which are typical for slow/fast systems. The theory has traditionally dealt only with perturbation problems near normally hyperbolic manifold of singularities and this manifold is supposed to isolated. We reduce the original singularly perturbed problem to a regularized one such that the existence of slow integral manifolds can be established by means of the standard theory. We illustrate our approach by several examples of control problems.Keywords: integral manifolds, singular perturbations, optimal control Citation: Sobolev VA. Critical cases in slow/fast control problems.

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Section: Mathematical Modellingmentioning

confidence: 99%

Critical Cases in Slow/Fast Control Problems

Sobolev

2016

Collection of Selected Papers of the II International Conference on Information Technology and Nanotechnology

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“…(29), using the results of [14], i.e., matrices q i (t, ε), i = 1, 2, and the components of the matrix q(t, ε) satisfying a system of equations differing from the system for Eq. (13) due to presence of the heterogeneity…”

Section: Analysis Of the Linear Equation (29)mentioning

confidence: 99%

Regularization of Cheap Periodic Control Problems

Smetannikova

Sobolev

2005

Autom Remote Control

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A regularization method for cheap periodic control problems is designed. The periodic problem for singularly perturbed Riccati matrix differential equations is reduced to an extended system of singularly perturbed initial problems. This system admits the application of a geometric approach based on the use of integral manifolds and, consequently, dimensional reduction. The solution is asymptotically expanded in fractional powers of a small parameter. An example on control of periodic oscillations of a mechanical system is given.

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“…The existence and uniqueness issues for the slow integral manifold of the linear problems (21) and (22) and an algorithm for its construction were described in [11]. After reducing the dimension of problem (21), we obtain a problem for the zero approximation to the matrix X(t, ε) in the forṁ…”

Section: From the Linear Equation Kbmentioning

confidence: 99%