Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Viktor KULYK; Ganna KULYK; Nataliya STEPANENKO

doi:https://doi.org/10.33434/cams.550428

Article Details

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Authors:

Viktor KULYK | Ganna KULYK | Nataliya STEPANENKO

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 3, pp.176-181

Document Type:

Article

Publication Date:

9/2019

Document Language:

English

Document ID (Org No.):

2000911696

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

https://doi.org/10.33434/cams.550428

Abstract:

 


It is considered linear systems of differential equations and investigated questions of regularity of these systems. To explore the regularity it is comfortable to use quadratic form whose derivative with respect to the adjoint system is positive definite. Sometimes it is possible to find such a quadratic form, the derivative of which with respect to the system is non-negative. There are examples showing that in this case we can't say anything about the exponential dichotomy of this system (that is, its regularity). The question arises whether it is possible to combine a certain set of quadratic forms to get such a form, the derivative of which with respect to the system is positive definite. This question is similar to the question that arises in the theory of control: having a set of certain data about an object, can one say something about this object as a whole. It turns out that this is possible, only a set of these quadratic forms should be special, in some sense complete. In the presented article the authors propose to write it with the help of some combination of specific symmetric matrices S1,S2,…S1,S2,…S_1, S_2, \dots . So we have a quadratic form Vp=p1⟨S1(t)x,x⟩+p2⟨S2(t)x,x⟩+⋯+pk−1⟨Sk−1(t)x,x⟩+⟨Sk(t)x,x⟩Vp=p1⟨S1(t)x,x⟩+p2⟨S2(t)x,x⟩+⋯+pk−1⟨Sk−1(t)x,x⟩+⟨Sk(t)x,x⟩V_{p} =p_{1} \left\langle S_{1} \left(t\right)x,x\right\rangle +p_{2} \left\langle S_{2} \left(t\right)x,x\right\rangle + \dots +p_{k-1} \left\langle S_{k-1} \left(t\right)x,x\right\rangle +\left\langle S_{k} \left(t\right)x,x\right\rangle  It is proved that the derivative of this quadratic form is positive definite for sufficiently large parameters p1,…,pk−1p1,…,pk−1p_1, \dots, p_{k-1}. The results are illustrated by examples.

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Article Details

Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Authors: Viktor KULYK | Ganna KULYK | Nataliya STEPANENKO

Publisher: Emrah Evren KARA

Main Topics: Mathematics

Journal: Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 3, pp.176-181

Document Type: Article

Publication Date: 9/2019

Document Language: English

Document ID (Org No.): 2000911696

Frequency: Quarterly

pISSN: 2651-4001

DOI: https://doi.org/10.33434/cams.550428

Issues

Authors:

Viktor KULYK | Ganna KULYK | Nataliya STEPANENKO

Publisher:

Emrah Evren KARA

Main Topics:

Mathematics

Journal:

Communications in Advanced Mathematical Sciences 2019 Vol.2 Issue 3, pp.176-181

Document Type:

Article

Publication Date:

9/2019

Document Language:

English

Document ID (Org No.):

2000911696

Frequency:

Quarterly

pISSN:

2651-4001

DOI:

https://doi.org/10.33434/cams.550428