Abstract
In this paper, we proposed a version of the Floquet theory for delay differential equations. We demonstrated that very natural assumptions for control in technical applications can lead us to a one-dimensional fundamental system. This approach allowed researchers to work with classical methods used in the case of ordinary differential equations. On this basis, new original unexpected results on the exponential stability were proposed. For example, in the equation x' (4)+a(t)x(t—-T(7)) = 0, t € [0, co), we avoided the assumption on the smallness of the product sup,j9,.) 41 SUP;< {9,00) TD) < 3/2 for asymptotic stability. We obtained that in the case of w-periodic coefficient and delay, the fact that the period w was situated in a corresponding interval can lead to exponential stability. We then applied our new tests of stability to the stabilization of a drone's flight, where smallness of the noted above product could not be achieved from a technical point of view. For an equation with periodic coefficient and delay, we got a formula of the solution's representation on the semiaxis.
Recommended Citation
M. Bohner et al., "Floquet Theory for First-order Delay Equations and an Application to Height Stabilization of a Drone’s Flight," Electronic Research Archive, vol. 33, no. 5, pp. 2840 - 2861, AIMS Press, Jan 2025.
The definitive version is available at https://doi.org/10.3934/era.2025125
Department(s)
Mathematics and Statistics
Publication Status
Open Access
Keywords and Phrases
delay equation; drone flight; exponential stability; Floquet theory
International Standard Serial Number (ISSN)
2688-1594
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2025 The Authors, All rights reserved.
Creative Commons Licensing
This work is licensed under a Creative Commons Attribution 4.0 License.
Publication Date
01 Jan 2025
Comments
Ministry of Innovation, Science and Technology, Grant None