4b68d0f90d585d28c0756fdbf270b70b mathematics-13-00365-v2.pdf 8e2788a3a1e57d72b7a2973fe92742965fbcca8c mathematics-13-00365-v2.pdf 08f89974b33a5d928bdab1761fd7162e2ecfd4e5d7b5b227d81d51a6e4b62a42 mathematics-13-00365-v2.pdf Title: On the Evolution Operators of a Class of Time-Delay Systems with Impulsive Parameterizations Subject: This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free and the delayed parts can undergo impulsive changes. Also, particular evolution operators are defined explicitly for the non-impulsive and impulsive time-varying delay-free case, and also for the case of impulsive delayed time-varying systems. In the impulsive cases, in general, the evolution operators are non-unique. The delays are assumed to be a finite number of constant delays that are not necessarily commensurate, that is, all of them being integer multiples of a minimum delay. On the other hand, the impulsive actions through time are assumed to be state-dependent and to take place at certain isolated time instants on the matrix functions that define the delay-free and the delayed dynamics. Some variants are also proposed for the cases when the impulsive actions are state-independent or state- and dynamics-independent. The intervals in-between consecutive impulses can be, in general, time-varying while subject to a minimum threshold. The boundedness of the state-trajectory solutions, which imply the system’s global stability, is investigated in the most general case for any given piecewise-continuous bounded function of initial conditions defined on the initial maximum delay interval. Such a solution boundedness property can be achieved, even if the delay-free dynamics is unstable, by an appropriate distribution of the impulsive actions. An illustrative first-order example is developed in detail to illustrate the impulsive stabilization results. Keywords: delay differential systems; point delays; evolution operator; impulsive actions; global stability Author: Manuel De la Sen, Asier Ibeas, Aitor J. Garrido and Izaskun Garrido Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25 CreationDate: Sat Jan 25 13:51:10 2025 CET ModDate: Sat Jan 25 14:11:00 2025 CET Custom Metadata: no Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 29 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 410974 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- MIKRSI+VnURWPalladioL Type 1 Custom yes yes yes 10 0 RLRPNZ+URWPalladioL-Roma Type 1 Custom yes yes yes 16 0 XEURBQ+URWPalladioL-Bold Type 1 Custom yes yes yes 22 0 TXAHVM+URWPalladioL-Ital Type 1 Custom yes yes yes 27 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 53 0 WYSWYO+CMSY10 Type 1 Builtin yes yes yes 58 0 ALMRZQ+PazoMath Type 1 Builtin yes yes yes 69 0 QOQJNH+PazoMath-Italic Type 1 Builtin yes yes yes 74 0 VUABVQ+CMMI10 Type 1 Builtin yes yes yes 79 0 PQSFML+URWPalladioL-BoldItal Type 1 Custom yes yes yes 84 0 HRQNPV+CMEX10 Type 1 Builtin yes yes yes 92 0 TMHVPV+MSAM10 Type 1 Builtin yes yes yes 103 0 ULZEWH+MSBM10 Type 1 Builtin yes yes yes 136 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2025-05-29 02:42:26 CEST RepresentationInformation: mathematics-13-00365-v2.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2025-05-28 11:09:42 CEST Size: 410974 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 298 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Destination: section.1 Item: Title: Time-Varying Linear Delay-Free Differential Systems and Their Evolution Operators for the Non-Impulsive and Impulsive Cases Destination: section.2 Item: Title: Impulsive Time-Varying Differential Systems with Constant Point Delays and Their Evolution Operator Destination: section.3 Children: Item: Title: Trajectory Solution of the Differential Impulsive System with Delays Destination: subsection.3.1 Item: Title: Some Results on the Solution Boundedness and the Global Stability Destination: subsection.3.2 Item: Title: Conclusions Destination: section.4 Item: Title: References Destination: section.5 Info: Title: On the Evolution Operators of a Class of Time-Delay Systems with Impulsive Parameterizations Author: Manuel De la Sen, Asier Ibeas, Aitor J. Garrido and Izaskun Garrido Subject: This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free and the delayed parts can undergo impulsive changes. Also, particular evolution operators are defined explicitly for the non-impulsive and impulsive time-varying delay-free case, and also for the case of impulsive delayed time-varying systems. In the impulsive cases, in general, the evolution operators are non-unique. The delays are assumed to be a finite number of constant delays that are not necessarily commensurate, that is, all of them being integer multiples of a minimum delay. On the other hand, the impulsive actions through time are assumed to be state-dependent and to take place at certain isolated time instants on the matrix functions that define the delay-free and the delayed dynamics. Some variants are also proposed for the cases when the impulsive actions are state-independent or state- and dynamics-independent. The intervals in-between consecutive impulses can be, in general, time-varying while subject to a minimum threshold. The boundedness of the state-trajectory solutions, which imply the system’s global stability, is investigated in the most general case for any given piecewise-continuous bounded function of initial conditions defined on the initial maximum delay interval. Such a solution boundedness property can be achieved, even if the delay-free dynamics is unstable, by an appropriate distribution of the impulsive actions. An illustrative first-order example is developed in detail to illustrate the impulsive stabilization results. 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