Stability analysis of impulsive fuzzy differential equations with finite delayed state

Document Type: research paper

Authors

1 PhD, Department of Electrical Engineering (Control), Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

2 Professor, Department of Electrical Engineering (Control), Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

3 Professor, Department of Applied Mathematics, Faculty of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

Abstract

In this paper we introduce some stability criteria for impulsive fuzzy system of differential equations with finite delay in states. Firstly, a new comparison principle for fuzzy differential system compared to crisp ordinary differential equation, based on a notion of upper quasi-monotone nondecreasing, in N dimentional state space is presented. Furthermore, in order to analyze the stability of fuzzy dynamical systems, vector Lyapunov like functions are defined. Then, by using these vector Lyapunov-like functions together with the new comparison theorem which is presented before, we will get results for some concepts of stability (eventual stability, asymptotic stability, strong stability, uniform stability and their combinations) for impulsive fuzzy delayed system of differential equations. Moreover, some theorems for practical stability in terms of two measures are introduced and are proved. Finally, an illustrating example for stability checking of a system of differential equations with fuzziness and time delay in states is given.

Keywords


Article Title [Persian]

تحلیل پایداری معادلات دیفرانسیل فازی ضربه‌ای دارای حالت تاخیری محدود

Authors [Persian]

  • داود ناصح 1
  • ناصر پریز 2
  • علی وحیدیان کامیاد 3
1 گروه مهندسی برق-کنترل، دانشکده مهندسی، دانشگاه فردوسی مشهد، مشهد، ایران
2 گروه مهندسی برق-کنترل، دانشکده مهندسی، دانشگاه فردوسی مشهد، مشهد، ایران
3 گروه ریاضی کاربردی، دانشکده ریاضی، دانشگاه فردوسی مشهد، مشهد، ایران
Abstract [Persian]

در این مقاله معیارهایی برای بررسی پایداری دستگاه معادلات دیفرانسیل فازی ضربه­ای دارای تاخیر محدود در حالت ارائه می­گردد. ابتدا، قضیه مقایسه جدیدی برای کرانداری پاسخ سیستم دیفرانسیل فازی در قیاس با سیستم دیفرانسیل معمولی تعینی در فضای N بعدی بر اساس مفهوم توابع غیرنزولی شبه یکنوای فوقانی بیان می­گردد؛ همچنین، برای تحلیل پایداری سیستم­های دینامیکی فازی، توابع شبه لیاپانوف برداری تعریف می­گردند. سپس با استفاده از این توابع برداری شبه لیاپانوف به همراه قضیه مقایسه جدیدی که مطرح شده است، برخی قضایا برای بررسی انواع مفاهیم پایداری (پایداری نهایی، پایداری مجانبی، پایداری قوی و پایداری یکنواخت) برای سیستم دیفرانسیل فازی ضربه­ای دارای حالت تاخیردار مطرح می­شوند. علاوه بر آن، قضایای پایداری کاربردی بر حسب دو معیار ارائه شده و به اثبات می­رسند. در انتها مثالی روشنگر برای نحوه بکارگیری قضایای پایداری مطرح و پایداری یک سیستم دیفرانسیل فازی دارای تاخیر بررسی می­گردد.

Keywords [Persian]

  • دستگاه معادلات دیفرانسیل فازی
  • قضیه مقایسه
  • توابع شبه لیاپانوف برداری
  • غیرنزولی شبه یکنوای فوقانی
  • پایداری کاربردی

 

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