Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
Volume 27, Issue 10, Pages 5631-5652Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcdsb.2021289
Keywords
Switched systems; Switching at variable times; Existence; Uniqueness; Continuous dependence; Differentiability; Stability
Categories
Funding
- Foundation of Postgraduate of Guizhou Province [2019032]
- National Natural Science Foundation of China [12061021, 11661020]
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This paper deals with the qualitative theory for a class of nonlinear differential equations with switching at variable times, including the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters, and the stability of the system. The global exponential stability is discussed, with necessary and sufficient conditions for SSVT. The paper also investigates the continuous dependence and differentiability of the solution with respect to the initial state and the switching line.
In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching k is an element of N times on bounded time intervals.
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