Derived numbers and stability of dynamical systems solutions
DOI:
https://doi.org/10.17072/1993-0550-2026-3-5-17Keywords:
stability of systems of ordinary differential equations, stability of the trivial solution, Lyapunov’s second method, derivative numbers, partial derivative numbersAbstract
This paper addresses the problem of expanding the class of investigated dynamical systems by relaxing the differentiability requirements for their right-hand sides. The objective is to develop stability criteria for systems where functions lack continuous derivatives.
The proposed approach is based on the application of the framework of partial derivate numbers (PDN). This mathematical tool enables the analysis of complex functions of several variables at points where the conventional gradient is undefined. The authors further develop the theory of PDN, building upon previously published theoretical works and results of applying this framework to stabilization problems. A key feature of this study is the deliberate avoidance of Dini derivatives. This approach circumvents excessive complexity in proofs and the need for cumbersome auxiliary constructions for functions of multiple variables.
In terms of the theory of partial derivate numbers, necessary and sufficient stability conditions for the zero solution of a system of ordinary differential equations (ODEs) are formulated and proven. The paper presents the relevant theorems along with their detailed substantiations. The obtained results allow for the efficient investigation of the stability of complex nonlinear systems without requiring classical smoothness of their mathematical models. This approach extends the capabilities of applied analysis for dynamical processes in engineering and control.
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Copyright (c) 2026 Геннадий Викторович Алфёров, Геннадий Григорьевич Иванов, Владимир Степанович Королёв, Ирина Юрьевна Потоцкая
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Articles are published under license Creative Commons Attribution 4.0 International (CC BY 4.0).
