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The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times. (English) Zbl 1505.34026

In this paper, the authors deal with the qualitative properties of a class of nonlinear differential equations with switching at variable times (SSVT for short). More precisely, let \(\mathcal{P}=\{1, 2, \dots, l\}\) (\(l\in \mathbb{N}^+\)), \(y_i: [t_0, +\infty]\to \mathbb{R}^n\) (\(i\in \mathcal{P}\)) is a switching line, \(Y=\{ y_i(\cdot) | i\in \mathcal{P}\}\) is the set of switching lines and the operating region is given by \(\tau_i(t)=\mathbb{R}^n \setminus \{ y_j(t) | i, j\in \mathcal{P}, i\neq j\}\). Consider a class of SSVT of the following form: \[ \dot{x}(t)=f_{\omega(t, x(t))}(t, x(t)), \quad t\geq t_0, \] where \(\omega: [t_0, +\infty]\times\mathbb{R}^n\to\mathcal{P}\) is the switching law given by \[ \omega(t, x(t))=i, \text{ if } x(\bar{t})=y_i(\bar{t}), \quad x(t)\in\tau_i(t), \quad t\geq \bar{t}\geq t_0, \quad i\in \mathcal{P}. \] For this kind of switching systems, the authors investigate the following problems: (1) the existence and uniqueness of the global solution; (2) the continuous dependence and differentiability of the solution with respect to the parameters of the system; (3) the global exponential stability of the solution of the system.

MSC:

34A36 Discontinuous ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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