Asymptotic stability of \(m\)-switched systems (Q2766036)

The authors consider the following \(m\)-switched differential system NEWLINE\[NEWLINE\dot x(t)= f_i\bigl(x(t) \bigr),\;x\in\mathbb{R}^n,\;i\in M=\{1,2, \dots, m\}, \tag{1}NEWLINE\]NEWLINE where \(f_i\) are globally Lipschitz-continuous functions, \(f_i(0) =0\). For an arbitrary initial time \(\tau_0\) and initial value \(x_0=x(\tau_0)\), define the switch sequence \(S\) by \(S=x_0\); \((i_0,\tau_0)\), \((i_1,\tau_1), \dots, (i_k,\tau_k), \dots\), where \(\{\tau_k\}\) is a monotonically increasing time sequence and all \(i_k\in M\). Then a switch time sequence \(S\mid i\) corresponding to the \(i\)th subsystem of (1) is defined. The main result of the paper can be restated as follows:NEWLINENEWLINENEWLINETheorem 1. Let \(V_i:\mathbb{R}^n \to\mathbb{R}^+\) \((i=1,2, \dots,m)\) be positive definite and radius-unbounded, and let all their first-order derivatives be continuous. If for every switch sequence \(S\) and all \(i\) the function \(V_i\) is a Lyapunov-like function of \(f_i\) and \(x(\cdot)\) on \(S\mid i\), then the system (1) is asymptotically stable in the sense of Lyapunov.NEWLINENEWLINENEWLINEHere, the definition of Lyapunov-like functions was given by \textit{P. Peleties} and \textit{R. A. Dearlo} [Proc. Am. Control Conf., Boston MA, 1679-1684 (1991)]. To convey the usefulness of the above theorem, a simulation example with \(m=2\) is also given; it does not satisfy the conditions of the corresponding result established in the above-mentioned paper.