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Discretized Lyapunov functional for uncertain systems with multiple time-delay. (English) Zbl 0959.93053
The author considers the uncertain linear system with delays \[ \dot x(t)=\sum_{i=0}^{K}A^{i}(t)x(t-r^{i}), \] where \(x\in\mathbb{R}^{n}\), \(0=r^{0}<r^{1}<\cdots<r^{K}\) are time-delays, \(A^{i}(t)\), \(i=0,\ldots,K\), are uncertain matrices, \((A^{0}(t),\ldots,A^{K}(t))\) belongs to a known set \(\Omega\) for any \(t\geq 0\). By using the quadratic Lyapunov functional \[ \begin{aligned} V(\phi)&=\frac{1}{2}\phi^{T}(0)P\phi(0)+ \sum_{i=1}^{K}\phi^{T}(0) \int_{-r}^{0}Q^{i}(\xi)\phi(\xi) d\xi+ \frac{1}{2}\sum_{i=1}^{K} \int_{-r^{i}}^{0}\phi^{T}(\xi)S^{i}(\xi)\phi(\xi) d\xi\\ &+\frac{1}{2}\sum_{i=1}^{K} \sum_{j=1}^{K}\int_{-r}^{0} d\xi \int_{-r^{i}}^{0}\phi^{T}(\xi)R^{ij}(\xi,\eta) \phi(\eta) d\eta, \end{aligned} \] sufficient conditions for asymptotic stability are formulated as a corollary of Theorem 2.1 in [J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag (1993; Zbl 0787.34002)].
The main purpose of the paper is to obtain more computable sufficient conditions by discretization of the above functional when the kernels \(Q(\xi)\), \(S^{i}(\xi)\), \(R^{ij}(\xi,\eta)\) are replaced by piecewise linear matrix functions. It gives an opportunity to rewrite the stability criterion as a set of linear matrix inequalities. Two numerical illustrative examples are presented.

93D30 Lyapunov and storage functions
93D09 Robust stability
93C23 Control/observation systems governed by functional-differential equations
34K20 Stability theory of functional-differential equations
93D20 Asymptotic stability in control theory
93C05 Linear systems in control theory
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