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Finite-time boundedness and $$L_{2}$$-gain analysis for switched delay systems with norm-bounded disturbance. (English) Zbl 1218.34082
Consider the class of linear switched delay systems with time-varying exogenous disturbances
\begin{aligned}\dot x(t)&=A_{\sigma(t)}x(t)+B_{\sigma(t)}x(t-h)+G_{\sigma(t)}\omega(t),\\ z(t)&= C_{\sigma(t)}x(t)+D_{\sigma(t)}\omega(t),\quad t\geq 0,\tag{1}\\ x(t)&=\varphi(t),\quad t\in[-h,0], \end{aligned}
where $$x(t)\in\mathbb R^n$$ is the state, $$z(t)\in\mathbb R^m$$ is the output, $$\omega(t)$$ is a time-varying exogenous disturbance meeting the constraint $$\int_0^{\infty}\omega^{\top}(t)\omega(t)\,dt\leq d$$, $$\sigma(t):[0,\infty)\to M=\{1,2,\dots,m\}$$ is the switching signal. Sufficient conditions are given which guarantee that (1) is finite-time bounded and has finite-time weighted $$L_2$$-gain. These conditions are delay-dependent and are given in terms of linear matrix inequalities and a condition of the form $$\tau_a>\tau_a^*$$ where $$\tau_a$$ is an average dwell-time. The proofs are based on Lyapunov functionals. A numerical example is given to verify the efficiency of the proposed method.

##### MSC:
 34K12 Growth, boundedness, comparison of solutions to functional-differential equations 34K35 Control problems for functional-differential equations
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