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A differential inequality with delay and impulse and its applications to the design of robust control. (Chinese. English summary) Zbl 0995.93063

The present paper is devoted to the following delay differential inequality with impulse \[ \begin{cases} \dot f(t)\leq-\alpha f(t)+ \beta \|f_t\|,\;t\neq t_k,\\ f(t_k)\leq a_kf(t_{k-})+b_k\|f_{t_k} \|, \end{cases} \tag{*} \] where \(\|f_\eta\|: =\sup_{\eta-\sigma\leq s\leq \eta}f (s)\), \(\sigma(>0)\) is the delay, \(\alpha,\beta, a_k,b_k\geq 0\), \(\alpha> \beta\). The set of all discontinuous points of the nonnegative function \(f(t)\) is \(t_1< t_2< \cdots<t_k <\dots\) with \(t_k\to \infty\) \((k\to\infty)\), and \(f(t_0+ \theta) \) is continuous for \(\theta\in [t_0-\sigma, t_0]\). Assume that \(t_k-t_{k-1}> \delta \sigma\) always holds for some number \(\delta>1\) and there exist some numbers \(\gamma>0\), \(M>0\) such that \(\rho_1\rho_2 \dots\rho_{k+1} (e^{\lambda \sigma})^k\leq Me^{\gamma (t_k-t_0)}\) holds for \(k=1,2,\dots\). Then it is proved that all solutions to inequality (*) obey an estimation of the exponential form \[ f(t)\leq M\|f_{t_0} \|\exp \bigl[(\gamma- \lambda)(t-t_0)\bigr],\;t< t_0, \] where \(\lambda\) is the unique positive root of the equation \(\lambda= \alpha-\beta e^{\lambda\sigma}\).
Using the well-known Riccati equation method and the last result, some results on a robust control problem for some measure delay differential systems with impulse are discussed. An illustrating numerical example is also indicated.

MSC:

93D20 Asymptotic stability in control theory
34K45 Functional-differential equations with impulses
34K20 Stability theory of functional-differential equations
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