A differential inequality with delay and impulse and its applications to the design of robust control (Q2744501)

The present paper is devoted to the following delay differential inequality with impulse NEWLINE\[NEWLINE\begin{cases} \dot f(t)\leq-\alpha f(t)+ \beta \|f_t\|,\;t\neq t_k,\\ f(t_k)\leq a_kf(t_{k-})+b_k\|f_{t_k} \|, \end{cases} \tag{*}NEWLINE\]NEWLINE 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 NEWLINE\[NEWLINEf(t)\leq M\|f_{t_0} \|\exp \bigl[(\gamma- \lambda)(t-t_0)\bigr],\;t< t_0,NEWLINE\]NEWLINE where \(\lambda\) is the unique positive root of the equation \(\lambda= \alpha-\beta e^{\lambda\sigma}\).NEWLINENEWLINENEWLINEUsing 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.