مرکز منطقه ای اطلاع رساني علوم و فناوري - Componentwise asymptotic stability of linear constant dynamical systems

Abstract :

This note deals with a special type of asymptotic stability, namely componentwise asymptotic stability with respect to the vector $\\gamma (t)$ (CWASγ) of system $S: \\dot{x} = Ax + Bu, t \\geq 0$ , where $\\gamma (t) > 0$ (componentwise inequality) and $\\gamma (t) \\rightarrow 0$ as $t \\rightarrow + \\infty$ . $S$ is CWASγ if for each $t_{0} \\geq 0$ and for each $|x(t_{0})| \\leq \\gamma (t_{0}) (|x (t_{0})|$ with the components $|x_{i}(t_{0})|$ the free response of $S$ satisfies $|x(t)| \\leq \\gamma (t)$ for each $t \\geq t_{0}$ . For $\\gamma (t){\\underline { \\underline \\delta } } \\alpha e^{-\\beta t}, t \\geq 0$ , with $\\alpha > 0$ and $\\beta > 0$ (scalar), the CWEAS ( $E$ = exponential) may be defined. $S$ is CWAS γ (CWEAS) if and only if $\\dot{\\gamma }(t) \\geq \\bar{A}\\gamma (t), t \\geq 0 (\\bar{A}\\alpha < 0); A {\\underline { \\underline \\delta } } (a_{ij})$ and $\\bar{A}$ has the elements aijand $|a_{ij}|, i \\neq j$ . These results may be used in order to evaluate in a more detailed manner the dynamical behavior of $S$ as well as to stabilize $S$ componentwise by a suitable linear state feedback.