The $L_1$-induced norm analysis for linear multivariable differential equations

In this paper, we consider the $L_1$-induced norm analysis for linear multivariable differential equations. Because such an analysis requires integrating the absolute value of the associated impulse response on the infinite-interval $[0,\mathrm{\infty })$, this interval was divided into $[0,H)$ and $[H,\mathrm{\infty })$, with the truncation parameter $H$. The former was divided into $M$ subintervals with an equal width, and the kernel function of the relevant input\slash output operator on each subinterval was approximated by a $p$th order polynomial with $p=0,1,2,3$. This derived to an upper bound and a lower bound on the $L_1$-induced norm for $[0,H)$, with the convergence rate of $1/M^{p+1}$. An upper bound on the $L_1$-induced norm for $[H,\mathrm{\infty })$ was also derived, with an exponential order of $H$. Combining these bounds led to an upper bound and a lower bound on the original $L_1$-induced norm on $[0,\mathrm{\infty })$, within the order of $1/M^{p+1}$. Furthermore, the $l_1$-induced norm of difference equations was tackled in a parallel fashion. Finally, numerical studies were given to demonstrate the overall arguments.