The $ L_1 $-induced norm analysis for linear multivariable differential equations - Junghoon Kim; Jung Hoon Kim (AIMS Mathematics)
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, \infty) $, this interval was divided into $ [0, H) $ and $ [H, \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, \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, \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.