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Quadratic Lyapunov function has been widely used in the analysis of lineartime invariant (LTI) systems ever since it has shown that the existence of suchquadratic Lyapunov function certifies the stability of the LTI system. In thiswork, the problem of finding upper and lower bounds for the$L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunovfunctions are used to find the star norm, the best upper on the$L_{\infty}$-induced norm, by bounding the unit peak input reachable sets byinescapable ellipsoids. Instead, a more general class of homogeneous Lyapunovfunctions is used to get less conservative upper bounds on the$L_{\infty}$-induced norm and better conservative approximations for thereachable sets than those obtained using standard quadratic Lyapunov functions.The homogeneous Lyapunov function for the LTI system is considered to be aquadratic Lyapunov function for a higher-order system obtained by Lifting theLTI system via Kronecker product. Different examples are provided to show thesignificant improvements on the bounds obtained by using Homogeneous Lyapunovfunctions.

Computing Bounds on $L_{\infty}$-induced Norm for Linear Time-Invariant Systems Using Homogeneous Lyapunov Functions