Matrix Bounds for the Solution of the Continuous Algebraic Riccati Equation

Table 1

Numerical results of the lower eigenvalue bounds.

Method	Eigenvalue bounds
Ours ($\varepsilon =0.0417$)	${\lambda }_n(P)\ge 0.2192,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\lambda }_1(P)\ge 0.5951,\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{tr}\hspace{0.17em}(P)\ge 0.8143,\mathrm{\hspace{0.17em}\hspace{0.17em}}\det \hspace{0.17em}(P)\ge 0.1304$
Choi and Kuc 2002 [20] ($\varepsilon =0.0417$)	${\lambda }_n(P)\ge 0.1879,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\lambda }_1(P)\ge 0.5831,\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{tr}\hspace{0.17em}(P)\ge 0.771,\mathrm{\hspace{0.17em}\hspace{0.17em}}\det \hspace{0.17em}(P)\ge 0.1095$
Chen and Lee 2009 [21]	${\lambda }_n(P)\ge 0.0807,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\lambda }_1(P)\ge 0.4996,\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{tr}\hspace{0.17em}(P)\ge 0.5803,\mathrm{\hspace{0.17em}\hspace{0.17em}}\det \hspace{0.17em}(P)\ge 0.1226$
Lee 1997 [19]	${\lambda }_n(P)\ge 0.1706,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\lambda }_1(P)\ge 0.5840,\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{tr}\hspace{0.17em}(P)\ge 0.7546,\mathrm{\hspace{0.17em}\hspace{0.17em}}\det \hspace{0.17em}(P)\ge 0.0996$
Kwon and Pearson 1977 [13]	${\lambda }_n(P)\ge 0.1705$
Yasuda and Hirai 1979 [15]	${\lambda }_1(P)\ge 0.2369$
Karanam 1983 [16]	${\lambda }_1(P)\ge 0.225$
Patel and Toda 1978 [14]	${\lambda }_1(P)\ge 0.5613$
Kwon et al. 1985 [17]	$\mathrm{tr}\hspace{0.17em}(P)\ge 0.4210$
Wang et al. 1986 [6]	$\mathrm{tr}\hspace{0.17em}(P)\ge 0.6568$