Stability radius for structured perturbations and the algebraic Riccati equation (Q581344)

Given a stable (i. e. with eigenvalues in the open left half plane) matrix A, and matrices B, C of appropriate sizes, define the stability radius for structured perturbations by \(r=r(A,B,C)=\inf \{\| D\|:\sigma (A+BDC)\cap \{z:\) Re \(z\geq 0\}=\emptyset \}\). In the paper various properties of r are studied. In particular, it is shown that r plays a role in the optimization problem with parameter \(\rho\) : min\(\int^{\infty}_{0}[\| v(s)\|^ 2-\rho \| y(s)\|^ 2]ds\) subject to \(v\in L^ 2\), \(\dot x=Ax+Bu\), \(x(0)=x_ 0\), \(y=Cx\). Consequently, r is characterized in terms of the parametrized algebraic Riccati equation (1) \(A^*P+PA-\rho C^*C-PBB^*P=0.\) One of main results: Suppose \(\rho <r^ 2<\infty\). Then there is unique stabilizing solution \(P_{\rho}\) of (1) and this solution is Hermitian; if \(\rho =r^ 2<\infty\), then there is unique solution \(P_{r^ 2}\) of (1) for which all eigenvalues of \(A-BB^*P_{r^ 2}\) are in the closed left half plane, and \(P_{r^ 2}=P^*_{r^ 2}\); conversely, if there is a solution \(P_{\rho}=P^*_{\rho}\) of (1), then \(\rho \leq r^ 2\). The behavior of \(P_{\rho}\) as a function of \(\rho\) is studied. Finally, a nonlinear system (2) \(\dot x=Ax+BN(C(x,t))\), \(t\geq 0\), \(x(0)=x_ 0\), where N(x,t) is continuously differentiable and \(N(0,t)=0\), is considered. The regions of asymptotic stability of the origin for (2) are studied in terms of \(r=r(A,B,C)\) and of the stabilizing solution \(P_{\rho}\) of (1).