Perturbation bounds and characterisation of the solution of the associated algebraic Riccati equation
The following complex algebraic matrix quadratic equation is studied: \[ Q+A^*X+X^*A-X^*MX=0,\tag{1} \] where \(Q,M,A\) are given complex \(n\times n\) matrices, and \(X\) is an unknown \(n\times n\) matrix. The standing hypotheses are: \(Q\) and \(M\) are positive semidefinite, the pair \((Q,A)\) is detectable, and the pair \((A,M)\) is stabilizable. The above equation is closely related to the ubiquitous algebraic matrix Riccati equation which has the same form (1), but only Hermitian solutions \(X\) are sought. An effective parametrization of the solutions of (1) is obtained: The parameter set is the group of \(n\times n\) unitary matrices with a complex algebraic set removed; in particular, the set of solutions of (1) has no isolated points. Allowing for a small perturbations of the coefficients \(Q,M\) and \(A\), local linear perturbation bounds are obtained for solutions of the perturbed equation (1). These bounds are based on computation of partial Fréchet derivatives, and are valid asymptotically when the perturbations of \(Q,M\), and \(A\) tend to zero. Another type of perturbation bounds is derived as well, which is applicable in well-defined neighborhoods of the coefficients.
complex algebraic matrix quadratic equation
scientific article; zbMATH DE number 1371030
scientific article