A parametrization of the nonnegative definite solutions of the algebraic Riccati equation
The author studies the set \(\mathcal P\) of Hermitian nonnegative definite solutions of the continuous time algebraic Riccati equation \(A^{*}P+PA-PBB^{*}P+C^{*}C=0\). Using the parametrization result of \textit{H. K. Wimmer} [SIAM J. Control Optim. 32, No. 4, 995-1007 (1994; Zbl 0799.93034)], he proves a theorem according to which any solution in the set \(\mathcal P\) can be represented as the product of the largest solution and a projection. This extends the results of \textit{F. M. Callier} and \textit{J. L. Willems} [IEEE Trans. Automat. Control AC-26, 1232-1242 (1981; Zbl 0472.93032)] and \textit{T. Ando} [Matrix quadratic equations (1988; Zbl 0645.34001)] proved under the hypothesis of stabilizability of the pair \((A,B)\). The analysis of the present paper only assumes the existence of a largest solution in \(\mathcal P\). The results are illustrated by a numerical example.
scientific article; zbMATH DE number 1192896