Isolated semidefinite solutions of the continuous-time algebraic Riccati equation
The author considers the continuous-time algebraic Riccati equation (CARE) \[ {\mathcal R} (X) = A^*X + XA + XBB^* X - C^* C = 0 \] where \(A\), \(B\) and \(C\) are complex matrices of dimensions \(n \times n\), \(n \times p\) and \(q \times n\), respectively, focuses on the set \({\mathcal T} = \{X |{\mathcal R} (X) = 0\), \(X \leq 0\}\) of negative-semidefinite solutions, and characterizes those elements \(X\) of \({\mathcal T}\) which are isolated (in the topology which \({\mathcal T}\) inherits as a subset of the normed space \(\mathbb{C}^{n \times n})\). Such an isolated \(X\) has the property that for a sufficiently small \(\varepsilon\) there is no solution \(Y \in {\mathcal T}\), \(Y \neq X\), with \(|X - Y |< \varepsilon\). The author proves the following main result. Theorem: Assume that CARE has a solution \(X \leq 0\). Then \({\mathcal T}\) has an isolated element iff all purely imaginary eigenvalues of \(A\) are \(B\)- controllable. A solution \(X \in {\mathcal T}\) is isolated in \({\mathcal T}\) iff its kernel has the following property: if \(\text{Re} \lambda > 0\) and \[ \text{rank} {A - \lambda I \choose C} \leq n - 2 \quad \text{then either} \quad \ker {A - \lambda I \choose C} \cap \ker X = \varnothing \] or \(V(A,C) \cap E_\lambda (A) \subseteq \ker X\). Here \(E_\lambda = \ker (A - \lambda I)^n\) denotes the generalized eigenspace for every eigenvalue \(\lambda \in \sigma (A)\) and \(V(A,C) = \ker O (A,C)\) where \(O(A,C)\) is the observable matrix of the pair \((A,C)\) (hence \(V(A,C)\) is an unobservable subspace). The study of the reviewed paper is based on the author's results [SIAM J. Control Optimization 32, No. 4, 995-1007 (1994; Zbl 0799.93034)].
continuous-time algebraic Riccati equation
scientific article; zbMATH DE number 765290