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On the generalized algebraic Riccati equation for continuous-time descriptor systems. (English) Zbl 0931.93059

The authors study the following generalized algebraic Riccati equation (GARE) for continuous-time descriptor systems \[ A^{T}X + X^{T}A + Q + X^{T}RX = 0, \quad E^{T}X = X^{T}E, \] where \(E, A, Q, R \in {\mathbb R}^{n \times n}\), \(Q = Q^{T}\), \(R = R^{T}\) and rank \(E = r \leq n\). No assumptions are made on the definiteness of \(Q\) and \(R\). Necessary and sufficient conditions for the existence of stabilizing solutions of the GARE are derived based on the Hamiltonian matrix pencil approach. It is shown that all stabilizing solutions of the GARE can be expressed by real solutions to an appropriate algebraic quadratic matrix equation. The main result of the paper can be applied to a wide class of control problems for continuous-time descriptor systems, including \(H_{2}/H_{\infty}\) control problems.

MSC:

93D15 Stabilization of systems by feedback
93C35 Multivariable systems, multidimensional control systems
15A24 Matrix equations and identities
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