Abstract
Stable and Lipschitz stable hermitian solutions of the discrete algebraic Riccati equations are characterized, in the complex as well as in the real case.
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B. D. O. Anderson and J. B. Moore,Optimal Filtering, Prentice-Hall, Englewood Cliffs, NJ, 1979.
H. Baumgartel,Analytic Perturbation Theory for Matrices and Operators, Operator Theory: Advances and Applications, Vol. 15, Birkhäuser, Basel, 1985.
A. N. Churilov, On solutions of a quadratic matrix equation, inNonlinear Vibrations and Control Theory, pp. 24–33, Udmurt State University, Izhevsk, 1978 (Russian).
A. N. Churilov, On solutions of a quadratic matrix equation encountered in investigation of discrete control systems,Izv. Vyssh. Uchebn. Zaved. Mat.,11 (1986), 59–65 (Russian).
P. Dorato, Theoretical developments in discrete-time control,Automatica,19 (1983), 95–400.
D. F. Delchamps, A note on the analyticity of the Riccati metric, inAlgebraic and Geometric Methods in Linear System Theory, pp. 37–41, Lectures in Applied Mathematics, Vol. 18 (C. I. Byrnes and C. F. Martin, eds.), American Mathematical Society, Providence, RI, 1980.
I. Gohberg, P. Lancaster and L. Rodman,Matrices and Indefinite Scalar Products. Birkhäuser, Basel, 1983.
I. Gohberg, P. Lancaster and L. Rodman,Invariant Subspaces of Matrices with Applications, Wiley, New York, 1986.
I. Gohberg, P. Lancaster and L. Rodman,Matrix Polynomials, Academic Press, New York, 1982.
I. Gohberg, P. Lancaster and L. Rodman, Spectral analysis of selfadjoint matrix polynomials,Ann. of Math.,112 (1980), 34–71.
W. L. Green and E. W. Kamen, Stabilizability of linear systems over a commutative normed algebra with applications to spatially distributed and parameter dependent systems,SIAM J. Control Optim.,23 (1985), 1–18.
V. Kučera, The discrete Riccati equation of optimal control,Kybernetika,8 (1972), 430–447.
E. W. Kamen and P. P. Khargonekar, On the control of linear systems whose coefficients are functions of parameters,IEEE Trans. Automat. Control,29 (1984), 25–33.
M. A. Kaashoek, C. V. M. Van Der Mee and L. Rodman, Analytic operator functions with compact spectrum II. Spectral pairs and factorizations,Integral Equations Operator Theory,5 (1982), 791–827.
P. Lancaster and L. Rodman, Existence and uniqueness theorems for the algebraic Riccati equation,Internat. J. Control,32 (1980), 285–310.
P. Lancaster and L. Rodman, Solutions of the continuous and discrete-time algebraic Riccati equations: a review, inThe Riccati Equation (S. Bittanti, A. J. Laub, and J. C. Willems, eds.), Springer-Verlag, New York, 1991, pp. 11–51.
P. Lancaster, A. C. M. Ran and L. Rodman, Hermitian solutions of the discrete algebraic Riccati equation,Internat. J. Control,44 (1986), 777–802.
P. Lancaster, and M. Tismenetsky,The Theory of Matrices: With Applications, 2nd edn., Academic Press, Orlando, 1985.
B. P. Molinari, The stabilizing solution of the discrete algebraic Riccati equation,IEEE Trans. Automat. Control,20 (1975), 396–399.
T. Pappas, A. J. Laub and N. R. Sandell, On the numerical solution of the discrete-time algebraic Riccati equations,IEEE Trans. Automat. Control,25 (1980), 631–641.
L. Rodman, On extremal solutions of the algebraic Riccati equation, inAlgebraic and Geometric Methods in Linear System Theory, pp. 311–327, Lectures in Applied Mathematics, Vol. 18, (C. I. Byrnes and C. F. Martin, eds.), American Mathematical Society, Providence, RI, 1980.
A. C. M. Ran and L. Rodman, Stability of invariant maximal semidefinite subspaces I,Linear Algebra Appl.,62 (1984), 51–86.
A. C. M. Ran and L. Rodman, Stability of invariant Lagrangian subspaces I,Operator Theory: Advances and Applications, Vol. 32, pp. 181–228, Birkhauser, Basel, 1988.
A. C. M. Ran and L. Rodman, Stability of invariant Lagrangian subspaces II,Operator Theory: Advances and Applications, Vol. 40, pp. 391–425, Birkhauser, Basel, 1989.
A. C. M. Ran and L. Rodman, Stable solutions of real algebraic matrix Riccati equations, to appear inSIAM J. Control Optim.
A. C. M. Ran and L. Rodman, On parameter dependence of solutions of algebraic Riccati equations,Math. Control Signals Systems 1 (1988), 269–284.
A. C. M. Ran and R. Vreugdenhil, Existence and comparison theorems for algebraic Riccati equations for continuous — and discrete — time systems,Linear Algebra Appl.,99 (1988), 63–83.
E. D. Sontag,Mathematical Control Theory: Deterministic Finite Dimensional Systems, Springer-Verlag, New York, 1990.
M. A. Shayman, Geometry of the algebraic Riccati equations I, II,SIAM J. Control Optim.,21 (1983), 375–394; 395–409.
D. R. Vaughan, A nonrecursive algebraic solution for the discrete Riccati equation,IEEE Trans. Automat. Control,15 (1970), 597–599.
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This paper was written while the first author visited The College of William and Mary.
Partially supported by NSF Grant DMS-8802836 and by the Binational United States-Israel Science Foundation.
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Ran, A.C.M., Rodman, L. Stable Hermitian solutions of discrete algebraic Riccati equations. Math. Control Signal Systems 5, 165–193 (1992). https://doi.org/10.1007/BF01215844
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DOI: https://doi.org/10.1007/BF01215844