Abstract
In this paper, we study close connections that exist between the Riccati operator (differential) equation that arises in linear control systems and the symplectic group and its subsemigroup of symplectic Hamiltonian operators. A canonical triple factorization is derived for the symplectic Hamiltonian operators, and their closure under multiplication is deduced from this property. This semigroup of Hamiltonian operators, which we call the symplectic semigroup, is studied from the viewpoint of Lie semigroup theory, and resulting consequences for the theory of the Riccati equation are delineated. Among other things, these developments provide an elementary proof for the existence of a solution of the Riccati equation for all t ≥ 0 under rather general hypotheses.
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References
1. P. Bougerol, Kalman filtering with random coefficients and contractions. SIAM J. Control Optim. 31 (1993), 942–959.
2. L. Dieci and T. Eirola, Preserving monotonicity in the numerical solution of Riccati differential equations. Numer. Math. 74 (1996), 35–47.
3. R. Hermann, Cartanian geometry, nonlinear waves, and control theory, Part A. Interdisciplinary Mathematics 20, Math. Sci. Press, Boorline, MA (1979).
4. J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie groups, convex cones, and semigroups. Oxford Univ. Press, Oxford (1989).
5. J. Hilgert and K.-H. Neeb, Basic theory of Lie semigroups and applications. Lect. Notes Math. 1552, Springer-Verlag (1993).
6. V. Jurdjevic, Geometric control theory. Cambridge Univ. Press, Cambridge (1997).
7. K. Koufany, Semi-groupe de Lie associe a un cone symmetrique. Ann. Inst. Fourier 45 (1995), 1–29.
8. J. Lawson, Geometric control and Lie semigroup theory. In: Differential Geometry and Control. Proc. Symp. Pure Math., Vol. 64, Amer. Math. Soc. (1999), pp. 207–221.
9. J. Lawson and Y. Lim, Lie semigroups with triple decomposition. Pacific J. Math. 194 (2000), 393–412.
10. ——, The sympletic semigroup and Riccati differential equations, II. Preprint.
11. J. J. Levin, On the matrix Riccati equation. Proc. Amer. Math. Soc. 10 (1959), 519–524.
12. W. T. Reid, Monotonity properties of solutions of Hermitian Riccati equations. SIAM J. Math. Anal. 1 (1970), 195–213.
13. Yu. Sachkov, Survey on controllability of invariant systems on solvable Lie groups. In: Differential Geometry and Control. Proc. Symp. Pure Math., Vol. 64, Amer. Math. Soc. (1999), pp. 297–318.
14. M. Shayman, Phase portrait of the matrix Riccati equation. SIAM J. Control Optim. 24 (1986), 1–65.
15. E. Sontag, Mathematical control theory. Springer-Verlag, Berlin (1998).
16. M. Wojtkowski, Invariant families of cones and Lyapunov exponents. Ergodic Theory Dynam. Systems 5 (1985), 145–161.
17. ——, Measure theoretic entropy of the system of hard spheres. Ergodic Theory Dynam. Systems 8 (1988), 133–153.
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2000 Mathematics Subject Classification. 49N10, 93B27, 93B03, 22E15.
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Lawson, J., Lim, Y. The Symplectic Semigroup and Riccati Differential Equations. J Dyn Control Syst 12, 49–77 (2006). https://doi.org/10.1007/s10450-006-9683-8
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DOI: https://doi.org/10.1007/s10450-006-9683-8