Abstract
We give a new proof for the convergence of the solution of a terminal value problem for the periodic Riccati differential equation towards its strong solution as t → −∞. The proof is mainly based on well-known comparison results and also on an explicit representation formula for the solution that reflects precisely the dependence on the terminal value. Moreover, we give sufficient conditions for the existence of a periodic solution of the differential equation. Similar results are derived for the discrete-time Riccati equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bitmead, R.R., Gevers, M.R., Petersen, I.R., Kaye, R.J., “Monotonicity and stabilizability properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples”, Systems Control Lett. 5 (1985), 309–315.
Bittanti, S., Colaneri, P., De Nicoiao, G., “The Periodic Riccati Equation”, in: “The Riccati Equation” (ed. by Bittanti, S., Laub, A.J., Willems, J.C.), Berlin: Springer, 1991.
Bittanti, S., Bolzern, P., “Stabilizability and detectability of linear periodic systems”, Systems Control Lett. 6 (1985), 141–145.
Bittanti, S., Laub, A.J., Willems, J.C., “The Riccati Equation”, Berlin: Springer Verlag, 1991.
Caines, P.E., Mayne, D.Q., “On the discrete time matrix Riccati equation of optimal control”, Internat. J. Control 12(1971), 785–794 (erratum: Internat. J. Control 14 (1971), 205-207).
Callier, F.M., Desoer, C.A., “Linear system theory”, New York: Springer-Verlag, 1991.
Callier, F.M. and Willems, J.K., “Criterion for the convergence of the solution of the Riccati differential equation”, IEEE Trans. Automat. Control 26 (1981), 1232–1242.
Callier, F.M., Winkin, J., “Convergence of the Time-Invariant Riccati Differential Equation towards Its Strong Solution for Stabilizable Systems”, J. Math. Anal. Appl. 192 (1995), 230–257.
Callier, F.M., Winkin, J., Willems, J.L., “Convergence of the time-invariant Riccati differential equation and LQ-problem: Mechanisms of attraction”, Internat. J. Control 59 (1994), 983–1000.
Chen, Y.Z., Liu, J.Q., Chen, S.B., “Comparison and uniqueness theorems for periodic Riccati differential equations”, Internat. J. Control 69 (1998), 467–473.
D’Alessandro, D., “Geometric Aspects of the Riccati Difference Equation in the Nonsymmetric Case”, Linear Algebra Appl. 255 (1997), 1–18.
De Nicoiao, G., “On the convergence to the strong solution of periodic Riccati equations”, Internat. J. Control 56 (1992), 87–97.
De Nicolao, G., “ Cyclomonotonicity and Stabilizability Properties of Solutions of the Difference Periodic Riccati Equation”, IEEE Trans. Automat. Control 37 (1992), 1405–1410.
De Nicolao, G., “On the time-varying Riccati difference equation of optimal filtering”, SIAM J. Control Optim. 30 (1992), 1251–1269.
De Nicoiao, G., “Cyclomonotonicity, Riccati Equations and Periodic Receding Horizon Control”, Automatica 30 (1994), 1375–1388.
De Nicolao, G., Gevers, M., “Difference and Differential Riccati Equations: A Note on the Convergence to the Strong Solution”, IEEE Trans. Automat. Control 37 (1992), 1055–1057.
Freiling, G., Ionescu, V., “Time-varying discrete Riccati equation: some monotonicity results”, Linear Algebra Appl. 286 (1999), 135–148.
Freiling, G., Ionescu, V., “Nonsymmetric discrete-time difference and algebraic Riccati equations: some representation formulae and comments”, Dynam. Systems Appl. 8 (1999), 421–437.
Freiling, G., Jank, G., “Non-Symmetric Matrix Riccati Equations”, Z. Anal. Anwendungen 14 (1995), 259–284.
Freiling, G., Jank, G., “Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations”, J. Dynam. Control Systems 2 (1996), 529–547.
Lancaster, P., Rodman, L., “Algebraic Riccati Equations”, New York: Oxford Science Publications, 1995.
Matson, J.B., Anderson, B.D.O., Laub, A.J., Clements, D.J., “Riccati difference equations for discrete time spectral factorization with unit circle zeros”, World Sci. Ser. Appl. Anal. 5 (1995), 311–326.
Pastor, A., Hernandez, V., “Differential Periodic Riccati Equations: Existence and Uniqueness of Nonnegative Definite Solutions”, Math. Control Signals Systems 6 (1993), 341–362.
Park, P.G., Kailath, T., “Convergence of the DRE Solution to the ARE Strong Solution”, IEEE Trans. Automat. Control 42 (1997), 573–578.
Reid, W.T., “Riccati Differential Equations”, New York: Academic Press, 1972.
Shayman, M.A., “On the phase portrait of the matrix Riccati equation arising from the periodic control problem”, SIAM J. Control Optim. 23 (1985), 717–751.
Willems, J.C., “Least squares stationary optimal control and the algebraic Riccati equation”, IEEE Trans. Automat. Control 16 (1971), 621–634.
Wimmer, H.K., Pavon, M., “A comparison theorem for matrix Riccati difference equations”, System Control Lett. 19 (1992), 233–239.
Yakubovich, V.A., Starzhinskii, V.M., “Linear differential equations with periodic coefficients”, New York: Wiley, 1975.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Freiling, G., Hochhaus, A. Convergence and existence results for continuous- and discrete-time Riccati equations. Results. Math. 42, 252–276 (2002). https://doi.org/10.1007/BF03322854
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322854
Keywords
- Periodic matrix Riccati equations
- periodic equilibria
- Riccati difference equation
- convergence to the strong solution