Properties of the solutions of rational matrix difference equations. (English) Zbl 1055.39033

Freiling, G.; Hochhaus, A.

Comput. Math. Appl. 45, No. 6-9, 1137-1154 (2003).

Consider the matrix difference equation \[ \begin{split} X_t=A^*X_{t+1}A+Q+\Pi_1(X_{t+1})-[S+A^*X_{t+1}B+ \Pi_{12}(X_{t+1})]\times\\ \times[R+B^*X_{t+1}B+\Pi_2(X_{t+1})]^+[S+A^*X_{t+1} B+\Pi_{12}(X_{t+1})]^*\end{split}\tag{A} \] and the corresponding algebraic equations \[ \begin{split} A^*XA-X+Q+\Pi_1(X)-[S+A^*XB+\Pi_{12}(X)]\times\\ \times [R+B^*XB+\Pi_2(X)]^+[S+A^*XB+\Pi_{12}(X)]^*= 0.\end{split}\tag{B} \] Sufficient conditions for the existence and uniqueness of stabilizing solutions for (A) are established. Finally, it is shown that under certain conditions on the coefficients the solutions of (A) converge for any positive value to the stabilizing solutions of (B).

Reviewer: B. M. Agrawal (Gwalior)

MSC:

39A20	Multiplicative and other generalized difference equations
39A70	Difference operators

Keywords:

rational matrix difference equations; generalized Riccati difference equations; comparison theorem; Convergence

References:

[1]	Lancaster, P.; Rodman, L., Algebraic Riccati equations, (1995), Clarendon Press Oxford · Zbl 0836.15005
[2]	Wonham, W.M., On a matrix Riccati equation of stochastic control, SIAM J. control, 6, 681-697, (1968) · Zbl 0182.20803
[3]	Abou-Kandil, H.; Freiling, G.; Jank, G., On the solution of discrete-time Markovian jump linear quadratic control problems, Automatica J. IFAC, 31, 5, 765-768, (1995) · Zbl 0822.93074
[4]	Fragoso, M.D.; Costa, O.L.V.; de Souza, C.E., A new approach to linearly perturbed Riccati equations arising in stochastic control, Appl. math. optim., 37, 1, 99-126, (1998) · Zbl 0895.93042
[5]	V. Dragan and T. Morozan, Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping, J. Differential Equations (to appear). · Zbl 1039.34052
[6]	Damm, T.; Hinrichsen, D., Newton’s method for a rational matrix equation occuring in stochastic control, Linear algebra appl., 332-334, 81-109, (2001) · Zbl 0982.65050
[7]	Yong, J.; Zhou, X.Y., Stochastic controls. Hamiltonian systems and HJB equations, (1999), Springer-Verlag New York · Zbl 0943.93002
[8]	G. Freiling and A. Hochhaus, On a class of rational matrix differential equations arising in stochastic control, Linear Algebra Appl. (to appear). · Zbl 1070.34054
[9]	Zeidler, E., Nonlinear functional analysis and its applications. I. fixed-point theorems, (1986), Springer-Verlag New York
[10]	Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1994), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 0815.15016
[11]	Krasnoselskii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
[12]	Berman, A.; Ben-Israel, A., Linear equations over cones with interior: A solvability theorem with applications to matrix theory, Linear algebra appl., 7, 139-149, (1973) · Zbl 0254.15010
[13]	Schneider, H., Positive operators and an inertia theorem, Numer. math., 7, 11-17, (1965) · Zbl 0158.28003
[14]	Lancaster, P.; Tismenetsky, M., The theory of matrices, (1985), Academic Press Orlando, FL · Zbl 0516.15018
[15]	Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian systems, (1996), Kluwer Academic Dordrecht · Zbl 1304.39002
[16]	Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses, SIAM J. appl. math., 17, 434-440, (1969) · Zbl 0265.15002
[17]	Clements, D.J.; Wimmer, H.K., Monotonicity of the optimal cost in the discrete-time regulator problem and Schur complements, Automatica J. IFAC, 37, 1779-1786, (2001) · Zbl 0990.93069
[18]	Wimmer, H.K.; Pavon, M., A comparison theorem for matrix Riccati difference equations, Systems control lett., 19, 3, 233-239, (1992) · Zbl 0774.49023
[19]	Freiling, G.; Jank, G., Existence and comparison theorems for algebraic Riccati equations and Riccati differential and difference equations, J. dynam. control systems, 2, 4, 529-547, (1996) · Zbl 0979.93091
[20]	Freiling, G.; Ionescu, V., Time-varying discrete Riccati equation: some monotonicity results, Linear algebra appl., 286, 1-3, 135-148, (1999) · Zbl 0939.93022
[21]	de Souza, C.E.; Fragoso, M.D., On the existence of maximal solution for generalized algebraic Riccati equations arising in stochastic control, Systems control lett., 14, 3, 233-239, (1990) · Zbl 0701.93106
[22]	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, 5, 309-315, (1985) · Zbl 0567.93059
[23]	Ait Rami, M.; El Ghaoui, L., LMI optimization for nonstandard Riccati equations arising in stochastic control, IEEE trans. automat. control, 41, 11, 1666-1671, (1996) · Zbl 0863.93087
[24]	do Val, J.B.R.; Geromel, J.C.; Costa, O.L.V., Uncoupled Riccati iterations for the linear quadratic control problem of discrete-time Markov jump linear systems, IEEE trans. automat. control, 43, 12, 1727-1733, (1998) · Zbl 1056.93537
[25]	El Bouhtouri, A.; Hinrichsen, D.; Pritchard, A.J., H∞-type control for discrete-time stochastic systems, Internat. J. robust nonlinear control, 9, 13, 923-948, (1999) · Zbl 0934.93022