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Explicit solutions for a system of coupled Lyapunov differential matrix equations

Published online by Cambridge University Press:  20 January 2009

L. Jodar  and
M. Mariton
L. Jodar
Affiliation:
Department of Applied Mathematics, Polytechnical University of Valencia, P.O. Box 22.012, Valencia, Spain
M. Mariton
Affiliation:
Laboratoire des Signaux et Systemes, C.N.R.S., E.S.E., Plateau du Moulon, 91.190 GIF, France
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This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the type

where Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 30 , Issue 3 , October 1987 , pp. 427 - 434
Copyright
Copyright © Edinburgh Mathematical Society 1987

References

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