×

zbMATH — the first resource for mathematics

Estimates and exact expressions for Lyapunov exponents of stochastic linear differential equations. (English) Zbl 0669.34060
The paper deals with the asymptotic behaviour of the solutions of a stochastic linear differential equation of the form \(\dot x(t)=A_{j_ t}x(t)\), \(t\geq 0\), \(x(0)=x_ 0\in R^ d\), where \(j_ t\) is a stationary, ergodic, Markov process with a finite state space \(\{1,...,N\}\), and \(A_ 1,...,A_ N\) are \(d\times d\) constant matrices. Both sample path Lyapunov exponent \(\lambda\) and p-moment Lyapunov exponents g(p) (for positive p) are considered. It is proved that g(p) does not depend on \(x_ 0\), is a convex function of p, \(g(0)=0\), \(g(p)/p\) is nondecreasing, and \(g(p)\) and \(\lambda\) are simply related. For g(2), using a Lyapunov function approach, an exact expression is obtained as the maximal real part of the eigenvalues of a certain \(Nd^ 2\) dimensional matrix. A similar method allows to compute upper and lower bounds for \(g(p)\). Therefore, exploiting the relation between \(\lambda\) and the \(g(p)\)’s, upper and lower bounds for \(\lambda\) are obtained.
Reviewer: F.Flandoli

MSC:
34F05 Ordinary differential equations and systems with randomness
93E15 Stochastic stability in control theory
34D20 Stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1137/0144057 · Zbl 0561.93063 · doi:10.1137/0144057
[2] Arnold L., Probabilistic Analysis and Related Topics 3 pp 1– (1983)
[3] Arnold L., Lecture Notes in Mathematics 1186 pp 129– (1985)
[4] Arnold L., Lecture Notes in Mathematics 1186 pp 129– (1985)
[5] DOI: 10.1137/0146030 · Zbl 0603.60051 · doi:10.1137/0146030
[6] DOI: 10.1109/TAC.1977.1101612 · Zbl 0362.93033 · doi:10.1109/TAC.1977.1101612
[7] DOI: 10.1137/0146053 · Zbl 0613.60053 · doi:10.1137/0146053
[8] Blankenship G. L., Lecture Notes on Mathematics 1186 pp 160– (1985)
[9] Crauel H., Stochastics 14 pp 11– (1984)
[10] DOI: 10.1137/0136009 · Zbl 0412.60069 · doi:10.1137/0136009
[11] DOI: 10.1137/1112019 · doi:10.1137/1112019
[12] Has’minskii R. Z., Stochastic Stability of Differential Equations (1980) · doi:10.1007/978-94-009-9121-7
[13] Infante E. F., Journal of Applied Mechanics 35 pp 7– (1969)
[14] Kliemann W., Lecture Notes on Control and Information Sciences 16 (1979)
[15] DOI: 10.1080/07362998308809016 · Zbl 0525.93070 · doi:10.1080/07362998308809016
[16] DOI: 10.1080/07362998308809005 · Zbl 0517.93061 · doi:10.1080/07362998308809005
[17] Oseledec V. I., Transactions of the Moscow MathematicalSociety 19 pp 197– (1968)
[18] DOI: 10.1002/cpa.3160270303 · Zbl 0295.60046 · doi:10.1002/cpa.3160270303
[19] DOI: 10.1137/0146031 · Zbl 0603.60052 · doi:10.1137/0146031
[20] Wihstutz V., Lecture Notes in Mathematics 1186 pp 200– (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.