## Transition matrix and generalized matrix exponential via the Peano-Baker series.(English)Zbl 1093.39017

Using generalized form of the Peano-Baker series, the author gives a closed form of the transition matrix for the linear system $$x^\Delta (t) = A(t) x(t)$$ on an arbitrary time scale $$\mathbb{T}$$ (i.e., some nonempty closed subset of the real axis $$\mathbb{R}$$), where $$A(t)$$ is an $$n \times n$$ matrix function, $$x^\Delta (t)$$ is so-called delta derivative. When $$A(t) \equiv A$$ is a constant matrix then the transition matrix has the form of a generalized matrix an exponential on the time scale. Using the generalized Laplace transform, an explicit representation of the time scale matrix exponential is obtained. It is shown that the matrix exponential can be expressed also as a finite sum of powers of the matrix $$A$$ with infinitely delta differentiable functions as coefficients.

### MSC:

 39A12 Discrete version of topics in analysis 34A30 Linear ordinary differential equations and systems

### Keywords:

time scale; linear system; transition matrix; matrix exponential
Full Text:

### References:

 [1] DOI: 10.1080/10236190412331303468 · Zbl 1066.34006 [2] DOI: 10.1016/S0377-0427(01)00432-0 · Zbl 1020.39008 [3] DOI: 10.1007/978-0-8176-8230-9 [4] Bohner M., Dynamic Equations on Time Scales: An Introduction with Applications (2001) · Zbl 0978.39001 [5] Hilger S., Results in Mathematics 18 pp 18– (1990) · Zbl 0722.39001 [6] Hilger S., Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (1988) · Zbl 0695.34001 [7] Kailath T., Linear Systems (1980) [8] Kelly W., Difference Equations: An Introduction with Applications (2001) [9] Kelly W., The Theory of Differential Equations Classical and Qualitative (2004) [10] Rugh W.J., Linear System Theory (1996) · Zbl 0892.93002
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.