State-transition matrix: Difference between revisions

Revision as of 12:47, 4 February 2015

Revision as of 17:18, 4 February 2015

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==Peano-Baker series==

The most general transition matrix is given by the Peano-Baker series

:<math> \mathbf{\Phi}(t,\tau) = \mathbf{I} + \int_\tau^t\mathbf{A}(\sigma_1)\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^t{\sigma_1}\mathbf{A}(\sigma_2)\,d\sigma_2\,d\sigma_1 + \int_\tau^t\mathbf{A}(\sigma_1)\int_\tau^t{\sigma_1}\mathbf{A}(\sigma_2)\int_\tau^t{\sigma_2}\mathbf{A}(\sigma_3)\,d\sigma_3\,d\sigma_2\,d\sigma_1 + ...</math>

where <math>\mathbf{I}</math> is the [[identity matrix]]. This matrix converges uniformly and absolutely to a solution that exists and is unique.<ref name=rugh />