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New integrating methods for time-varying linear systems and Lie-group computations. (English) Zbl 1152.93340

Summary: In many engineering applications the Lie group calculation is very important. With this in mind, the subject of this paper is an in-depth investigation of time-varying linear systems and the accompanied Lie group calculations. In terms of system matrix \(A\) and a one-order lower fundamental solution matrix associated with the sub-state matrix function \( A^s_s\), we propose two methods to transform the system into one with a nilpotent system matrix. As a consequence, we obtain two different calculations of the general linear group. Then the nilpotent system is further transformed into a unique new system \(\dot{ Z}(t)={ B}(t){ Z}(t)\), which has a special simple \({ B}(t)\), whose \({ B}^s_s\) and \(B^0_0\) are vanishing. Correspondingly, we get a third calculation of the general linear group. By using the nilpotent property we can develop a very simple numerical scheme of nilpotent type to calculate the state transition matrix. We also develop a Lie group solver in terms of the exponential mapping of \({ B}\). Several numerical examples are employed to assess the performance of the proposed schemes. The new Lie group solver proves to be very stable and highly accurate.

MSC:

93B29 Differential-geometric methods in systems theory (MSC2000)
34A26 Geometric methods in ordinary differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
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