Abstract
We consider products of matrix exponentials under the assumption that the matrices span a nilpotent Lie algebra. In 1995, Gurvits conjectured that nilpotency implies that these products are, in some sense, simple. More precisely, there exists a uniform bound ι such that any product can be represented as a product of no more than ι matrix exponentials. This conjecture has important applications in the analysis of linear switched systems, as it is closely related to the problem of reachability using a uniformly bounded number of switches. It is also closely related to the concept of nice reachability for bilinear control systems. The conjecture is trivially true for the case of first-order nilpotency. Gurvits proved the conjecture for the case of second-order nilpotency using the Baker-Campbell-Hausdorff formula. We show that the conjecture is false for the third-order nilpotent case using an explicit counterexample. Yet, the underlying philosophy behind Gurvits' conjecture is valid in the case of third-order nilpotency. Namely, such systems do satisfy the following nice reachability property: any point in the reachable set can be reached using a piecewise constant control with no more than four switches. We show that even this form of finite reachability is no longer true for the case of fifth-order nilpotency.
Original language | English |
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Pages (from-to) | 1123-1126 |
Number of pages | 4 |
Journal | IEEE Transactions on Automatic Control |
Volume | 52 |
Issue number | 6 |
DOIs | |
State | Published - Jun 2007 |
Keywords
- Bang-bang control
- Bilinear control systems
- Differential inclusions
- Fuller's problem
- Lie algebra
- Lie's product formula
- Optimal control
- Singular control
- Switched linear systems