Stabilization of linear systems by rotation

Abstract

We introduce the concept of "stabilization by rotation" for deterministic linear systems with negative trace. This concept encompasses the well-known concept of "vibrational stabilization" introduced by Meerkov in the 1970s and is a deterministic version of 'stabilization by noise' for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called "gain function" (possibly a constant) which is sufficiently large. To overcome the problem of what is "sufficiently large", we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.

Simultaneous hollowisation, joint numerical range, and stabilization by noise

“…The case b < 0 is slightly more subtle. We first derive a lower bound for s in (9). To this end note that the norm A 4 2 = ∆ is invariant under orthogonal transformations and a ≤ ∆.…”

“…Vice versa, one can ask, whether for any matrix A with trace A < 0, there exists a zero trace matrix M of a certain type, such that σ(A + M) ⊂ C − . In [9] it has been shown, that such a matrix M can always be chosen to be skew-symmetric. Then we say that M stabilizes A or by rotation, see e.g.…”

“…Hence for µ > 3.7 the system ẋ = (A + µ M0 )x is asymptotically stable. In [9] a servo-mechanism was described, which chooses a suitable gain µ adaptively via the feedback equation…”

Interplay of Analysis and Probability in Physics

“…A celebrated example is Kapiza's problem of the inverted pendulum [13]. This averaging is also effective in case of deterministic rotation of the system [10]. But there are very few examples due to additive noise.…”