seminar: Stabilization by rotation
speaker: Hans Crauel (Technische Universität Ilmenau)abstract: It is well known that a linear system $\dot x=Ax$ can be stabilized by ``stochastic vibration'', provided the matrix, or operator, $A$ has negative trace. More precisely, one can prove that the maximal Lyapunov exponent of the linear stochastic differential equation $dx=Ax\,dt+\sigma\sum{j=1}m Bjx\circ dWj(t)$ is negative for $\sigma$ sufficiently large, provided the span of the matrices $Bj$, $1\leq j\leq m$, is sufficiently large (e.\,g., in finite dimensions the space of skew symmetric matrices suffices). This has subsequently been generalized in several directions, in particular allowing for more degeneracy of the noise term. Here we discuss the problem whether this can be `generalized' to deterministic systems in the sense that the maximal real part of the eigenvalues for $\dot x=(A+\sigma B)x$ can be made negative by a suitable choice of a skew symmetric matrix $B$, and $\sigma$ sufficiently large. Then we discuss how to achieve this in an adaptive manner, which amounts to investigating the time inhomogeneous system $\dot x=(A+\sigma(t)B)x$. On first view this may appear to be obvious. However, after realizing that there exist functions $t\mapsto\sigma(t)$ with $\lim\sigma(t)=\infty$ for $t\to\infty$, for which the time inhomogeneous system is unstable, one may agree that it is not. Finally, we look at periodic $\sigma$, generalizing a result of Meerkov's on vibrational control. (joint work with Tobias Damm, Braunschweig, and Achim Ilchmann, Ilmenau)
timetable:
Tue 28 Mar, 15:00 - 15:40, Sala Conferenze Centro De Giorgi
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