Dispersiveness, higher stability and controllability of linear control systems on the Heisenberg group

This paper presents sufficient conditions for dispersiveness of linear control systemns on the Heisenberg group. The dispersiveness implies the absolute stability of the orbits, which means stability of all orders. Necessary conditions for the existence of a control set are derived. A linear control system is determined by a derivation D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {D}}$$\end{document} of the Heisenberg algebra h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {h}}$$\end{document} and invariant vector fields X1,& mldr;,Xm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{1},\ldots ,X_{m}$$\end{document}. The main result assures that the system is dispersive with respect to a compact set K in the control range, if the mean limits limtn ->+infinity 1tn integral 0tn & sum;i=1muinsXids\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\lim }\limits _{{t_{n} \rightarrow + \infty }} \frac{1}{{t_{n} }}\int _{0}<^>{{t_{n} }} {\sum \nolimits _{{i = 1}}<^>{m} {u_{i}<^>{n} \left( s \right) X_{i} {\hspace{1.0pt}} ds} } $$\end{document} do not reach the subspace h2+ImD subset of h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {h}}<^>{2}+\textrm{Im}\left( {\mathcal {D}}\right) \subset {\mathfrak {h}}$$\end{document}. Consequently, the system is K-dispersive, if 0 is not an element of K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\notin K$$\end{document} and the sum h2+ImD\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {h}}<^>{2}+\textrm{Im}\left( {\mathcal {D}}\right) $$\end{document} meets SpanX1,& mldr;,Xm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Span\left\{ X_{1},\ldots ,X_{m}\right\} $$\end{document} trivially. The mean limit criterion is applied to the linear Heisenberg flywheel and other three-dimensional systems.