Control sets for bilinear and affine systems

For homogeneous bilinear control systems, the control sets are characterized using a Lie algebra rank condition for the induced systems on projective space. This is based on a classical Diophantine approximation result. For affine control systems, the control sets around the equilibria for constant controls are characterized with particular attention to the question when the control sets are unbounded.

[1]  W. Kliemann,et al.  Controllability properties of bilinear systems in dimension 2 , 2016 .

[2]  Velimir Jurdjevic,et al.  Controllability properties of affine systems , 1984, The 23rd IEEE Conference on Decision and Control.

[3]  Luiz A. B. San Martin.  Invariant control sets on flag manifolds , 1993, Math. Control. Signals Syst..

[4]  D. Elliott.  Bilinear Control Systems: Matrices in Action , 2009 .

[5]  Wolfgang Kliemann,et al.  The dynamics of control , 2000 .

[6]  Wolfgang Kliemann,et al.  Asymptotic null controllability of bilinear systems , 1995 .

[7]  Y. Sachkov.  On Invariant Orthants of Bilinear Systems , 1998 .

[8]  Velimir Jurdjevic,et al.  Control systems on semi-simple Lie groups and their homogeneous spaces , 1981 .

[9]  Bernard Bonnard,et al.  Contrôlabilité des Systèmes Bilinéaires , 1981, Mathematical systems theory.

[10]  R. R. Mohler,et al.  Completely Controllable Bilinear Systems , 1968 .

[11]  C. J. Barros,et al.  On the number of control sets on projective spaces , 1996 .

[12]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[13]  William M. Boothby,et al.  Determination of the Transitivity of Bilinear Systems , 1979 .

[14]  G. Teschl.  Ordinary Differential Equations and Dynamical Systems , 2012 .

[15]  J. Gauthier,et al.  Contrôlabilité des Systèmes Bilinéaires , 1982 .

[16]  Osvaldo G. do Rocio,et al.  Semigroups of Affine Groups, Controllability of Affine Systems and Affine Bilinear Systems in $\mathrmSl(2, R)\rtimesR2$ , 2009, SIAM J. Control. Optim..

[17]  Nitakshi Goyal,et al.  General Topology-I , 2017 .

[18]  S. Lang,et al.  Introduction to Diophantine Approximations , 1995 .

[19]  Bernard Bonnard,et al.  Transitivity of families of invariant vector fields on the semidirect products of Lie groups , 1982 .

[20]  Andrea Bacciotti,et al.  On radial and directional controllability of bilinear systems , 2013, Syst. Control. Lett..

[21]  David L. Elliott,et al.  Bilinear Control Systems , 2009 .

[22]  David L. Elliott,et al.  Geometric control theory , 2000, IEEE Trans. Autom. Control..

[23]  H. Hermes,et al.  Foundations of optimal control theory , 1968 .

[24]  Mario Sigalotti,et al.  Approximately controllable finite-dimensional bilinear systems are controllable , 2021, Syst. Control. Lett..