Ongoing and Future Work

In this chapter we will give a sketchy, but we hope compelling, idea of how the tautological control system framework can be used to say new things about control systems. This will also provide an illustration of how, in practice, one can do control theory within the confines of the tautological control system framework, without reverting to the comforting control parameterisations with which one is familiar. We will emphasise that some of these ideas are in the preliminary stages of investigation, so the final word on what results will look like has yet to be uttered. Nonetheless, we believe that even the clear problem formulations we give make it apparent that there is something “going on” here.

[1]  H. Cartan,et al.  Variétés analytiques réelles et variétés analytiques complexes , 1957 .

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

[3]  A. D. Lewis,et al.  Geometric local controllability: second-order conditions , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[4]  Paulo Tabuada,et al.  Quotients of Fully Nonlinear Control Systems , 2005, SIAM J. Control. Optim..

[5]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .

[6]  Andrei A. Agrachev,et al.  Local controllability and semigroups of diffeomorphisms , 1993 .

[7]  P. Stefan.  Accessible Sets, Orbits, and Foliations with Singularities , 1974 .

[8]  L. Praly,et al.  Necessary conditions for stability and attractivity of continuous systems , 2003 .

[9]  Gianna Stefani,et al.  On the relationship between global and local controllability , 1983, Mathematical systems theory.

[10]  J. Coron.  A necessary condition for feedback stabilization , 1990 .

[11]  H. Hermes.  On Local and Global Controllability , 1974 .

[12]  H. Sussmann.  Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems , 1983 .

[13]  M. Kawski.  The complexity of deciding controllability , 1990 .

[14]  Yu. S. Ledyaev,et al.  Asymptotic controllability implies feedback stabilization , 1997, IEEE Trans. Autom. Control..

[15]  Andrew D. Lewis,et al.  Locally convex topologies and control theory , 2016, Mathematics of Control, Signals, and Systems.

[16]  Pantelis Isaiah,et al.  Feedback stabilisation of locally controllable systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[17]  Henry Hermes,et al.  Local controllability and sufficient conditions in singular problems , 1976 .

[18]  H. Sussmann.  A general theorem on local controllability , 1987 .

[19]  Henry Hermes.  On Local Controllability , 1982 .

[20]  H. Hermes,et al.  Nonlinear Controllability via Lie Theory , 1970 .

[21]  Gianna Stefani,et al.  Controllability along a trajectory: a variational approach , 1993 .

[22]  Gianna Stefani,et al.  Normal local controllability of order one , 1984 .

[23]  M. Kawski.  Control variations with an increasing number of switchings , 1988 .

[24]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[25]  J. Basto-Gonçalves,et al.  Second-order conditions for local controllability , 1998 .

[26]  H. Sussmann.  Orbits of families of vector fields and integrability of distributions , 1973 .

[27]  Rosa-Maria Bianchini,et al.  Needle Variations that Cannot be Summed , 2003, SIAM J. Control. Optim..

[28]  R. Montgomery.  A Tour of Subriemannian Geometries, Their Geodesics and Applications , 2006 .

[29]  J. Zabczyk.  Some comments on stabilizability , 1989 .

[30]  H. Sussmann.  A Sufficient Condition for Local Controllability , 1978 .

[31]  Eduardo Sontag.  Controllability is harder to decide than accessibility , 1988 .

[32]  H. Sussmann,et al.  Controllability of nonlinear systems , 1972 .

[33]  High order conditions for local controllability and controlled stability , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[34]  Raymond O. Wells,et al.  Differential analysis on complex manifolds , 1980 .

[35]  Local controllability of affine distributions , 2010 .