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August  2013, 33(8): 3329-3353. doi: 10.3934/dcds.2013.33.3329

On the control of non holonomic systems by active constraints

Alberto Bressan 1, , Ke Han 2, and  Franco Rampazzo 3,

1. 

Department of Mathematics, Penn State University, University Park, Pa.16802
2. 

Department of Mathematics, Pennsylvania State University, University Park, PA 16802, United States
3. 

Dipartimento di Matematica Pura ed Applicata, Università di Padova, Padova 35141, Italy

Received  April 2012 Revised  August 2012 Published  January 2013

The paper is concerned with mechanical systems which are controlled by implementing a number of time-dependent, frictionless holonomic constraints. The main novelty is due to the presence of additional non-holonomic constraints. We develop a general framework to analyze these problems, deriving the equations of motion and studying the continuity properties of the "control-to-trajectory" maps. Various geometric characterizations are provided, in order that the equations be affine w.r.t. the time derivative of the control. In this case the system is fit for jumps, and the evolution is well defined also in connection with discontinuous control functions. The classical Roller Racer provides an example where the non-affine dependence of the equations on the derivative of the control is due only to the non-holonomic constraint. This is a case where the presence of quadratic terms in the equations can be used for controllability purposes.
Keywords: Impulsive control, non-holonomic system, active constraints..
Mathematics Subject Classification: Primary: 93C15, 37J60; Secondary: 70F25, 34A3.
Citation: Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
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show all references

References:
[1]

in "Mathematical Control Theory" (Eds. J. Baillieul and J. C. Willems), Springer Verlag, New York, (1998), 322-354.  Google Scholar

[2]

Springer Verlag, 2003. doi: 10.1007/b97376.  Google Scholar

[3]

Notices Amer. Math. Soc., 52 (2005), 324-333.  Google Scholar

[4]

Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.  Google Scholar

[5]

Boll. Un. Matem. Italiana, 2-B (1988), 641-656.  Google Scholar

[6]

J. Optim. Theory & Appl., 71 (1991), 67-84. doi: 10.1007/BF00940040.  Google Scholar

[7]

SIAM J. Control, 31 (1993), 1205-1220. doi: 10.1137/0331057.  Google Scholar

[8]

Arch. Rational Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6.  Google Scholar

[9]

Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 19 (1989), 195–-246. (1991).  Google Scholar

[10]

Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Mem., 1 (1991), 147-196.  Google Scholar

[11]

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 9 (1993), 5-30.  Google Scholar

[12]

Springer Verlag, 2004.  Google Scholar

[13]

Dynam. Cont. Discr. Imp. Systems, 4 (1998), 1-21.  Google Scholar

[14]

Ann. Soc. Polonaise Math., 6 (1927), 1-7. Google Scholar

[15]

Dynamical Systems, 16 (2001), 347-397. doi: 10.1080/14689360110090424.  Google Scholar

[16]

Int. J. Bifurc. Chaos, 15 (2005), 2747-2756. doi: 10.1142/S0218127405013745.  Google Scholar

[17]

Nonlinearity, 18 (2005), 2737-2743. doi: 10.1088/0951-7715/18/6/017.  Google Scholar

[18]

in "Proc. 30-th IEEE Conference on Decision and Control" IEEE Publications, New York, (1991), 437-442. Google Scholar

[19]

in "Symplectic Geometry and Mathematical Physics" (1991), 260-287, (Eds. P. Donato, C. Duval, J. Elhadad and G. M. Tuynman), Birkhäuser, Boston.  Google Scholar

[20]

J. Math. Syst. Estim. Control, 4 (1994), 385-388.  Google Scholar

[21]

Annals of Math., 63 (1956), 20-63.  Google Scholar

[22]

Springer Verlag, New York, 1990.  Google Scholar

[23]

Atti Accad. Naz. Lincei, Classe di Scienze Mat. Fis. Nat. Serie 8, 82 (1988), 685-695.  Google Scholar

[24]

European J. Mechanics A/Solids, 10 (1991), 405-431.  Google Scholar

[25]

Differential Geometry and Control, Proc. Sympos. Pure Math., (AMS, Providence) (1999), 279-296.  Google Scholar

[26]

Annals of Math., 69 (1959), 119-132.  Google Scholar

[27]

Springer-Verlag, Berlin, 1983.  Google Scholar

[28]

Ann. Prob., 6 (1978), 17-41.  Google Scholar

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