# Singular Limits for Impulsive Lagrangian Systems with Dissipative Sources

## Abstract

Consider a mechanical system, described by finitely many Lagrangian coordinates. Assume that an external controller can influence the evolution of the system by directly assigning the values of some of the coordinates. If these assignments are implemented by means of frictionless constraints, one obtains a set of ordinary differential equations where the right hand side depends also on the time derivatives of the control functions. Some basic aspects of the mathematical theory for these equations are reviewed here. We then consider a system with an additional dissipative term, which vanishes on a stable submanifold *N*. As the coefficient of the source term approaches infinity, we show that the limiting impulsive dynamics on the reduced state space *N* can be modelled by two different systems, depending on the order in which two singular limits are taken. These results are motivated by the analysis of impulsive systems with non-holonomic constraints.

## Keywords

Cauchy Problem Control Function Bound Variation Impulsive Control Singular Limit## Preview

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## References

- [1]J. P. Aubin and A. Cellina,
*Differential Inclusions*. Springer-Verlag, Berlin, 1984.MATHGoogle Scholar - [2]A. Bressan,
*On differential systems with impulsive controls*, Rend. Semin. Mat. Univ. Padova**78**(1987), 227–236.MathSciNetGoogle Scholar - [3]A. Bressan,
*Impulsive control of Lagrangian systems and locomotion in fluids*, Discrete Cont. Dynam. Syst., to appear.Google Scholar - [4]A. Bressan and F. Rampazzo,
*On differential systems with vector-valued impulsive controls*, Boll. Un. Mat. Ital.**2-B**(1988), 641–656.MathSciNetGoogle Scholar - [5]A. Bressan and F. Rampazzo,
*Impulsive control systems with commutative vector fields*, J. Optim. Theory Appl.**71**(1991), 67–83.MATHCrossRefMathSciNetGoogle Scholar - [6]A. Bressan and F. Rampazzo,
*On differential systems with quadratic impulses and their applications to Lagrangian mechanics*, SIAM J. Control Optim.**31**(1993), 1205–1220.MATHCrossRefMathSciNetGoogle Scholar - [7]A. Bressan and F. Rampazzo,
*Impulsive control systems without commutativity assumptions*, J. Optim. Theory Appl.**81**(1994), 435–457.MATHCrossRefMathSciNetGoogle Scholar - [8]A. Bressan and F. Rampazzo,
*Stabilization of Lagrangian systems by moving constraints*, to appear.Google Scholar - [9]A. Bressan,
*Hyper-impulsive motions and controllizable coordinates for Lagrangean systems*, Atti Accad. Naz. Lincei, Memorie, Serie VIII, Vol. XIX (1990), 197–246.Google Scholar - [10]A. Bressan,
*On some control problems concerning the ski or swing*, Atti Accad. Naz. Lincei, Memorie, Serie IX, Vol. I, (1991), 147–196.MathSciNetGoogle Scholar - [11]F. Cardin and M. Favretti,
*Hyper-impulsive motion on manifolds*. Dynam. Contin. Discrete Impuls. Systems**4**(1998), 1–21.MATHMathSciNetGoogle Scholar - [12]F.H. Clarke, Yu S. Ledyaev, R.J. Stern, and P.R. Wolenski,
*Nonsmooth Analysis and Control Theory*, Springer-Verlag, New York, 1998.MATHGoogle Scholar - [13]G. Dal Maso and F. Rampazzo,
*On systems of ordinary differential equations with measures as controls*, Differential Integral Equat.**4**(1991), 739–765.MATHGoogle Scholar - [14]G. Dal Maso, P. Le Floch, and F. Murat,
*Definition and weak stability of nonconservative products*, J. Math. Pures Appl. (9)**74**(1995), 483–548.MATHMathSciNetGoogle Scholar - [15]V. Jurdjevic,
*Geometric Control Theory*, Cambridge University Press, 1997.Google Scholar - [16]P. LeFloch.
*Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves*, Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2002.MATHGoogle Scholar - [17]W. S. Liu and H. J. Sussmann,
*Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories*, In Proc. 30-th IEEE Conference on Decision and Control, IEEE Publications, New York, 1991, pp. 437–442.Google Scholar - [18]C. Marle,
*Géométrie des systèmes mécaniques à liaisons actives*, In Symplectic Geometry and Mathematical Physics, 260–287, P. Donato, C. Duval, J. Elhadad, and G. M. Tuynman (Eds.), Birkhäuser, Boston, 1991.Google Scholar - [19]B. M. Miller,
*The generalized solutions of ordinary differential equations in the impulse control problems*, J. Math. Systems Estim. Control**4**(1994).Google Scholar - [20]B. M. Miller,
*Generalized solutions in nonlinear optimization problems with impulse controls. I. The problem of the existence of solutions*, Automat. Remote Control**56**(1995), 505–516.MATHMathSciNetGoogle Scholar - [21]B. M. Miller,
*Generalized solutions in nonlinear optimization problems with impulse controls. II. Representation of solutions by means of differential equations with a measure*, Automat. Remote Control**56**(1995), 657–669.MATHMathSciNetGoogle Scholar - [22]M. Motta and F. Rampazzo,
*Dynamic programming for nonlinear systems driven by ordinary and impulsive controls*, SIAM J. Control Optim.**34**(1996), 199–225.MATHCrossRefMathSciNetGoogle Scholar - [23]H. Nijmeijer and A. van der Schaft,
*Nonlinear dynamical control systems*, Springer-Verlag, New York, 1990.MATHGoogle Scholar - [24]F. Rampazzo,
*On the Riemannian structure of a Lagrangian system and the problem of adding time-dependent coordinates as controls*, European J. Mechanics A/Solids**10**(1991), 405–431.MATHMathSciNetGoogle Scholar - [25]B. L. Reinhart,
*Foliated manifolds with bundle-like metrics*, Annals of Math.**69**(1959), 119–132.CrossRefMathSciNetGoogle Scholar - [26]B. L. Reinhart,
*Differential geometry of foliations. The fundamental integrability problem*. Springer-Verlag, Berlin, 1983.MATHGoogle Scholar - [27]G. V. Smirnov,
*Introduction to the theory of differential inclusions*, American Mathematical Society, Graduate studies in mathematics, vol. 41 (2002).Google Scholar - [28]E. D. Sontag,
*Mathematical control theory. Deterministic finite-dimensional systems*. Second edition. Springer-Verlag, New York, 1998.MATHGoogle Scholar