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.
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References
J. P. Aubin and A. Cellina, Differential Inclusions. Springer-Verlag, Berlin, 1984.
A. Bressan, On differential systems with impulsive controls, Rend. Semin. Mat. Univ. Padova 78 (1987), 227–236.
A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discrete Cont. Dynam. Syst., to appear.
A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital. 2-B (1988), 641–656.
A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl. 71 (1991), 67–83.
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.
A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl. 81 (1994), 435–457.
A. Bressan and F. Rampazzo, Stabilization of Lagrangian systems by moving constraints, to appear.
A. Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei, Memorie, Serie VIII, Vol. XIX (1990), 197–246.
A. Bressan, On some control problems concerning the ski or swing, Atti Accad. Naz. Lincei, Memorie, Serie IX, Vol. I, (1991), 147–196.
F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds. Dynam. Contin. Discrete Impuls. Systems 4 (1998), 1–21.
F.H. Clarke, Yu S. Ledyaev, R.J. Stern, and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.
G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equat. 4 (1991), 739–765.
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.
V. Jurdjevic, Geometric Control Theory, Cambridge University Press, 1997.
P. LeFloch. Hyperbolic systems of conservation laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2002.
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.
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.
B. M. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Systems Estim. Control 4 (1994).
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.
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.
M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim. 34 (1996), 199–225.
H. Nijmeijer and A. van der Schaft, Nonlinear dynamical control systems, Springer-Verlag, New York, 1990.
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.
B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math. 69 (1959), 119–132.
B. L. Reinhart, Differential geometry of foliations. The fundamental integrability problem. Springer-Verlag, Berlin, 1983.
G. V. Smirnov, Introduction to the theory of differential inclusions, American Mathematical Society, Graduate studies in mathematics, vol. 41 (2002).
E. D. Sontag, Mathematical control theory. Deterministic finite-dimensional systems. Second edition. Springer-Verlag, New York, 1998.
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Dedicated to Arrigo Cellina and James Yorke
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Bressan, A. (2007). Singular Limits for Impulsive Lagrangian Systems with Dissipative Sources. In: Staicu, V. (eds) Differential Equations, Chaos and Variational Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 75. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8482-1_6
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DOI: https://doi.org/10.1007/978-3-7643-8482-1_6
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