Impulsive control of lagrangian systems and locomotion in fluids

Discrete and Continuous Dynamical Systems

Volumn 20, Issue 1, 2008, Pages 1-35

  Bressan, Alberto ^a  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

Author keywords

Controlled stability; Impulsive control; Lagrangian system

Indexed keywords

EID: 41649094383     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2008.20.1     Document Type: Article

References (55)

1. R. Abraham and J. E. Marsden, "Foundations of Mechanics," Benjamin/Cummings Publishing Co., Reading, Mass., 1978.
2. A. A. Agrachev and Y. L. Sachkov, "Control Theory form the Geometric Viewpoint," Springer-Verlag, 2004.
3. J. P. Aubin and A. Cellina, "Differential Inclusions", Springer-Verlag, Berlin, 1984.
4. G. Barles, Deterministic impulse control problems, SIAM J. Control Optim., 23 (1985), 419-432.
5. A. Bensoussan and J. L. Lions, "Impulse Control and Quasivariational Inequalities," Heyden & Son, Philadelphia, 1984.
6. U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Springer SMAI series, 43, Heidelberg, 2004.
7. A. Bressan, On differential systems with impulsive controls, Rend. Semin. Mat. Univ. Padova, 78 (1987), 227-236.
8. A. Bressan, Singular limits for impulsive Lagrangian systems with dissipative sources, in "Progress in Nonlinear Differential Equations and Their Applications," 75, 79-103, Birkhäuser, Basel, 2007.
9. A. Bressan and B. Piccoli, "Introduction to the Mathematical Theory of Control," A.I.M.S. Series on Applied Mathematics, 2, Springfield MO, 2007.
10. A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Mat. Ital., 2-B (1988), 641-656.
11. A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl., 71 (1991), 67-83.
12. 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.
13. A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457.
14. A. Bressan and F. Rampazzo, Stabilization of Lagrangian systems by moving constraints, to appear.
15. Aldo Bressan, Hyper-impulsive motions and controllizable coordinates for Lagrangean systems, Atti Accad. Naz. Lincei, Memorie, Serie VIII, XIX (1990), 197-246.
16. Aldo Bressan, On some control problems concerning the ski or swing, Atti Accad. Naz. Lincei, Memorie, Serie IX, I (1991), 147-196.
17. F. Cardin and M. Favretti, Hyper-impulsive motion on manifolds, Dynam. Contin. Discrete Impuls. Systems, 4 (1998), 1-21.
18. J. Carling, T. Williams and G. Bowtell, Self-propelled anguilliform swimming: simultaneous solutions of the two-dimensional Navier-Stokes equations and Newton's laws of motion, J. Experimental Biology, 201 (1998), 3143-3166.
19. S. Childress, "Mechanics of Swimming and Flying," Cambridge University Press, 1981.
20. A. J. Chorin and J. E. Marsden, "A Mathematical Introduction to Fluid Mechanics," Springer-Verlag, New York, 1979.
21. G. Dal Maso and F. Rampazzo, On systems of ordinary differential equations with measures as controls, Differential Integral Equat., 4 (1991), 739-765.
22. G. Dal Maso, P. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), 483-548.
23. G. P. Galdi, On the motion of a rigid body in a viscous fluid: a mathematical analysis with applications, in "Handbook of Mathematical Fluid Dynamics," 1, North-Holland, Amsterdam 2002, 653-791.
24. E. Kanso, J. E. Marsden, C. W. Rowley and J. B. Melli-Huber, Locomotion of articulated bodies in a perfect fluid, J. Nonlinear Sci., 15 (2005), 255-289.
25. P. L. Kapitsa, Dynamic stability of a pendulum when its point of suspension vibrates, Collected works by P. L. Kapitsa, II, 714-725, Pergamon Press, London, 1965.
26. P. L. Kapitsa, Pendulum with a vibrating suspension, Collected works by P. L. Kapitsa, II, 726-737. Pergamon Press, London, 1965.
27. S. D. Kelly and R. Murray, Modelling efficient pisciform swimming for control, Intern. J. Robust Nonlinear Control, 201 (1998), 3143-3166.
28. V. V. Kozlov and D. A. Onishchenko, Motion of a body with undeformable shell and variable mass geometry in an unbounded perfect fluid, J. Dynam. Diff. Equat., 15 (2003), 553-570.
29. V. V. Kozlov and S. M. Ramodanov, On the motion of a variable body in an ideal fluid, J. Appl. Math. Mech., 65 (2001), 579-587.
30. V. V. Kozlov and S. M. Ramodanov, On the motion of a body with rigid shell and variable mass geometry in a perfect fluid, Doklady Physics, 47 (2002), 132-135.
31. J. Kulkarni and B. Piccoli, Pumping a swing by standing and squatting: do children pump time optimally? IEEE Control Systems Magazine, 25 (2005), 8-56.
32. V. Jurdjevic, "Geometric Control Theory," Cambridge University Press, 1997.
33. P. LeFloch, "Hyperbolic Systems of Conservation Laws, The Theory of Classical and Non-classical Shock Waves," Lectures in Mathematics ETH Zrich. Birkhuser, Basel, 2002.
34. M. Levi, Geometry and physics of averaging with applications, Physica D, 132 (1999), 150-164.
35. M. Levi, Geometry of vibrational stabilization and some applications, Int. J. Bifurc. Chaos, 15 (2005), 2747-2756.
36. M. Levi and Q. Ren, Geodesics on vibrating surfaces and curvature of the normal family, Nonlinearity, 18 (2005), 2737-2743.
37. J. Lighthill, "Mathematical Biofluiddynamics," Society for Industrial and Applied Mathematics, Philadelphia, 1975.
38. 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, 437-442.
39. C. Marie, 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.
40. C. Marchioro and M. Pulvirenti, "Mathematical Theory of Incompressible Nonviscous Fluids," Springer-Verlag, New York, 1994.
41. B. M. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Systems Estim. Control, 4 (1994).
42. 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.
43. 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.
44. M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), 199-225.
45. H. Nijmeijer and A. van der Schaft, "Nonlinear Dynamical Control Systems," Springer-Verlag, New York, 1990.
46. B. Piccoli, Time-optimal control problems for the swing and the ski, International Journal of Control, 62 (1995), 1409-1429.
47. F. Rampazzo, On the Riemannian structure of a Lagrangean system and the problem of adding time-dependent coordinates as controls, European J. Mechanics A/Solids, 10 (1991), 405-431.
48. B. L. Reinhart, Foliated manifolds with bundle-like metrics, Annals of Math., 69 (1959), 119-132.
49. B. L. Reinhart, "Differential Geometry of Foliations. The Fundamental Integrability Problem," Springer-Verlag, Berlin, 1983.
50. P. G. Saffman, The self-propulsion of a deformable body in a perfect fluid, J. Fluid Mech., 28 (1967), 385-389.
51. G. V. Smirnov, "Introduction to the Theory of Differential Inclusions," American Mathematical Society, Graduate studies in mathematics, 41 (2002).
52. J. San Martin, J. F. Scheid, T. Takahashi and M. Tucsnak, An initial and boundary value problem modeling fish-like swimming, Arch. Rational Mech. Anal., to appear.
53. A. Stephenson, On a new type of dynamical stability, Manchester Memoirs, 52 (1908) 1-10.
54. H. J. Sussmann, On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6 (1978), 17-41.
55. T. Y. Wu, Mathematical biofluiddynamics and mechanophysiology of fish locomotion, Math. Methods Appl. Sci., 24 (2001), 1541-1564.