Skip to main content

Advertisement

SpringerLink
Log in

Differential variational inequalities

  • FULL LENGTH PAPER
  • Published:
Mathematical Programming Submit manuscript

Abstract

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acary, V., Brogliato, B., Goeleven, D.: Higher order Moreau’s sweeping process: Mathematical formulation and numerical simulation. INRIA Report No. 5236, Version 2 (Mai 2005)

  2. Adly S. and Goeleven D. (2004). A stability theory for second order nonsmooth dynamical systems with applications to friction problems. J. Math. Pures Appl. 83: 17–51

    MATH  MathSciNet  Google Scholar 

  3. Anitescu M. and Hart G.D. (2004). A constraint-stabilized time-stepping for multi-body dynamics with contact and friction. Int. J. Numer. Methods Eng. 60: 2335–2371

    Article  MATH  MathSciNet  Google Scholar 

  4. Anitescu M. and Potra F. (1997). Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems. ASME Nonlinear Dynam. 4: 231–247

    Article  MathSciNet  Google Scholar 

  5. Anitescu M., Potra F.A. and Stewart D. (1999). Time-stepping for three-dimensional rigid-body dynamics. Comput. Methods Appl. Mech. Eng. 177: 183–197

    Article  MATH  MathSciNet  Google Scholar 

  6. Anitescu M. and Potra F. (2002). A time-stepping method for stiff multi-body dynamics with friction and contact. Int. J. Numer. Methods Eng. 55: 753–784

    Article  MATH  MathSciNet  Google Scholar 

  7. Ascher U.M., Mattheij R.M. and Russell R.D. (1988). Numerical Solution of Ordinary Value Problems for Ordinary Differential Equations. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  8. Ascher U.M., Mattheij R.M. and Russell R.D. (1995). Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  9. Ascher U.M. and Petzold L.R. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  10. Aubin J.P. and Cellina A. (1984). Differential Inclusions. Springer, New York

    MATH  Google Scholar 

  11. Ban, X.J.: Quasi-variational inequality formulations and solution approaches for dynamic user equilibria. Ph.D Dissertation, Department of Civil and Environmental Engineering, University of Wisconsin-Madison (2005)

  12. Başar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. SIAM Series in Classics in Applied Mathematics (Philadelphia 1999) [Revised, updated version of the 1995 Academic Press book with the same title.]

  13. Begle E.G. (1950). A fixed point theorem. Ann. Math. 51: 544–550

    Article  MathSciNet  Google Scholar 

  14. Billups S. and Ferris M.C. (1999). Solutions to affine generalized equations using proximal mappings. Math. Oper. Res. 24: 219–236

    MATH  MathSciNet  Google Scholar 

  15. Bounkhel M. (2003). General existence results for second order nonconvex sweeping process with unbounded perturbations. Portugaliae Mathe. Nova Série 60: 269–304

    MATH  MathSciNet  Google Scholar 

  16. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing, Amsterdam (1973)

  17. Brogliato B. (1999). Nonsmooth Mechanics. Models, Dynamics and Control. Springer, London

    MATH  Google Scholar 

  18. Brogliato B., ten Dam A.A., Paoli L. and Abadie M. (2002). Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. Appl. Mech. Rev. 55: 107–150

    Article  Google Scholar 

  19. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. vol. 14, SIAM Publications Classics in Applied Mathematics, Philadelphia (1996)

  20. Castaing C., Dúc Hà T.X. and Valadier M. (1993). Evolution equations governed by the sweeping process. Set-Valued Anal. 1: 109–139

    Article  MATH  MathSciNet  Google Scholar 

  21. Çamlibel, M.K.: Complementarity methods in the analysis of piecewise linear dynamical systems. Ph.D. Thesis, Center for Economic Research, Tilburg University, The Netherlands (May 2001)

  22. Çamlibel M.K., Pang J.S. and Shen J.L. (2006). Lyapunov stability of linear complementarity systems. SIAM J. Optimi. 17: 1056–1101

    Article  MATH  Google Scholar 

  23. Çamlibel M.K., Pang J.S. and Shen J.L. (2006). Conewise linear systems. SIAM J. Control Optimi. 45: 1769–1800

    Article  MATH  Google Scholar 

  24. Chen C.H. and Mangasarian O.L. (1995). Smoothing methods for convex inequalities and linear problems. Math. Programm. 71: 51–69

    Article  MathSciNet  Google Scholar 

  25. Chen C.H. and Mangasarian O.L. (1996). A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optimi. Appl. 5: 97–138

    MATH  MathSciNet  Google Scholar 

  26. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. Tata–McGraw Hill, New Delhi (1972) [Originally published by McGraw Hill in 1955]

  27. Cojocaru, M.G.: Projected dynamical systems on Hilbert spaces. Ph.D. Thesis, Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada (August 2002)

  28. Cojocaru M.G., Daniele P. and Nagurney A. (2005). Projected dynamics and evolutionary variational inequalities via Hilbert spaces with applications. J. Optimi. Theory Appl. 127: 549–563

    Article  MATH  MathSciNet  Google Scholar 

  29. Cojocaru M.G. and Jonker L.B. (2004). Existence of solutions to projected differential equations in Hilbert space. Proc. Am. Mathe. Soc. 132: 183–193

    Article  MATH  MathSciNet  Google Scholar 

  30. Cottle R.W., Pang J.S. and Stone R.E. (1992). The Linear Complementarity Problem. Academic Press, Boston

    MATH  Google Scholar 

  31. Jong H. (2002). Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9: 69–105

    Article  Google Scholar 

  32. Deimling K. (1992). Multivalued Differential Equations. Walter de Gruyter, Berlin

    MATH  Google Scholar 

  33. Dirkse S.P. and Ferris M.C. (1995). The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimi. Methods Software 5: 123–156

    Article  Google Scholar 

  34. Dockner E., Jergensen S., Long N.V. and Sorger G. (2000). Differential Games in Economics and Management Science. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  35. Dontchev A. and Lempio F. (1992). Difference methods for differential inclusions: A survey. SIAM Rev. 34: 263–294

    Article  MATH  MathSciNet  Google Scholar 

  36. Dupuis P. and Nagurney A. (1993). Dynamical systems and variational inequalities. Ann. Oper. Res. 44: 9–42

    Article  MATH  MathSciNet  Google Scholar 

  37. Eilenberg S. and Montgomery D. (1946). Fixed point theorems for multi-valued transformations. Am. J. Math. 68: 214–222

    Article  MATH  MathSciNet  Google Scholar 

  38. Facchinei F. and Pang J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York

    Google Scholar 

  39. Faik L.A. and Syam A. (2001). Differential inclusions governed by a nonconvex sweeping process. J. Nonlinear Convex Anal. 2: 381–392

    MATH  MathSciNet  Google Scholar 

  40. Ferris M.C. and Munson T.S. (1999). Interfaces to PATH 3.0: Design, implementation and usage. Comput. Optimi. Appl. 12: 207–227

    Article  MATH  MathSciNet  Google Scholar 

  41. Filippov, A.F.: Differential equations with discontinuous right-hand side. Matematicheskiu Sbornik. Novaya Seriya 5, 99–127 (1960) Also: American Mathematical Society Translation 42, 199–231 (1964)

  42. Filippov A.F. (1962). On certain questions in the theory of optimal control. SIAM J. Control 1: 76–84

    Article  MATH  Google Scholar 

  43. Filippov A.F. (1988). Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  44. Goeleven D. and Brogliato B. (2004). Stability and instability matrices for linear evolution variational inequalities. IEEE Trans. Automat. Control 49: 521–534

    Article  MathSciNet  Google Scholar 

  45. Goeleven D. and Brogliato B. (2005). Necessary conditions of asymptotic stability for unilateral dynamical systems.. Nonlinear Analy. 61: 961–1004

    Article  MATH  MathSciNet  Google Scholar 

  46. Goeleven D., Motreanu M. and Motreanu V. (2003). On the stability of stationary solutions of evolution variational inequalities. Adv. Nonlinear Variat. Inequal. 6: 1–30

    MATH  MathSciNet  Google Scholar 

  47. Górniewicz, L.: Homological methods in fixed-point theory of multi-valued maps. Dissertationes Mathematicae. Rozprawy Matematyczne 129 (1976) pp. 71

    Google Scholar 

  48. Górniewicz L. (1999). Topological Fixed Point Theory of Multivalued Mappings. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  49. Heemels, W.P.H.: Linear complementarity systems: a study in hybrid dynamics. Ph.D. Thesis, Department of Electrical Engineering, Eindhoven University of Technology (November 1999)

  50. Heemels, W.P.M.H., Çamlibel, M.K., van der Schaft, A.J., Schumacher, J.M.: Well-posedness of hybrid systems. In: Unbehauen, H. (ed.), Control Systems, Robotics and Automation, Theme 6.43 of Encyclopedia of Life Support Systems (developed under the auspices of UNESCO), EOLSS Publishers, Oxford (2004)

  51. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Well-posedness of linear complementarity systems. In: 38th IEEE Conference on Decision and Control, Phoenix, pp. 3037–3042 (1999)

  52. Heemels W.P.M.H., Schumacher J.M. and Weiland S. (2000). Linear complementarity systems. SIAM J. Appl. Math. 60: 1234–1269

    Article  MATH  MathSciNet  Google Scholar 

  53. Henry C. (1972). Differential equations with discontinuous right-hand side for planning procedures. J. Econ. Theory 4: 545–551

    Article  MathSciNet  Google Scholar 

  54. Henry C. (1973). An existence theorem for a class of differential equations with multivalued right-hand side. J. Mathe. Anal. Appl. 41: 179–186

    Article  MATH  MathSciNet  Google Scholar 

  55. Hipfel, D.: The nonlinear differential complementarity problem. Ph.D. Thesis, Department of Mathematical Sciences, Rensselaer Polytechnic Institute (1993)

  56. Hoffman A.J. (1952). On approximate solutions of systems of linear inequalities. J. Rese. Nat. Bureau Standards 49: 263–265

    Google Scholar 

  57. Jean M. (1999). The nonsmooth contact dynamics approach. Comput. Methods Appl. Mech. Eng. 177: 235–277

    Article  MATH  MathSciNet  Google Scholar 

  58. Kakutani S. (1941). A generalizations of Brouwer’s fixed point theorem. Duke Math. J. 8: 457–458

    Article  MATH  MathSciNet  Google Scholar 

  59. Kakhu A.I. and Pantelides C.C. (2003). Dynamic modeling of acqueous electrolyte systems. Comput. Chem. Eng. 27: 869–882

    Article  Google Scholar 

  60. Keller H.B. (1968). Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell Publishing Company, Waltham

    MATH  Google Scholar 

  61. Kunze M. and Monteiro Marques M.D.P. (1997). Existence of solutions for degenerate sweeping processes. J. Convex Anal. 4: 165–176

    MATH  MathSciNet  Google Scholar 

  62. Kunze, M., Monteiro Marques, M.D.P.: An introduction to Moreau’s sweeping process. In: Brogliato, B. (ed.) Impacts in Mechanical Systems. Lecture Notes in Physics, vol. 551. pp. 1–60 Springer, New York, (2000)

  63. Lang S. (1993). Real and Functional Analysis. Springer, Berlin

    MATH  Google Scholar 

  64. Liu, H.X., Ban, X., Ran, B., Mirchandani, P.: An analytical dynamic traffic assignment model with probabilistic travel times and perceptions. UCI-ITS-WP-01-14, Institute of Transportation Studies, University of California, Irvine (December 2001)

  65. Luo Z.Q. and Tseng P. (1991). A decomposition property for a class of square matrices. Appl. Math. Lett. 4: 67–69

    Article  MATH  MathSciNet  Google Scholar 

  66. Monteiro Marques M.D.P. (1993). Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  67. Moreau J.J. (1977). Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equations 26: 347–374

    Article  MATH  MathSciNet  Google Scholar 

  68. Moreau J.J. (1988). Bounded variation in time. In: Moreau, J.J., Panagiotopoulos, P.D. and Strang, G. (eds) Topics in Nonsmooth Mechanics., pp 1–74. Birkhäuser Verlag, Massachusetts

    Google Scholar 

  69. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Nonsmooth Mechanics and Applications (CISM Courses and Lectures No. 302 International Center for Mechanical Sciences). pp. 1–82 Springer, Heidelberg, (1988)

  70. Moreau J.J. (1999). Numerical aspects of the sweeping process. Computational modeling of contact and friction. Comput. Methods Appl. Mech. Eng. 177: 329–349

    Article  MATH  MathSciNet  Google Scholar 

  71. Ortega J.M. and Rheinboldt W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York

    MATH  Google Scholar 

  72. Pang, J.S., Shen, J.: Strongly regular variational systems, IEEE Trans. Automat. Control (forthcoming)

  73. Pang J.S., Song P. and Kumar V. (2005). Convergence of time-stepping methods for initial and boundary value frictional compliant contact problems. SIAM J. Numer. Anal. 43: 2200–2226

    Article  MATH  MathSciNet  Google Scholar 

  74. Pang, J.S., Stewart, D.E.: Solution dependence on initial conditions in differential variational inequalities Math. Programm. Ser. B (forthcoming)

  75. Pantelides C.C., Morrison K.R., Gritsis D. and Sargent R.W.H. (1988). The mathematical modeling of transient systems using differential=algebraic equations. Comput. Chem. Eng. 12: 449–454

    Article  Google Scholar 

  76. Petrov A. and Schatzman M. (2005). A pseudodifferential linear complementarity problem related to a one-dimensional viscoeleastic model with Signorini conditions. Arch. Rat. Mech. Anal. 334: 983–988

    MathSciNet  Google Scholar 

  77. Petzold L.R. and Ascher U.M. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  78. Ran B. and Boyce D. (1996). Modeling Dynamic Transportation Networks. Springer, Heidelberg

    MATH  Google Scholar 

  79. Roberts S.M. and Shipman J.S. (1972). Two-Point Boundary Problems: Shooting Methods. American Elsevier Publishing Company Inc, New York

    MATH  Google Scholar 

  80. Robinson S.M. (1979). Generalized equations and their solutions.. I. Basic theory. Math. Programm. Stud. 10: 128–141

    MATH  Google Scholar 

  81. Robinson S.M. (1980). Strongly regular generalized equations. Math. Oper. Res. 5: 43–62

    Article  MATH  MathSciNet  Google Scholar 

  82. Robinson S.M. (1981). Some continuity properties of polyhedral multifunctions. Math. Programm. Stud. 14: 206–214

    MATH  Google Scholar 

  83. Robinson S.M. (1982). Generalized equations and their solutions. II. Applications to nonlinear programming. Math. Programm. Stud. 19: 200–221

    MATH  Google Scholar 

  84. Robinson S.M. (1992). Normal maps induced by linear transformations. Math. Oper. Res. 17: 691–714

    MATH  MathSciNet  Google Scholar 

  85. Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  86. Saveliev P. (2000). Fixed points and selections of set-valued maps on spaces with convexity. Int. J. Math. Math. Sci. 24: 595–612

    Article  MATH  MathSciNet  Google Scholar 

  87. Schumacher J.M. (2004). Complementarity systems in optimization. Math. Programm. Ser. B 101: 263–295

    MATH  MathSciNet  Google Scholar 

  88. Shen J.L. and Pang J.S. (2005). Linear complementarity systems:Zeno states. SIAM J. Control Optimi. 44: 1040–1066

    Article  MATH  MathSciNet  Google Scholar 

  89. Smirnov, G.V.: Introduction to the Theory of Differential Inclusions. Graduate Studies in Mathematics, vol. 41. American Mathematical Society, Providence (2002)

  90. Song P., Krauss P., Kumar V. and Dupont P. (2001). Analysis of rigid-body dynamic models for simulation of systems with frictional contacts. J. Appl. Mech. 68: 118–128

    Article  MATH  Google Scholar 

  91. Song P., Pang J.S. and Kumar V. (2004). Semi-implicit time-stepping models for frictional compliant contact problems. Int. J. Numer. Methods Eng. 60: 2231–2261

    Article  MATH  MathSciNet  Google Scholar 

  92. Spanier, E.H.: Algebraic Topology. Springer, New York (1966) [Reprinted 1989]

  93. Stewart D.E. (1990). A high accuracy method for solving ODEs with discontinuous right-hand side. Numer. Math. 58: 299–328

    Article  MATH  MathSciNet  Google Scholar 

  94. Stewart D.E. (1998). Convergence of a time-stepping scheme for rigid-body dynamics and resolution of Painlevé’s problem. Arch. Rat. Mech. Anal. 145: 215–260

    Article  MATH  Google Scholar 

  95. Stewart D.E. (2000). Rigid-body dynamics with friction and impact. SIAM Rev. 42: 3–39

    Article  MATH  MathSciNet  Google Scholar 

  96. Stewart D.E. (2001). Reformulations of measure differential inclusions and their closed graph property. J. Differential Equations 175: 108–129

    Article  MATH  MathSciNet  Google Scholar 

  97. Stewart D.E. (2006). Convolution complementarity problems with application to impact problems. IMA J. Appl. Math. 71: 92–119

    Article  MATH  MathSciNet  Google Scholar 

  98. Stewart D.E. and Trinkle J.C. (1996). An implicit time-stepping scheme for rigid body dynamics with inelastic collisions and Coulomb friction. Int. J. Numer. Methods Eng. 39: 2673–2691

    Article  MATH  MathSciNet  Google Scholar 

  99. Stromberg, K.R.: Introduction to Classical Real Analysis. Wadsworth, Inc. (Belmont, CA 1981)

  100. Thibault L. (2003). Sweeping process with regular and nonregular sets. J. Differential Equations 193: 1–26

    Article  MATH  MathSciNet  Google Scholar 

  101. Trinkle J.C., Tzitzouris J.A. and Pang J.S. (2001). Dynamic multi-rigid-systems with concurrent distributed contacts. Roy. Soc. Philos. Trans. Math. Phys. Eng. Sci. 359: 2575–2593

    Article  MATH  MathSciNet  Google Scholar 

  102. Tzitzouris, J.A.: Numerical Resolution of Frictional Multi-Rigid-Body Systems via Fully-Implicit Time-Stepping and Nonlinear Complementarity. Ph.D. Thesis, Department of Sciences, The Johns Hopkins University (September 2001)

  103. Tzitzouris J. and Pang J.S. (2002). A time-stepping complementarity approach for frictionless systems of rigid bodies. SIAM J. Optimi. 12: 834–860

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences and Department of Decision Science and Engineering Systems, Rensselaer Polytechnic Institute, Troy, NY, 12180-3590, USA

    Jong-Shi Pang

  2. Department of Mathematics, University of Iowa, Iowa City, IA, 52242, USA

    David E. Stewart

Authors
  1. Jong-Shi Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David E. Stewart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jong-Shi Pang.

Additional information

The work of J.-S. Pang is supported by the National Science Foundation under grants CCR-0098013 CCR-0353074, and DMS-0508986, by a Focused Research Group Grant DMS-0139715 to the Johns Hopkins University and DMS-0353016 to Rensselaer Polytechnic Institute, and by the Office of Naval Research under grant N00014-02-1-0286. The work of D. E. Stewart is supported by the National Science Foundation under a Focused Research Group grant DMS-0138708.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pang, JS., Stewart, D.E. Differential variational inequalities. Math. Program. 113, 345–424 (2008). https://doi.org/10.1007/s10107-006-0052-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0052-x

Keywords

Search

Navigation