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Differential variational inequalities. (English) Zbl 1139.58011
Summary: 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.

58E35Variational inequalities (global problems)
[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 · Zbl 1103.34054 · doi:10.1016/S0021-7824(03)00071-0
[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 · Zbl 1075.70501 · doi:10.1002/nme.1047
[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 · Zbl 0899.70005 · doi:10.1023/A:1008292328909
[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 · Zbl 0967.70003 · doi:10.1016/S0045-7825(98)00380-6
[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 · Zbl 1027.70001 · doi:10.1002/nme.512
[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
[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
[9]Ascher U.M. and Petzold L.R. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia
[10]Aubin J.P. and Cellina A. (1984). Differential Inclusions. Springer, New York
[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 · Zbl 0036.38901 · doi:10.2307/1969367
[14]Billups S. and Ferris M.C. (1999). Solutions to affine generalized equations using proximal mappings. Math. Oper. Res. 24: 219–236 · Zbl 0977.90055 · doi:10.1287/moor.24.1.219
[15]Bounkhel M. (2003). General existence results for second order nonconvex sweeping process with unbounded perturbations. Portugaliae Mathe. Nova Série 60: 269–304
[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
[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 · doi:10.1115/1.1454112
[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 · Zbl 0813.34018 · doi:10.1007/BF01027688
[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 · Zbl 1124.93042 · doi:10.1137/050629185
[23]Çamlibel M.K., Pang J.S. and Shen J.L. (2006). Conewise linear systems. SIAM J. Control Optimi. 45: 1769–1800 · Zbl 1126.93030 · doi:10.1137/050645166
[24]Chen C.H. and Mangasarian O.L. (1995). Smoothing methods for convex inequalities and linear problems. Math. Programm. 71: 51–69 · Zbl 0855.90124 · doi:10.1007/BF01592244
[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
[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 · Zbl 1093.49004 · doi:10.1007/s10957-005-7502-0
[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 · Zbl 1055.34118 · doi:10.1090/S0002-9939-03-07015-1
[30]Cottle R.W., Pang J.S. and Stone R.E. (1992). The Linear Complementarity Problem. Academic Press, Boston
[31]Jong H. (2002). Modeling and simulation of genetic regulatory systems: a literature review. J. Comput. Biol. 9: 69–105
[32]Deimling K. (1992). Multivalued Differential Equations. Walter de Gruyter, Berlin
[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 · doi:10.1080/10556789508805606
[34]Dockner E., Jergensen S., Long N.V. and Sorger G. (2000). Differential Games in Economics and Management Science. Cambridge University Press, Cambridge
[35]Dontchev A. and Lempio F. (1992). Difference methods for differential inclusions: A survey. SIAM Rev. 34: 263–294 · Zbl 0757.34018 · doi:10.1137/1034050
[36]Dupuis P. and Nagurney A. (1993). Dynamical systems and variational inequalities. Ann. Oper. Res. 44: 9–42 · Zbl 0785.93044 · doi:10.1007/BF02073589
[37]Eilenberg S. and Montgomery D. (1946). Fixed point theorems for multi-valued transformations. Am. J. Math. 68: 214–222 · Zbl 0060.40203 · doi:10.2307/2371832
[38]Facchinei F. and Pang J.S. (2003). Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York
[39]Faik L.A. and Syam A. (2001). Differential inclusions governed by a nonconvex sweeping process. J. Nonlinear Convex Anal. 2: 381–392
[40]Ferris M.C. and Munson T.S. (1999). Interfaces to PATH 3.0: Design, implementation and usage. Comput. Optimi. Appl. 12: 207–227 · Zbl 1040.90549 · doi:10.1023/A:1008636318275
[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
[43]Filippov A.F. (1988). Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic Publishers, Dordrecht
[44]Goeleven D. and Brogliato B. (2004). Stability and instability matrices for linear evolution variational inequalities. IEEE Trans. Automat. Control 49: 521–534 · doi:10.1109/TAC.2004.825654
[45]Goeleven D. and Brogliato B. (2005). Necessary conditions of asymptotic stability for unilateral dynamical systems.. Nonlinear Analy. 61: 961–1004 · Zbl 1068.49004 · doi:10.1016/j.na.2005.01.037
[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
[47]Górniewicz, L.: Homological methods in fixed-point theory of multi-valued maps. Dissertationes Mathematicae. Rozprawy Matematyczne 129 (1976) pp. 71
[48]Górniewicz L. (1999). Topological Fixed Point Theory of Multivalued Mappings. Kluwer Academic Publishers, Dordrecht
[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 · Zbl 0954.34007 · doi:10.1137/S0036139997325199
[53]Henry C. (1972). Differential equations with discontinuous right-hand side for planning procedures. J. Econ. Theory 4: 545–551 · doi:10.1016/0022-0531(72)90138-X
[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 · Zbl 0262.49019 · doi:10.1016/0022-247X(73)90192-3
[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
[57]Jean M. (1999). The nonsmooth contact dynamics approach. Comput. Methods Appl. Mech. Eng. 177: 235–277 · Zbl 0959.74046 · doi:10.1016/S0045-7825(98)00383-1
[58]Kakutani S. (1941). A generalizations of Brouwer’s fixed point theorem. Duke Math. J. 8: 457–458 · Zbl 0061.40304 · doi:10.1215/S0012-7094-41-00838-4
[59]Kakhu A.I. and Pantelides C.C. (2003). Dynamic modeling of acqueous electrolyte systems. Comput. Chem. Eng. 27: 869–882 · doi:10.1016/S0098-1354(03)00002-4
[60]Keller H.B. (1968). Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell Publishing Company, Waltham
[61]Kunze M. and Monteiro Marques M.D.P. (1997). Existence of solutions for degenerate sweeping processes. J. Convex Anal. 4: 165–176
[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
[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 · Zbl 0733.15006 · doi:10.1016/0893-9659(91)90148-O
[66]Monteiro Marques M.D.P. (1993). Differential Inclusions in Nonsmooth Mechanical Problems. Shocks and Dry Friction. Birkhäuser Verlag, Basel
[67]Moreau J.J. (1977). Evolution problem associated with a moving convex set in a Hilbert space. J. Differential Equations 26: 347–374 · Zbl 0356.34067 · doi:10.1016/0022-0396(77)90085-7
[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
[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 · Zbl 0968.70006 · doi:10.1016/S0045-7825(98)00387-9
[71]Ortega J.M. and Rheinboldt W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York
[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 · Zbl 1145.74027 · doi:10.1137/040612269
[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 · doi:10.1016/0098-1354(88)85062-2
[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
[77]Petzold L.R. and Ascher U.M. (1998). Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia
[78]Ran B. and Boyce D. (1996). Modeling Dynamic Transportation Networks. Springer, Heidelberg
[79]Roberts S.M. and Shipman J.S. (1972). Two-Point Boundary Problems: Shooting Methods. American Elsevier Publishing Company Inc, New York
[80]Robinson S.M. (1979). Generalized equations and their solutions.. I. Basic theory. Math. Programm. Stud. 10: 128–141
[81]Robinson S.M. (1980). Strongly regular generalized equations. Math. Oper. Res. 5: 43–62 · Zbl 0437.90094 · doi:10.1287/moor.5.1.43
[82]Robinson S.M. (1981). Some continuity properties of polyhedral multifunctions. Math. Programm. Stud. 14: 206–214
[83]Robinson S.M. (1982). Generalized equations and their solutions. II. Applications to nonlinear programming. Math. Programm. Stud. 19: 200–221
[84]Robinson S.M. (1992). Normal maps induced by linear transformations. Math. Oper. Res. 17: 691–714 · Zbl 0777.90063 · doi:10.1287/moor.17.3.691
[85]Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton
[86]Saveliev P. (2000). Fixed points and selections of set-valued maps on spaces with convexity. Int. J. Math. Math. Sci. 24: 595–612 · Zbl 0968.47016 · doi:10.1155/S0161171200004403
[87]Schumacher J.M. (2004). Complementarity systems in optimization. Math. Programm. Ser. B 101: 263–295
[88]Shen J.L. and Pang J.S. (2005). Linear complementarity systems:Zeno states. SIAM J. Control Optimi. 44: 1040–1066 · Zbl 1092.90052 · doi:10.1137/040612270
[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 · Zbl 1110.74686 · doi:10.1115/1.1331060
[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 · Zbl 1072.74062 · doi:10.1002/nme.1049
[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 · Zbl 0693.65049 · doi:10.1007/BF01385627
[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 · Zbl 0922.70004 · doi:10.1007/s002050050129
[95]Stewart D.E. (2000). Rigid-body dynamics with friction and impact. SIAM Rev. 42: 3–39 · Zbl 0962.70010 · doi:10.1137/S0036144599360110
[96]Stewart D.E. (2001). Reformulations of measure differential inclusions and their closed graph property. J. Differential Equations 175: 108–129 · Zbl 0990.34014 · doi:10.1006/jdeq.2000.3968
[97]Stewart D.E. (2006). Convolution complementarity problems with application to impact problems. IMA J. Appl. Math. 71: 92–119 · Zbl 1138.90485 · doi:10.1093/imamat/hxh087
[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 · doi:10.1002/(SICI)1097-0207(19960815)39:15<2673::AID-NME972>3.0.CO;2-I
[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 · Zbl 1037.34007 · doi:10.1016/S0022-0396(03)00129-3
[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 · Zbl 1014.70007 · doi:10.1098/rsta.2001.0911
[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 · Zbl 1009.70007 · doi:10.1137/S1052623400370369