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On a new class of differential variational inequalities and a stability result

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Abstract

This paper addresses a new class of differential variational inequalities that have recently been introduced and investigated in finite dimensions as a new modeling paradigm of variational analysis to treat many applied problems in engineering, operations research, and physical sciences. This new subclass of general differential inclusions unifies ordinary differential equations with possibly discontinuous right-hand sides, differential algebraic systems with constraints, dynamic complementarity systems, and evolutionary variational systems. The purpose of this paper is two-fold. Firstly, we show that these differential variational inequalities, when considering slow solutions and the more general level of a Hilbert space, contain projected dynamical systems, another recent subclass of general differential inclusions. This relation follows from a precise geometric description of the directional derivative of the metric projection in Hilbert space, which is based on the notion of the quasi relative interior. Secondly we are concerned with stability of the solution set to this class of differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data, including perturbations of the associated set-valued maps and the constraint set. Here we impose weak convergence assumptions on the perturbed set-valued maps, use the monotonicity method of Browder and Minty, and employ Mosco convergence as set convergence. Also as a consequence, we obtain a stability result for linear complementarity systems.

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References

  1. Adly, S., Goeleven, D.: A stability theory for second-order nonsmooth dynamical systems with application to friction problems. J. Math. Pures Appl. 83(9), 17–51 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Attouch, H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984)

    MATH  Google Scholar 

  3. Aubin, J.-P., Cellina, A.: Differential Inclusions. Set-valued Maps and Viability Theory. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  4. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston, Basel (1990)

    MATH  Google Scholar 

  5. Beer, G., Borwein, J.M.: Mosco convergence and reflexivity. Proc. Am. Math. Soc. 109, 427–436 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borwein, J., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. N.Y. 115, 2542–2553 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55, 45–51 (2006)

    Article  MATH  Google Scholar 

  9. Carstensen, C., Gwinner, J.: A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems. Ann. Mat. Pura Appl. IV Ser. 177, 363–394 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cojocaru, M.G., Jonker, L.B.: Existence of solutions to projected differential equations in Hilbert spaces. Proc. Am. Math. Soc. 132, 183–193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5, Evolution Problems I. Springer, Berlin (1992)

    MATH  Google Scholar 

  12. Dür, A.: On the optimality of the discrete Karhunen-Loève expansion. SIAM J. Control Optim. 36, 1937–1939 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dupuis, P.: Large deviations analysis of reflected diffusions and constrained stochastic approximation algorithms in convex sets. Stochastics 21, 63–96 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  16. Flåm, S.D., Seeger, A.: Solving cone-constrained convex programs by differential inclusions. Math. Program. 65A, 107–121 (1994)

    Article  Google Scholar 

  17. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    Book  MATH  Google Scholar 

  18. Goeleven, D., Brogliato, B.: Necessary conditions of asymptotic stability for unilateral dynamical systems. Nonlinear Anal. Theory Methods Appl. A 61, 961–1004 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goeleven, D., Motreanu, D., Motreanu, V.V.: On the stability of stationary solutions of first order evolution variational inequalities. Adv. Nonlinear Var. Inequal. 6, 1–30 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Gwinner, J.: A class of random variational inequalities and simple random unilateral boundary value problems—existence, discretization, finite element approximation. Stoch. Anal. Appl. 18, 967–993 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gwinner, J.: On differential variational inequalities and projected dynamical systems—equivalence and a stability result. Discret. Contin. Dyn. Syst. 2007, 467–476 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Gwinner, J., Raciti, F.: On a class of random variational inequalities on random sets. Numer. Funct. Anal. Optim. 27, 619–636 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gwinner, J., Raciti, F.: On monotone variational inequalities with random data. J. Math. Inequal. 3, 443–453 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Projected dynamical systems in a complementarity formalism. Oper. Res. Lett. 27, 83–91 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Hu, S., Papageorgiou, N.S.: Time-dependent subdifferential evolution inclusions and optimal control. Mem. Am. Math. Soc. 632, 81 (1998)

    MathSciNet  Google Scholar 

  27. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis, vol. I: Theory. Kluwer Boston, Dordrecht (1997)

    Google Scholar 

  28. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1984)

    Google Scholar 

  29. Kunze, M.: Non-Smooth Dynamical Systems. In: Lecture Notes in Mathematics, vol. 1744. Springer, Berlin (2000)

  30. Nagurney, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Boston, Dordrecht (1996)

    Book  Google Scholar 

  31. Pang, J.-S., Stewart, D.E.: Differential variational inequalities. Math. Program. 113, 345–424 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pang, J.-S., Stewart, D.E.: Solution dependence on initial conditions in differential variational inequalities. Math. Program. 116, 429–460 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Papageorgiou, N.S.: A stability result for differential inclusions in Banach spaces. J. Math. Anal. Appl. 118, 232–246 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. Papageorgiou, N.S.: On parametric evolution inclusions of the subdifferential type with applications to optimal control problems. Trans. Am. Math. Soc. 347, 203–231 (1995)

    Article  MATH  Google Scholar 

  35. Papageorgiou, N.S., Papalini, F.: Random evolution inclusions of the subdifferential type. Stoch. Anal. Appl. 15, 801–821 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shillor, M., Sofonea, M., Telega, J.J.: Models and Analysis of Quasistatic Contact. Variational Methods. In: Lecture Notes in Physics, vol. 655. Springer, Berlin (2004)

  37. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Madison, USA (1971)

    Google Scholar 

  38. Zeidler, E.: Nonlinear Functional Analysis and its Applications, vol. II/B, Nonlinear Monotone Operators. Springer, New York (1990)

    Book  Google Scholar 

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Acknowledgments

The author is indebted to two anonymous referees for their constructive suggestions.

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  1. Fakultät für Luft- und Raumfahrttechnik, Institut für Mathematik, Universität der Bundeswehr München, 85577, Neubiberg/München, Germany

    Joachim Gwinner

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  1. Joachim Gwinner
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Correspondence to Joachim Gwinner.

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Dedicated to Jonathan M. Borwein on the occasion of his sixtieth birthday.

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Gwinner, J. On a new class of differential variational inequalities and a stability result. Math. Program. 139, 205–221 (2013). https://doi.org/10.1007/s10107-013-0669-5

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