Advanced Search
Article Contents
Article Contents

On smoothness of solutions to projected differential equations

  • * Corresponding author: David Salas

    * Corresponding author: David Salas 
Abstract Full Text(HTML) Figure(5) Related Papers Cited by
  • Projected differential equations are known as fundamental mathematical models in economics, for electric circuits, etc. The present paper studies the (higher order) derivability as well as a generalized type of derivability of solutions of such equations when the set involved for projections is prox-regular with smooth boundary.

    Mathematics Subject Classification: Primary: 49J53, 34A60; Secondary: 49N60, 47J35, 37N40.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Two prox-regular sets with smooth and non-smooth boundary

    Figure 2.  Sets of Definition 4.1 for a trajectory in $ \mathbb{R} ^2 $

    Figure 3.  A circuit with an ideal diode, an inductor and a current source

    Figure 4.  Functions $ f_1 $ and $ f_2 $

    Figure 5.  Trajectory for $ f_2 $

  • [1] V. Acary, O. Bonnefon and B. Brogliato, Nonsmooth Modeling and Simulation for Switched Circuits, Springer, 2011. doi: 10.1007/978-90-481-9681-4.
    [2] S. Adly and L. Bourdin, Sensitivity analysis of variational inequalities via twice epi-differentiability and proto-differentiability of the proximity operator, SIAM J. Optim., 28 (2018), 1699-1725.  doi: 10.1137/17M1135013.
    [3] C. Arroud and G. Colombo, A maximum principle for the controlled sweeping process, Set-Valued Var. Anal., 26 (2018), 607-629.  doi: 10.1007/s11228-017-0400-4.
    [4] J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.
    [5] J.-P. Bressoud and  A. CellinaDifferential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.  doi: 10.1007/978-3-642-69512-4.
    [6] D. BressoudA Radical Approach to Lebesgue's Theory of Integration, Cambridge University Press, Cambridge, 2008. 
    [7] B. Brogliato and L. Thibault, Existence and uniqueness of solutions for non-autonomous complementary dynamical systems, J. Convex Anal., 17 (2010), 961-990. 
    [8] M. Brokate and P. Krejčí, Optimal control of ODE systems involving a rate independent variational inequality, Discrete Continuous Dynam. Systems - B, 18 (2013), 331-348.  doi: 10.3934/dcdsb.2013.18.331.
    [9] T. H. Cao and B. Mordukhovich, Optimal control of a perturbed sweeping process via discrete approximations, Discrete Continuous Dynam. Systems - B, 21 (2016), 3331-3358.  doi: 10.3934/dcdsb.2016100.
    [10] T. H. Cao and B. Mordukhovich, Optimality conditions for a controlled sweeping process with applications to the crowd motion model, Discrete Continuous Dynam. Systems - B, 22 (2017), 267-306.  doi: 10.3934/dcdsb.2017014.
    [11] T. H. Cao and B. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, J. Differential Equations, in Press (2018).
    [12] C. Christof and G. Wachsmuth, Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, preprint, arXiv: 1711.02720
    [13] F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.
    [14] M.-G. Cojocaru and L.-B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 132 (2004), 183-193.  doi: 10.1090/S0002-9939-03-07015-1.
    [15] G. ColomboR. HenrionN. Hoang and B. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 117-159. 
    [16] G. ColomboR. HenrionN. Hoang and B. Mordukhovich, Discrete approximations of a controlled sweeping process, Set-Valued Var. Anal., 23 (2015), 69-86.  doi: 10.1007/s11228-014-0299-y.
    [17] G. ColomboR. HenrionN. Hoang and B. Mordukhovich, Optimal control of the sweeping process over polyhedral controlled sets, J. Differential Equations, 260 (2016), 3397-3447.  doi: 10.1016/j.jde.2015.10.039.
    [18] G. Colombo and L. Thibault, Prox-regular sets and applications, in Handbook of Nonconvex Analysis and Applications (eds. D. Gao and D. Motreanu), International Press, Somerville, Mass, (2010), 99-182.
    [19] B. Cornet, Existence of slow solutions for a class of differential inclusions, J. Math. Anal. Appl., 96 (1983), 130-147.  doi: 10.1016/0022-247X(83)90032-X.
    [20] R. CorreaD. Salas and L. Thibault, Smoothness of the metric projection onto nonconvex bodies in Hilbert spaces, J. Math. Anal. Appl., 457 (2018), 1307-1322.  doi: 10.1016/j.jmaa.2016.08.064.
    [21] J.-F. Edmond and L. Thibault, Relaxation of an optimal control problem involving a perturbed sweeping process, Math. Program. Ser. B, 104 (2005), 347-373.  doi: 10.1007/s10107-005-0619-y.
    [22] G. FollandIntroduction to Partial Differential Equations, Princeton University Press, Princeton, N.J, 1995. 
    [23] C. Henry, An existence theorem for a class of differential equations with multivalued right-hand side, J. Math. Anal. Appl., 41 (1973), 179-186.  doi: 10.1016/0022-247X(73)90192-3.
    [24] P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Volume 8 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho Co., Ltd., Tokyo, 1996.
    [25] J. Lee, Introduction to Smooth Manifolds, Springer, New York London, 2013.
    [26] B. Maury and J. Venel, Un modéle de mouvements de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812.
    [27] B. Mordukhovich, Variational analysis and optimization of sweeping processes with controlled moving sets, Invest. Oper., 39 (2018), 283-302. 
    [28] B. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Application, Springer, Berlin New York, 2006.
    [29] J.-J. Moreau, Rafle par un convexe variable Ⅰ, expo. 15, Sém, Anal. Conv. Mont., (1971), 1-43.
    [30] J. Nash, Real Algebraic Manifolds, Ann. of Math., 56 (1952), 405-421.  doi: 10.2307/1969649.
    [31] R. A. PoliquinR. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.  doi: 10.1090/S0002-9947-00-02550-2.
    [32] D. Salas and L. Thibault, Characterizations of nonconvex sets with smooth boundary in terms of the metric projection in Hilbert spaces, preprint.
    [33] L. Thibault, Sweeping process with regular and nonregular sets, J. Differential Equations, 193 (2003), 1-26.  doi: 10.1016/S0022-0396(03)00129-3.
    [34] A. Tolstonogov, Control sweeping processes, J. Convex Anal., 23 (2016), 1099-1123. 
    [35] H. Toruńczyk, Smooth partitions of unity on some non-separable Banach spaces, Studia Math., 46 (1973), 43-51.  doi: 10.4064/sm-46-1-43-51.
  • 加载中



Article Metrics

HTML views(317) PDF downloads(415) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint