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April  2019, 39(4): 2255-2283. doi: 10.3934/dcds.2019095

On smoothness of solutions to projected differential equations

David Salas 1,, , Lionel Thibault 2, and  Emilio Vilches 3,

1. 

Lab. PROMES UPR CNRS 8521, Université de Perpignan Via Domitia, Rambla de la Thermodynamique, Tecnosud, F- 66100 Perpignan, France
2. 

Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France
3. 

Instituto de Ciencias de la Educación, Universidad de O'Higgins, Av. Libertador Bernardo O'Higgins 611, Rancagua, Chile

* Corresponding author: David Salas

Received  August 2018 Revised  October 2018 Published  January 2019

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.

Keywords: Projected differential equations, trajectory, $ \Omega $-derivability, submanifold, prox-regular set, collisions.
Mathematics Subject Classification: Primary: 49J53, 34A60; Secondary: 49N60, 47J35, 37N40.
Citation: David Salas, Lionel Thibault, Emilio Vilches. On smoothness of solutions to projected differential equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2255-2283. doi: 10.3934/dcds.2019095
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.  Google Scholar

[5] J.-P. Bressoud and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.  doi: 10.1007/978-3-642-69512-4.  Google Scholar
[6] D. Bressoud, A Radical Approach to Lebesgue's Theory of Integration, Cambridge University Press, Cambridge, 2008.   Google Scholar
[7]

B. Brogliato and L. Thibault, Existence and uniqueness of solutions for non-autonomous complementary dynamical systems, J. Convex Anal., 17 (2010), 961-990.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. H. Cao and B. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, J. Differential Equations, in Press (2018). Google Scholar

[12]

C. Christof and G. Wachsmuth, Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, preprint, arXiv: 1711.02720 Google Scholar

[13]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[15]

G. Colombo, R. Henrion, N. Hoang and B. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 117-159.   Google Scholar

[16]

G. Colombo, R. Henrion, N. 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.  Google Scholar

[17]

G. Colombo, R. Henrion, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Correa, D. 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.  Google Scholar

[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.  Google Scholar

[22] G. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, N.J, 1995.   Google Scholar
[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.  Google Scholar

[24]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Volume 8 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho Co., Ltd., Tokyo, 1996.  Google Scholar

[25]

J. Lee, Introduction to Smooth Manifolds, Springer, New York London, 2013.  Google Scholar

[26]

B. Maury and J. Venel, Un modéle de mouvements de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812.  Google Scholar

[27]

B. Mordukhovich, Variational analysis and optimization of sweeping processes with controlled moving sets, Invest. Oper., 39 (2018), 283-302.   Google Scholar

[28]

B. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Application, Springer, Berlin New York, 2006. Google Scholar

[29]

J.-J. Moreau, Rafle par un convexe variable Ⅰ, expo. 15, Sém, Anal. Conv. Mont., (1971), 1-43.  Google Scholar

[30]

J. Nash, Real Algebraic Manifolds, Ann. of Math., 56 (1952), 405-421.  doi: 10.2307/1969649.  Google Scholar

[31]

R. A. Poliquin, R. 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.  Google Scholar

[32]

D. Salas and L. Thibault, Characterizations of nonconvex sets with smooth boundary in terms of the metric projection in Hilbert spaces, preprint. Google Scholar

[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.  Google Scholar

[34]

A. Tolstonogov, Control sweeping processes, J. Convex Anal., 23 (2016), 1099-1123.   Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J.-P. Aubin, Viability Theory, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1991.  Google Scholar

[5] J.-P. Bressoud and A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.  doi: 10.1007/978-3-642-69512-4.  Google Scholar
[6] D. Bressoud, A Radical Approach to Lebesgue's Theory of Integration, Cambridge University Press, Cambridge, 2008.   Google Scholar
[7]

B. Brogliato and L. Thibault, Existence and uniqueness of solutions for non-autonomous complementary dynamical systems, J. Convex Anal., 17 (2010), 961-990.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

T. H. Cao and B. Mordukhovich, Optimal control of a nonconvex perturbed sweeping process, J. Differential Equations, in Press (2018). Google Scholar

[12]

C. Christof and G. Wachsmuth, Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators, preprint, arXiv: 1711.02720 Google Scholar

[13]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[15]

G. Colombo, R. Henrion, N. Hoang and B. Mordukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 117-159.   Google Scholar

[16]

G. Colombo, R. Henrion, N. 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.  Google Scholar

[17]

G. Colombo, R. Henrion, N. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. Correa, D. 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.  Google Scholar

[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.  Google Scholar

[22] G. Folland, Introduction to Partial Differential Equations, Princeton University Press, Princeton, N.J, 1995.   Google Scholar
[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.  Google Scholar

[24]

P. Krejčí, Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Volume 8 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho Co., Ltd., Tokyo, 1996.  Google Scholar

[25]

J. Lee, Introduction to Smooth Manifolds, Springer, New York London, 2013.  Google Scholar

[26]

B. Maury and J. Venel, Un modéle de mouvements de foule, ESAIM Proc., 18 (2007), 143-152.  doi: 10.1051/proc:071812.  Google Scholar

[27]

B. Mordukhovich, Variational analysis and optimization of sweeping processes with controlled moving sets, Invest. Oper., 39 (2018), 283-302.   Google Scholar

[28]

B. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Application, Springer, Berlin New York, 2006. Google Scholar

[29]

J.-J. Moreau, Rafle par un convexe variable Ⅰ, expo. 15, Sém, Anal. Conv. Mont., (1971), 1-43.  Google Scholar

[30]

J. Nash, Real Algebraic Manifolds, Ann. of Math., 56 (1952), 405-421.  doi: 10.2307/1969649.  Google Scholar

[31]

R. A. Poliquin, R. 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.  Google Scholar

[32]

D. Salas and L. Thibault, Characterizations of nonconvex sets with smooth boundary in terms of the metric projection in Hilbert spaces, preprint. Google Scholar

[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.  Google Scholar

[34]

A. Tolstonogov, Control sweeping processes, J. Convex Anal., 23 (2016), 1099-1123.   Google Scholar

[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.  Google Scholar

Figure 1.  Two prox-regular sets with smooth and non-smooth boundary
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Figure 2.  Sets of Definition 4.1 for a trajectory in $ \mathbb{R} ^2 $
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Figure 3.  A circuit with an ideal diode, an inductor and a current source
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Figure 4.  Functions $ f_1 $ and $ f_2 $
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Figure 5.  Trajectory for $ f_2 $
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