On smoothness of solutions to projected differential equations

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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.
[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. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[6]	D. Bressoud, A 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. 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.
[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.
[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.
[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. 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.
[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. Folland, Introduction 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. 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.
[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.

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.
[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. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[6]	D. Bressoud, A 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. 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.
[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.
[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.
[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. 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.
[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. Folland, Introduction 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. 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.
[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.

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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