Figure 1.
Two prox-regular sets with smooth and non-smooth boundary
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April
2019, 39(4): 2255-2283.
doi: 10.3934/dcds.2019095
On smoothness of solutions to projected differential equations
| 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 |
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 and Continuous Dynamical Systems,
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 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 5.
Trajectory for $ f_2 $
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