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Specified homogenization of a discrete traffic model leading to an effective junction condition
Local inversion of a class of piecewise regular maps
Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy |
This paper provides sufficient conditions for any map L, that is strongly piecewise linear relatively to a decomposition of $\mathbb{R}^k$ in admissible cones, to be invertible. Namely, via a degree theory argument, we show that when there are at most four convex pieces (or three pieces with at most a non convex one), the map is invertible. Examples show that the result cannot be plainly extended to a greater number of pieces. Our result is obtained by studying the structure of strongly piecewise linear maps. We then extend the results to the PC1 case.
References:
[1] |
P. Benevieri, M. Furi, M. P. Pera and M. Spadini, An Introduction to Topological Degree in Euclidean Spaces, Technical Report n. 42, Gennaio 2003, Università di Firenze, Dipartimento di Matematica Applicata, 2003. Google Scholar |
[2] |
F. E. Browder,
Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39.
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[3] |
S. A. Burden, S. S. Sastry, D. E. Koditschek and S. Revzen,
Event-selected vector field discontinuities yield piecewise-differentiable flows, SIAM J. Appl. Dyn. Syst., 15 (2016), 1227-1267.
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[4] |
F. H. Clarke,
On the inverse function theorem, Pacific J. Mathematics, 64 (1976), 97-102.
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[5] |
F. H. Clarke,
Optimization And Nonsmooth Analysis Unrev. reprinting of the orig., publ. 1983 by Wiley, Montréal: Centre de Recherches Mathématiques, Université de Montréal, 1989. |
[6] |
K. Deimling.
Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. |
[7] |
L. Kuntz and S. Scholtes,
Structural analysis of nonsmooth mappings, inverse functions and metric projections, Journal of Mathematical Analysis and Applications, 188 (1994), 346-386.
|
[8] |
N. G. Lloyd,
Degree Theory Cambridge Tracts in Math. 73, Cambridge University Press, Cambridge, 1978. |
[9] |
J. Milnor,
Topology from The Differentiable Viewpoint , The University Press of Virginia, 1965. |
[10] |
J. S. Pang and D. Ralph,
Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Mathematics of Operations Research, 21 (1996), 401-426.
|
[11] |
Sufficient optimality conditions for a bang-bang trajectory in a
Bolza problem. In, Mathematical Control Theory and Finance, (eds. A. Sarychev, A. Shiryaev, M. Guerra, and M. do Rosário Grossinho), Springer, Berlin Heidelberg, (2008), 337-357. |
[12] |
L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem,
SIAM Journal on Control and Optimization 49 (2011), 140-161, |
[13] |
L. Poggiolini and M. Spadini,
Local inversion of planar maps with nice nondifferentiability structure, Adv. Nonlin. Studies, 13 (2013), 411-430.
|
[14] |
L. Poggiolini and M. Spadini,
Bang-bang trajectories with a double switching time in the minimum time problem, ESAIM: Control Optimization and Calculus of Variations, 22 (2016), 688-709.
|
[15] |
E. Rosset, Topological degree in $\mathbb{R}^n$, Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics, 20 (1988), 319-329. Available from: http://hdl.handle.net/10077/4865. |
[16] |
S. Scholtes,
Introduction to Piecewise Differentiable Equations, Springer briefs in optimization. Springer, New York, 2012. |
show all references
References:
[1] |
P. Benevieri, M. Furi, M. P. Pera and M. Spadini, An Introduction to Topological Degree in Euclidean Spaces, Technical Report n. 42, Gennaio 2003, Università di Firenze, Dipartimento di Matematica Applicata, 2003. Google Scholar |
[2] |
F. E. Browder,
Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39.
|
[3] |
S. A. Burden, S. S. Sastry, D. E. Koditschek and S. Revzen,
Event-selected vector field discontinuities yield piecewise-differentiable flows, SIAM J. Appl. Dyn. Syst., 15 (2016), 1227-1267.
|
[4] |
F. H. Clarke,
On the inverse function theorem, Pacific J. Mathematics, 64 (1976), 97-102.
|
[5] |
F. H. Clarke,
Optimization And Nonsmooth Analysis Unrev. reprinting of the orig., publ. 1983 by Wiley, Montréal: Centre de Recherches Mathématiques, Université de Montréal, 1989. |
[6] |
K. Deimling.
Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985. |
[7] |
L. Kuntz and S. Scholtes,
Structural analysis of nonsmooth mappings, inverse functions and metric projections, Journal of Mathematical Analysis and Applications, 188 (1994), 346-386.
|
[8] |
N. G. Lloyd,
Degree Theory Cambridge Tracts in Math. 73, Cambridge University Press, Cambridge, 1978. |
[9] |
J. Milnor,
Topology from The Differentiable Viewpoint , The University Press of Virginia, 1965. |
[10] |
J. S. Pang and D. Ralph,
Piecewise smoothness, local invertibility, and parametric analysis of normal maps, Mathematics of Operations Research, 21 (1996), 401-426.
|
[11] |
Sufficient optimality conditions for a bang-bang trajectory in a
Bolza problem. In, Mathematical Control Theory and Finance, (eds. A. Sarychev, A. Shiryaev, M. Guerra, and M. do Rosário Grossinho), Springer, Berlin Heidelberg, (2008), 337-357. |
[12] |
L. Poggiolini and M. Spadini, Strong local optimality for a bang-bang trajectory in a Mayer problem,
SIAM Journal on Control and Optimization 49 (2011), 140-161, |
[13] |
L. Poggiolini and M. Spadini,
Local inversion of planar maps with nice nondifferentiability structure, Adv. Nonlin. Studies, 13 (2013), 411-430.
|
[14] |
L. Poggiolini and M. Spadini,
Bang-bang trajectories with a double switching time in the minimum time problem, ESAIM: Control Optimization and Calculus of Variations, 22 (2016), 688-709.
|
[15] |
E. Rosset, Topological degree in $\mathbb{R}^n$, Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics, 20 (1988), 319-329. Available from: http://hdl.handle.net/10077/4865. |
[16] |
S. Scholtes,
Introduction to Piecewise Differentiable Equations, Springer briefs in optimization. Springer, New York, 2012. |





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