September  2018, 17(5): 2207-2224. doi: 10.3934/cpaa.2018105

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

* Corresponding author

Received  July 2017 Revised  January 2018 Published  April 2018

Fund Project: The authors were partially supported by the "Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni" of the Istituto Nazionale di Alta Matematica "F. Severi".

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.

Citation: Laura Poggiolini, Marco Spadini. Local inversion of a class of piecewise regular maps. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2207-2224. doi: 10.3934/cpaa.2018105
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.

[2]

F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39.

[3]

S. A. BurdenS. S. SastryD. 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.

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.

[2]

F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 1-39.

[3]

S. A. BurdenS. S. SastryD. 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.

Figure 1.  The image of the unit disk centered at the origin in the plane $z = 0$ under the map $G$ of Example 2.4.
Figure 2.  The three cones in Example 3.2 for different choices of α and β. The z axis is not shown because it is assumed perpendicular to the page.
Figure 3.  Image of the unit sphere under $\mathcal{G}$ as in Example 3.2 for different choices of the parameters $\alpha$, $\beta$, $s_1$, $s_2$, $s_3$. The dark continuous line represents the image of the circle of radius $1$ centered at the origin.
Figure 4.  The image of the unit sphere centered at the origin under the map G of Example 3.5 The green lines on the left are the intersections of the cones C1, …, C4 with the unit sphere, those on the right are their images.
Figure 5.  The decomposition of $\mathbb{R}^3$ of Example 3.12
[1]

Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533

[2]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91

[3]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105

[4]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

[5]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451

[6]

Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753

[7]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[8]

Lluís Alsedà, Sylvie Ruette. On the set of periods of sigma maps of degree 1. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4683-4734. doi: 10.3934/dcds.2015.35.4683

[9]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040

[10]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055

[11]

Iryna Sushko, Anna Agliari, Laura Gardini. Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 881-897. doi: 10.3934/dcdsb.2005.5.881

[12]

Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739

[13]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877

[14]

Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365

[15]

Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843

[16]

Rodrigo P. Pacheco, Rafael O. Ruggiero. On C1, β density of metrics without invariant graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 247-261. doi: 10.3934/dcds.2018012

[17]

Shu-Guang Shao, Shu Wang, Wen-Qing Xu, Yu-Li Ge. On the local C1, α solution of ideal magneto-hydrodynamical equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2103-2113. doi: 10.3934/dcds.2017090

[18]

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915

[19]

Xu Zhang. Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2873-2886. doi: 10.3934/dcds.2016.36.2873

[20]

Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (30)
  • HTML views (127)
  • Cited by (0)

Other articles
by authors

[Back to Top]