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

Local inversion of a class of piecewise regular maps

Laura Poggiolini and  Marco Spadini ,

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.

Keywords: Piecewise linear maps, piecewise C1 maps, Bouligand derivative, degree theory.
Mathematics Subject Classification: 26B10, 47H11, 47J07.
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. Google Scholar

[2]

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

[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.   Google Scholar

[4]

F. H. Clarke, On the inverse function theorem, Pacific J. Mathematics, 64 (1976), 97-102.   Google Scholar

[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.  Google Scholar

[6]

K. Deimling. Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985.  Google Scholar

[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.   Google Scholar

[8]

N. G. Lloyd, Degree Theory Cambridge Tracts in Math. 73, Cambridge University Press, Cambridge, 1978.  Google Scholar

[9]

J. Milnor, Topology from The Differentiable Viewpoint , The University Press of Virginia, 1965.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[13]

L. Poggiolini and M. Spadini, Local inversion of planar maps with nice nondifferentiability structure, Adv. Nonlin. Studies, 13 (2013), 411-430.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer briefs in optimization. Springer, New York, 2012.  Google Scholar

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.   Google Scholar

[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.   Google Scholar

[4]

F. H. Clarke, On the inverse function theorem, Pacific J. Mathematics, 64 (1976), 97-102.   Google Scholar

[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.  Google Scholar

[6]

K. Deimling. Nonlinear Functional Analysis Springer-Verlag, Berlin, 1985.  Google Scholar

[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.   Google Scholar

[8]

N. G. Lloyd, Degree Theory Cambridge Tracts in Math. 73, Cambridge University Press, Cambridge, 1978.  Google Scholar

[9]

J. Milnor, Topology from The Differentiable Viewpoint , The University Press of Virginia, 1965.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[13]

L. Poggiolini and M. Spadini, Local inversion of planar maps with nice nondifferentiability structure, Adv. Nonlin. Studies, 13 (2013), 411-430.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[16]

S. Scholtes, Introduction to Piecewise Differentiable Equations, Springer briefs in optimization. Springer, New York, 2012.  Google Scholar

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.
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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.
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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.
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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.
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Figure 5.  The decomposition of $\mathbb{R}^3$ of Example 3.12
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