# American Institute of Mathematical Sciences

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. 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

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##### 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
The image of the unit disk centered at the origin in the plane $z = 0$ under the map $G$ of Example 2.4.
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.
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.
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.
The decomposition of $\mathbb{R}^3$ of Example 3.12
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