## The Aronsson equation, Lyapunov functions, and local Lipschitz regularity of the minimum time function.(English)Zbl 1456.49022

Summary: We define and study $$C^1$$-solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that $$C^1$$-solutions are absolutely minimizing functions. We discuss how $$C^1$$-supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct $$C^1$$-solutions (even classical) explicitly.

### MSC:

 49K40 Sensitivity, stability, well-posedness 49K15 Optimality conditions for problems involving ordinary differential equations 93B27 Geometric methods 37B25 Stability of topological dynamical systems
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### References:

 [1] Evans, L. C.; Savin, O., $$C^{1,α}$$ regularity for infinity harmonic functions in two dimensions, Calculus of Variations and Partial Differential Equations, 32, 3, 325-347 (2008) · Zbl 1151.35096 [2] Bardi, M.; Capuzzo-Dolcetta, I., Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. With Appendices by Maurizio Falcone and Pierpaolo Soravia, Systems & Control: Foundations & Applications (1997), Boston, MA: Birkhäuser Boston Inc., Boston, MA [3] Petrov, N. N., Controllability of autonomous systems, Differencial’nye Uravnenija, 4, 4, 606-617 (1968) · Zbl 0174.40302 [4] Liverovski, A. A., A Hölder condition for Bellman’s function, Differencial’nye Uravnenija, 13, 12, 2180-2187, 2301 (1977) · Zbl 0379.49029 [5] Soravia, P., Hölder continuity of the minimum-time function for C1-manifold targets, Journal of Optimization Theory and Applications, 75, 2, 401-421 (1992) · Zbl 0792.93058 [6] Krastanov, M. I.; Quincampoix, M., On the small-time controllability of discontinuous piece-wise linear systems systems, Systems & Control Letters, 62, 2, 218-223 (2013) · Zbl 1259.93026 [7] Marigonda, A., Second order conditions for the controllability of nonlinear systems with drift, Communications on Pure and Applied Analysis, 5, 4, 861-885 (2006) · Zbl 1175.93038 [8] Marigonda, A.; Le, T. T., Sufficient conditions for small time local attainability for a class of control systems, Large-scale Scientific Computing 9374, 117-125 (2015), Cham: Springer, Cham [9] Le Thuy, T. T.; Marigonda, A., Small-time local attainability for a class of control systems with state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 23, 3, 1003-1021 (2017) · Zbl 1369.49016 [10] Albano, P.; Cannarsa, P.; Scarinci, T., Partial regularity for solutions to subelliptic eikonal equations, Comptes Rendus Mathematique, 356, 2, 172-176 (2018) · Zbl 1383.35040 [11] Albano, P.; Cannarsa, P.; Scarinci, T., Regularity results for the minimum time function with Hörmander vector fields, Journal of Differential Equations, 264, 5, 3312-3335 (2018) · Zbl 1380.35057 [12] Motta, M.; Rampazzo, F., Asymptotic controllability and Lyapunov-like functions determined by Lie brackets, SIAM Journal on Control and Optimization, 56, 2, 1508-1534 (2018) · Zbl 1401.93039 [13] Aronsson, G., Minimization problems for the functional $$sup_xF(x, f(x), f$$′$$(x))$$, Arkiv för Matematik, 6, 1, 33-53 (1965) · Zbl 0156.12502 [14] Jensen, R. R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Archive for Rational Mechanics and Analysis, 123, 1, 51-74 (1993) · Zbl 0789.35008 [15] Barron, E. N.; Jensen, R. R.; Wang, C. Y., The Euler equation and absolute minimizers of $$L^∞$$ functionals, Archive for Rational Mechanics and Analysis, 157, 4, 255-283 (2001) · Zbl 0979.49003 [16] Juutinen, P., Annales Academiae Scientiarum Fennicae. Mathematica. Dissertationes, 115, 1998, p. 53., Jyvaskula: University of Jyvaskula, Jyvaskula [17] Crandall, M. G., An efficient derivation of the Aronsson equation, Archive for Rational Mechanics and Analysis, 167, 4, 271-279 (2003) · Zbl 1090.35067 [18] Aronsson, G.; Crandall, M. G.; Juutinen, P., A tour of the theory of absolutely minimizing functions, Bulletin of the American Mathematical Society, 41, 4, 439-505 (2004) · Zbl 1150.35047 [19] Soravia, P., Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation, Mathematical Control and Related Fields, 2, 4, 399-427 (2012) · Zbl 1264.35097 [20] Soravia, P., On Aronsson equation and deterministic optimal control, Applied Mathematics and Optimization, 59, 2, 175-201 (2009) · Zbl 1167.49025 [21] Soravia, P., Absolute minimizers, Aronsson equation and Eikonal equations with Lipschitz continuous vector fields, International conference for the 25th anniversary of viscosity solutions. Tokyo, 4-6 June 2007, Gakuto Int. Series, 30, 175-193 (2008), Gakkotosho Co., Ltd [22] Soravia, P., Existence of absolute minimizers for noncoercive Hamiltonians and viscosity solutions of the Aronsson equation, Mathematical Control and Related Fields, 2, 4, 399-427 (2012) · Zbl 1264.35097 [23] Bieske, T.; Capogna, L., The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Caratheodory metrics, Transactions of the American Mathematical Society, 357, 795-823 (2005) · Zbl 1058.35067 [24] Bieske, T., Properties of infinite harmonic functions of Grushin-type spaces, Rocky Mountain Journal of Mathematics, 39, 729-756 (2009) · Zbl 1172.35366 [25] Wang, C., The Aronsson equation for absolute minimizers of $$L^∞$$-functionals associated with vector fields satisfying Hörmander’s condition, Transactions of the American Mathematical Society, 359, 91-113 (2007) · Zbl 1192.35038 [26] Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, P. R., Qualitative properties of trajectories of control systems: a survey, Journal of Dynamical and Control Systems, 1, 1, 1-48 (1995) · Zbl 0951.49003 [27] Clarke, F. H.; Ledyaev, Yu. S.; Stern, R. J.; Wolenski, P. R., Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, 178 (1998), New York: Springer-Verlag, New York · Zbl 1047.49500 [28] Soravia, P., Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints, Differential Integral Equations, 12, 2, 275-293 (1999) · Zbl 1007.49016 [29] Donchev, T.; Rios, V.; Wolenski, P., Strong invariance and one-sided Lipschitz multifunctions, Nonlinear Analysis: Theory, Methods & Applications, 60, 5, 849-862 (2005) · Zbl 1068.34010 [30] Bardi, M.; Dragoni, F., Convexity and semiconvexity along vector fields, Convexity and Semiconvexity Along Vector Fields, 42, 3-4, 405-427 (2011) · Zbl 1232.26012 [31] Soravia, P., Some results on second order controllability conditions, https://arxiv.org/abs/1905.09585
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