We prove the continuity and the Hölder equivalence w.r.t. an Euclidean distance of the value function associated with the L^{1} cost of the control-affine system q̇ = f_{0}(q) + ∑_{j=1}^{m} u_{j} f_{j}(q), satisfying the strong Hörmander condition. This is done by proving a result in the same spirit as the Ball-Box theorem for driftless (or sub-Riemannian) systems. The techniques used are based on a reduction of the control-affine system to a linear but time-dependent one, for which we are able to define a generalization of the nilpotent approximation and through which we derive estimates for the shape of the reachable sets. Finally, we also prove the continuity of the value function associated with the L^{1} cost of time-dependent systems of the form q̇ = ∑_{j=1}^{m} u_{j} f_{j}^{t}(q).

Keywords: control-affine systems, time-dependent systems, sub-riemannian geometry, value function, Ball-Box theorem, nilpotent approximation

@article{COCV_2014__20_4_1224_0, author = {Prandi, Dario}, title = {H\"older equivalence of the value function for control-affine systems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1224--1248}, publisher = {EDP-Sciences}, volume = {20}, number = {4}, year = {2014}, doi = {10.1051/cocv/2014014}, mrnumber = {3264241}, zbl = {1301.53029}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2014014/} }

TY - JOUR AU - Prandi, Dario TI - Hölder equivalence of the value function for control-affine systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 1224 EP - 1248 VL - 20 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2014014/ DO - 10.1051/cocv/2014014 LA - en ID - COCV_2014__20_4_1224_0 ER -

%0 Journal Article %A Prandi, Dario %T Hölder equivalence of the value function for control-affine systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 1224-1248 %V 20 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2014014/ %R 10.1051/cocv/2014014 %G en %F COCV_2014__20_4_1224_0

Prandi, Dario. Hölder equivalence of the value function for control-affine systems. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 4, pp. 1224-1248. doi : 10.1051/cocv/2014014. http://archive.numdam.org/articles/10.1051/cocv/2014014/

[1] Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes). http://people.sissa.it/agrachev/agrachev˙files/notes.htm (2012).

, and ,[2] The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256 (2009) 2621-2655. | MR | Zbl

, , and ,[3] Continuity of optimal control costs and its application to weak KAM theory. Calc. Var. Partial Differ. Equ. 39 (2010) 213-232. | MR | Zbl

and ,[4] Two-dimensional almost-Riemannian structures with tangency points. Ann. Inst. Henri Poincaré Anal. Non Linéaire 27 (2010) 793-807. | Numdam | MR | Zbl

, , , and ,[5] Control theory from the geometric viewpoint. Control Theory and Optimization II. Vol. 87 of Encycl. Math. Sci. Springer-Verlag, Berlin (2004). | MR | Zbl

and ,[6] A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete Contin. Dyn. Syst. 20 (2008) 801-822. | MR | Zbl

, and ,[7] High-order sufficient conditions for configuration tracking of affine connection control systems. Syst. Control Lett. 59 (2010) 491-503. | MR | Zbl

and ,[8] The tangent space in sub-Riemannian geometry. In Sub-Riemannian geometry. Vol. 144 of Progr. Math. Birkhäuser, Basel (1996) 1-78. | MR | Zbl

,[9] Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28 (1990) 903-924. | MR | Zbl

and ,[10] Time-optimal problem and time-optimal map. Rend. Sem. Mat. Univ. Politec. Torino 48 (1992) 401-429 (1990). | MR | Zbl

and .[11] Time minimal trajectories for a spin 1 / 2 particle in a magnetic field. J. Math. Phys. 47 (2006) 062101, 29. | MR | Zbl

and ,[12] Resonance of minimizers for n-level quantum systems with an arbitrary cost. ESAIM: COCV 10 (2004) 593-614. | Numdam | MR | Zbl

and ,[13] Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43 (2002) 2107-2132. | MR | Zbl

, , , , and ,[14] Geometric control of mechanical systems. Modeling, analysis, and design for simple mechanical control systems. Vol. 49 of Texts Appl. Math. Springer-Verlag, New York (2005). | MR | Zbl

and ,[15] Singular trajectories of control-affine systems. SIAM J. Control Optim. 47 (2008) 1078-1095. | MR | Zbl

, and ,[16] Control and nonlinearity. Vol. 136 of Math. Surv. Monogr. American Mathematical Society, Providence, RI (2007). | MR | Zbl

,[17] Introduction to quantum control and dynamics. Chapman & Hall/CRC Appl. Math. Nonl. Sci. Series. Chapman & Hall/CRC, Boca Raton, FL (2008). | MR | Zbl

,[18] Optimal control of two-level quantum systems. IEEE Trans. Automat. Control 46 (2001) 866-876. | MR | Zbl

and ,[19] Smooth distributions are finitely generated. Ann. Global Anal. Geom. 41 (2012) 357-369. | MR | Zbl

, , and ,[20] Estimates for the | MR | Zbl

and ,[21] Value function in optimal control, in Mathematical control theory, Part 1, 2 (Trieste, 2001), vol. 8 of ICTP Lect. Notes. Abdus Salam Int. Cent. Theoret. Phys., Trieste (2002) 516-653 (electronic). | MR | Zbl

,[22] On the codimension one motion planning problem. J. Dyn. Control Syst. 11 (2005) 73-89. | MR | Zbl

and ,[23] On the motion planning problem, complexity, entropy, and nonholonomic interpolation. J. Dyn. Control Syst. 12 (2006) 371-404. | MR | Zbl

and ,[24] Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33 (1991) 238-264. | MR | Zbl

,[25] Complexity of nonholonomic motion planning. Int. J. Control 74 (2001) 776-782. | MR | Zbl

,[26] Uniform estimation of sub-Riemannian balls. J. Dynam. Control Systems 7 (2001) 473-500. | MR | Zbl

,[27] Entropy and complexity of a path in sub-Riemannian geometry. ESAIM: COCV 9 (2003) 485-508. | Numdam | MR | Zbl

,[28] Subelliptic, second order differential operators, in Complex analysis, III (College Park, Md., 1985-86). Vol. 1277 of Lect. Notes Math. Springer, Berlin (1987) 46-77. | MR | Zbl

and ,[29] Geometric control theory. Vol. 52 of Cambridge Stud. Adv. Math. Cambridge University Press, Cambridge (1997). | MR | Zbl

,[30] On Carnot−Carathéodory metrics. J. Differ. Geom. 21 35-45, 1985. | MR | Zbl

,[31] On complexity and motion planning for co-rank one sub-Riemannian metrics. ESAIM: COCV 10 (2004) 634-655. | Numdam | MR | Zbl

, and ,[32] Hypoelliptic differential operators and nilpotent groups. Acta Math. 137 (1976) 247-320. | MR | Zbl

and ,[33] A control theoretic approach to the swimming of microscopic organisms. Quart. Appl. Math. 65 (2007) 405-424. | MR | Zbl

, and ,[34] Some properties of vector field systems that are not altered by small perturbations. J. Differ. Equ. 20 (1976) 292-315. | MR | Zbl

,[35] Some recent results on the regularity of optimal cost functions, in Proc. of the Berkeley-Ames conference on nonlinear problems in control and fluid dynamics (Berkeley, Calif. (1983)), Lie Groups: Hist., Frontiers and Appl. Ser. B: Systems Inform. Control, II. Math Sci Press. Brookline, MA (1984) 429-434. | MR | Zbl

,[36] A general theorem on local controllability. SIAM J. Control Optim. 25 (1987) 158-194. | MR | Zbl

,[37] Regular synthesis for time-optimal control of single-input real analytic systems in the plane. SIAM J. Control Optim. 25 (1987) 1145-1162. | MR | Zbl

,[38] Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dyn. Control Systems 6 (2000) 511-541. | MR | Zbl

,*Cited by Sources: *