# American Institute of Mathematical Sciences

March  2013, 3(1): 51-82. doi: 10.3934/mcrf.2013.3.51

## On the minimum time function around the origin

 1 Università di Padova, Dipartimento di Matematica, via Trieste 63, 35121 Padova, Italy, Italy

Received  March 2012 Revised  November 2012 Published  February 2013

We deal with finite dimensional linear and nonlinear control systems. If the system is linear and autonomous and satisfies the classical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving a polynomial estimate. The result is based on a careful analysis of the switching function. We extend this result to nonautonomous linear systems, provided the time dependent system is not too far from the autonomous system obtained by taking the time to be $0$ in the dynamics.
Using a linearization approach, we prove a bang-bang principle, valid in dimensions $2$ and $3$ for a class of nonlinear systems, affine and symmetric with respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the origin satisfies the same polynomial strict convexity property as for the linearized dynamics, provided the nonlinearity is small enough. Finally, under the same assumptions we show that the epigraph of the minimum time function has positive reach, hence proving the first result of this type in a nonlinear setting. In all the above results, we require that the linearization at the origin be normal. We provide examples showing the sharpness of our assumptions.
Citation: Giovanni Colombo, Khai T. Nguyen. On the minimum time function around the origin. Mathematical Control & Related Fields, 2013, 3 (1) : 51-82. doi: 10.3934/mcrf.2013.3.51
##### References:
 [1] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems & Control: Foundations & Applications, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar [2] U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds,", Mathématiques & Applications (Berlin), 43 (2004). Google Scholar [3] P. Brunovský, Every normal linear system admits a regular time-optimal synthesis,, Math. Slovaca, 28 (1978), 81. Google Scholar [4] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems,, ESAIM Control Optim. Cal. Var., 12 (2006), 350. doi: 10.1051/cocv:2006002. Google Scholar [5] P. Cannarsa, F. Marino and P. R. Wolenski, Semiconcavity of the minimum time function for differential inclusions,, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 19 (2012), 187. Google Scholar [6] P. Cannarsa and Khai T. Nguyen, Exterior sphere condition and time optimal controlv for differential inclusions,, SIAM J. Control Optim., 49 (2011), 2558. doi: 10.1137/110825078. Google Scholar [7] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function,, Calc. Var. Partial Differential Equations, 3 (1995), 273. doi: 10.1007/BF01189393. Google Scholar [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004). Google Scholar [9] L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations,", Applications of Mathematics (New York), 17 (1983). Google Scholar [10] F. H. Clarke, "Optimization and Nonsmooth Analysis,", Second edition, 5 (1990). doi: 10.1137/1.9781611971309. Google Scholar [11] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998). Google Scholar [12] G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions,, Calc. Var. Partial Differential Equations, 25 (2006), 1. doi: 10.1007/s00526-005-0352-7. Google Scholar [13] G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function,, SIAM J. Control Optim., 44 (2006), 2285. doi: 10.1137/050630076. Google Scholar [14] G. Colombo and Khai T. Nguyen, On the structure of the minimum time function,, SIAM J. Control Optim., 48 (2010), 4776. doi: 10.1137/090774549. Google Scholar [15] H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. Google Scholar [16] H. Hermes and J. P. LaSalle, "Functional Analysis and Time Optimal Control,", Mathematics in Science and Engineering, (1969). Google Scholar [17] S. Łojasiewicz, Jr., Some properties of accessible sets in nonlinear control systems,, Annal. Polon. Math., 36 (1979), 123. Google Scholar [18] Khai T. Nguyen, Hypographs satisfying an external sphere condition and the regularity of the minimum time function,, J. Math. Anal. Appl., 372 (2010), 611. doi: 10.1016/j.jmaa.2010.07.010. Google Scholar [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$,, Nonlinear Analysis, 3 (1979), 145. doi: 10.1016/0362-546X(79)90044-0. Google Scholar [20] H. Sussmann, A bang-bang theorem with bounds on the number of switchings,, SIAM J. Control Optim., 17 (1979), 629. doi: 10.1137/0317045. Google Scholar

show all references

##### References:
 [1] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Systems & Control: Foundations & Applications, (1997). doi: 10.1007/978-0-8176-4755-1. Google Scholar [2] U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds,", Mathématiques & Applications (Berlin), 43 (2004). Google Scholar [3] P. Brunovský, Every normal linear system admits a regular time-optimal synthesis,, Math. Slovaca, 28 (1978), 81. Google Scholar [4] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems,, ESAIM Control Optim. Cal. Var., 12 (2006), 350. doi: 10.1051/cocv:2006002. Google Scholar [5] P. Cannarsa, F. Marino and P. R. Wolenski, Semiconcavity of the minimum time function for differential inclusions,, Discrete Contin. Dyn. Syst. Ser. B Appl. Algorithms, 19 (2012), 187. Google Scholar [6] P. Cannarsa and Khai T. Nguyen, Exterior sphere condition and time optimal controlv for differential inclusions,, SIAM J. Control Optim., 49 (2011), 2558. doi: 10.1137/110825078. Google Scholar [7] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function,, Calc. Var. Partial Differential Equations, 3 (1995), 273. doi: 10.1007/BF01189393. Google Scholar [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Progress in Nonlinear Differential Equations and their Applications, 58 (2004). Google Scholar [9] L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations,", Applications of Mathematics (New York), 17 (1983). Google Scholar [10] F. H. Clarke, "Optimization and Nonsmooth Analysis,", Second edition, 5 (1990). doi: 10.1137/1.9781611971309. Google Scholar [11] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998). Google Scholar [12] G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions,, Calc. Var. Partial Differential Equations, 25 (2006), 1. doi: 10.1007/s00526-005-0352-7. Google Scholar [13] G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function,, SIAM J. Control Optim., 44 (2006), 2285. doi: 10.1137/050630076. Google Scholar [14] G. Colombo and Khai T. Nguyen, On the structure of the minimum time function,, SIAM J. Control Optim., 48 (2010), 4776. doi: 10.1137/090774549. Google Scholar [15] H. Federer, Curvature measures,, Trans. Amer. Math. Soc., 93 (1959), 418. Google Scholar [16] H. Hermes and J. P. LaSalle, "Functional Analysis and Time Optimal Control,", Mathematics in Science and Engineering, (1969). Google Scholar [17] S. Łojasiewicz, Jr., Some properties of accessible sets in nonlinear control systems,, Annal. Polon. Math., 36 (1979), 123. Google Scholar [18] Khai T. Nguyen, Hypographs satisfying an external sphere condition and the regularity of the minimum time function,, J. Math. Anal. Appl., 372 (2010), 611. doi: 10.1016/j.jmaa.2010.07.010. Google Scholar [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$,, Nonlinear Analysis, 3 (1979), 145. doi: 10.1016/0362-546X(79)90044-0. Google Scholar [20] H. Sussmann, A bang-bang theorem with bounds on the number of switchings,, SIAM J. Control Optim., 17 (1979), 629. doi: 10.1137/0317045. Google Scholar
 [1] H. O. Fattorini. The maximum principle in infinite dimension. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 557-574. doi: 10.3934/dcds.2000.6.557 [2] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [3] Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042 [4] Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 [5] Carlo Orrieri. A stochastic maximum principle with dissipativity conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5499-5519. doi: 10.3934/dcds.2015.35.5499 [6] Chadi Nour, Ron J. Stern, Jean Takche. Generalized exterior sphere conditions and $\varphi$-convexity. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 615-622. doi: 10.3934/dcds.2011.29.615 [7] Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135 [8] Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789 [9] Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571 [10] Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055 [11] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [12] Alex Kontorovich. The local-global principle for integral Soddy sphere packings. Journal of Modern Dynamics, 2019, 15: 209-236. doi: 10.3934/jmd.2019019 [13] Chiun-Chuan Chen, Li-Chang Hung, Hsiao-Feng Liu. N-barrier maximum principle for degenerate elliptic systems and its application. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 791-821. doi: 10.3934/dcds.2018034 [14] Yunkyong Hyon, Do Young Kwak, Chun Liu. Energetic variational approach in complex fluids: Maximum dissipation principle. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1291-1304. doi: 10.3934/dcds.2010.26.1291 [15] Zaidong Zhan, Shuping Chen, Wei Wei. A unified theory of maximum principle for continuous and discrete time optimal control problems. Mathematical Control & Related Fields, 2012, 2 (2) : 195-215. doi: 10.3934/mcrf.2012.2.195 [16] Yan Wang, Yanxiang Zhao, Lei Wang, Aimin Song, Yanping Ma. Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer. Journal of Industrial & Management Optimization, 2018, 14 (2) : 653-671. doi: 10.3934/jimo.2017067 [17] Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581 [18] H. O. Fattorini. The maximum principle for linear infinite dimensional control systems with state constraints. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 77-101. doi: 10.3934/dcds.1995.1.77 [19] Isabeau Birindelli, Francoise Demengel. Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. Communications on Pure & Applied Analysis, 2007, 6 (2) : 335-366. doi: 10.3934/cpaa.2007.6.335 [20] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395

2018 Impact Factor: 1.292