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 and 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, Birkhäuser Boston, Inc., Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [2] U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Mathématiques & Applications (Berlin), 43, Springer-Verlag, Berlin, 2004. [3] P. Brunovský, Every normal linear system admits a regular time-optimal synthesis, Math. Slovaca, 28 (1978), 81-100. [4] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Cal. Var., 12 (2006), 350-370. doi: 10.1051/cocv:2006002. [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), 187-206. [6] P. Cannarsa and Khai T. Nguyen, Exterior sphere condition and time optimal controlv for differential inclusions, SIAM J. Control Optim., 49 (2011), 2558-2576. doi: 10.1137/110825078. [7] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations, 3 (1995), 273-298. doi: 10.1007/BF01189393. [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, 2004. [9] L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983. [10] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309. [11] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. [12] G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var. Partial Differential Equations, 25 (2006), 1-31. doi: 10.1007/s00526-005-0352-7. [13] G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: 10.1137/050630076. [14] G. Colombo and Khai T. Nguyen, On the structure of the minimum time function, SIAM J. Control Optim., 48 (2010), 4776-4814. doi: 10.1137/090774549. [15] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. [16] H. Hermes and J. P. LaSalle, "Functional Analysis and Time Optimal Control," Mathematics in Science and Engineering, Vol. 56, Academic Press, New York-London, 1969. [17] S. Łojasiewicz, Jr., Some properties of accessible sets in nonlinear control systems, Annal. Polon. Math., 36 (1979), 123-137. [18] Khai T. Nguyen, Hypographs satisfying an external sphere condition and the regularity of the minimum time function, J. Math. Anal. Appl., 372 (2010), 611-628. doi: 10.1016/j.jmaa.2010.07.010. [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$, Nonlinear Analysis, Theory, Methods and Applications, 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0. [20] H. Sussmann, A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim., 17 (1979), 629-651. doi: 10.1137/0317045.

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References:
 [1] M. Bardi and I. Capuzzo-Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, 1997. doi: 10.1007/978-0-8176-4755-1. [2] U. Boscain and B. Piccoli, "Optimal Syntheses for Control Systems on 2-D Manifolds," Mathématiques & Applications (Berlin), 43, Springer-Verlag, Berlin, 2004. [3] P. Brunovský, Every normal linear system admits a regular time-optimal synthesis, Math. Slovaca, 28 (1978), 81-100. [4] P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Cal. Var., 12 (2006), 350-370. doi: 10.1051/cocv:2006002. [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), 187-206. [6] P. Cannarsa and Khai T. Nguyen, Exterior sphere condition and time optimal controlv for differential inclusions, SIAM J. Control Optim., 49 (2011), 2558-2576. doi: 10.1137/110825078. [7] P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var. Partial Differential Equations, 3 (1995), 273-298. doi: 10.1007/BF01189393. [8] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, 2004. [9] L. Cesari, "Optimization-Theory and Applications. Problems with Ordinary Differential Equations," Applications of Mathematics (New York), 17, Springer-Verlag, New York, 1983. [10] F. H. Clarke, "Optimization and Nonsmooth Analysis," Second edition, Classics in Applied Mathematics, 5, SIAM, Philadelphia, 1990. doi: 10.1137/1.9781611971309. [11] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998. [12] G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var. Partial Differential Equations, 25 (2006), 1-31. doi: 10.1007/s00526-005-0352-7. [13] G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: 10.1137/050630076. [14] G. Colombo and Khai T. Nguyen, On the structure of the minimum time function, SIAM J. Control Optim., 48 (2010), 4776-4814. doi: 10.1137/090774549. [15] H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. [16] H. Hermes and J. P. LaSalle, "Functional Analysis and Time Optimal Control," Mathematics in Science and Engineering, Vol. 56, Academic Press, New York-London, 1969. [17] S. Łojasiewicz, Jr., Some properties of accessible sets in nonlinear control systems, Annal. Polon. Math., 36 (1979), 123-137. [18] Khai T. Nguyen, Hypographs satisfying an external sphere condition and the regularity of the minimum time function, J. Math. Anal. Appl., 372 (2010), 611-628. doi: 10.1016/j.jmaa.2010.07.010. [19] R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\mathbbR^n$, Nonlinear Analysis, Theory, Methods and Applications, 3 (1979), 145-154. doi: 10.1016/0362-546X(79)90044-0. [20] H. Sussmann, A bang-bang theorem with bounds on the number of switchings, SIAM J. Control Optim., 17 (1979), 629-651. doi: 10.1137/0317045.
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