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

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