June  2013, 3(2): 185-208. doi: 10.3934/mcrf.2013.3.185

Time optimal control for a nonholonomic system with state constraint

1. 

Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France, France

Received  June 2012 Revised  December 2012 Published  March 2013

The aim of this paper is to tackle the time optimal controllability of an $(n+1)$-dimensional nonholonomic integrator. In the optimal control problem we consider, the state variables are subject to a bound constraint. We give a full description of the optimal control and optimal trajectories are explicitly obtained. The optimal trajectories we construct, lie in a 2-dimensional plane and they are composed of arcs of circle.
Citation: Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
References:
[1]

A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry,, Calc. Var. Partial Differential Equations, 43 (2012), 355. doi: 10.1007/s00526-011-0414-y. Google Scholar

[2]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 (2004). Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar

[4]

R. Beals, B. Gaveau and P. C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups,, J. Math. Pures Appl. (9), 79 (2000), 633. doi: 10.1016/S0021-7824(00)00169-0. Google Scholar

[5]

A. M. Bloch, Nonholonomic mechanics and control,, With the collaboration of J. Baillieul, 24 (2003). doi: 10.1007/b97376. Google Scholar

[6]

J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM J. Control Optim., 46 (2007), 1398. doi: 10.1137/06065756X. Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry,, in, (1982), 11. Google Scholar

[8]

A. E. Bryson, Jr. and Y. C. Ho, Applied optimal control. Optimization, estimation, and control,, Revised printing, (1975). 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]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007). Google Scholar

[12]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar

[13]

R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Rev., 37 (1995), 181. doi: 10.1137/1037043. Google Scholar

[14]

R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985). Google Scholar

[15]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", Studies in Mathematics and its Applications, 6 (1979). Google Scholar

[16]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers,, Acta Applicandae Mathematicae, 123 (2013), 175. doi: 10.1007/s10440-012-9760-9. Google Scholar

[17]

C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback,, Math. Control Signals Systems, 17 (2005), 201. doi: 10.1007/s00498-005-0152-9. Google Scholar

[18]

A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number,, J. Fluid. Mech., 198 (1989), 587. doi: 10.1017/S0022112089000261. Google Scholar

show all references

References:
[1]

A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry,, Calc. Var. Partial Differential Equations, 43 (2012), 355. doi: 10.1007/s00526-011-0414-y. Google Scholar

[2]

A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint,", Encyclopaedia of Mathematical Sciences, 87 (2004). Google Scholar

[3]

L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Mathematical Monographs, (2000). Google Scholar

[4]

R. Beals, B. Gaveau and P. C. Greiner, Hamilton-Jacobi theory and the heat kernel on Heisenberg groups,, J. Math. Pures Appl. (9), 79 (2000), 633. doi: 10.1016/S0021-7824(00)00169-0. Google Scholar

[5]

A. M. Bloch, Nonholonomic mechanics and control,, With the collaboration of J. Baillieul, 24 (2003). doi: 10.1007/b97376. Google Scholar

[6]

J. F. Bonnans and A. Hermant, Well-posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control,, SIAM J. Control Optim., 46 (2007), 1398. doi: 10.1137/06065756X. Google Scholar

[7]

R. W. Brockett, Control theory and singular Riemannian geometry,, in, (1982), 11. Google Scholar

[8]

A. E. Bryson, Jr. and Y. C. Ho, Applied optimal control. Optimization, estimation, and control,, Revised printing, (1975). 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]

J.-M. Coron, "Control and Nonlinearity,", Mathematical Surveys and Monographs, 136 (2007). Google Scholar

[12]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992). Google Scholar

[13]

R. F. Hartl, S. P. Sethi and R. G. Vickson, A survey of the maximum principles for optimal control problems with state constraints,, SIAM Rev., 37 (1995), 181. doi: 10.1137/1037043. Google Scholar

[14]

R. A. Horn and C. R. Johnson, "Matrix Analysis,", Cambridge University Press, (1985). Google Scholar

[15]

A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems,", Studies in Mathematics and its Applications, 6 (1979). Google Scholar

[16]

J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers,, Acta Applicandae Mathematicae, 123 (2013), 175. doi: 10.1007/s10440-012-9760-9. Google Scholar

[17]

C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback,, Math. Control Signals Systems, 17 (2005), 201. doi: 10.1007/s00498-005-0152-9. Google Scholar

[18]

A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number,, J. Fluid. Mech., 198 (1989), 587. doi: 10.1017/S0022112089000261. Google Scholar

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