# American Institute of Mathematical Sciences

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 and 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), 355-388. doi: 10.1007/s00526-011-0414-y. [2] A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Encyclopaedia of Mathematical Sciences, 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004. [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [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), 633-689. doi: 10.1016/S0021-7824(00)00169-0. [5] A. M. Bloch, Nonholonomic mechanics and control, With the collaboration of J. Baillieul, P. Crouch and J. Marsden, With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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), 1398-1430. doi: 10.1137/06065756X. [7] R. W. Brockett, Control theory and singular Riemannian geometry, in "New Directions in Applied Mathematics" (Cleveland, Ohio, 1980), Springer, New York-Berlin, (1982), 11-27. [8] A. E. Bryson, Jr. and Y. C. Ho, Applied optimal control. Optimization, estimation, and control, Revised printing, Hemisphere Publishing Corp. Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975. [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, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309. [11] J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. [12] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [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), 181-218. doi: 10.1137/1037043. [14] R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [15] A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems," Studies in Mathematics and its Applications, 6, North-Holland Publishing Co., Amsterdam-New York, 1979. [16] J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers, Acta Applicandae Mathematicae, 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9. [17] C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback, Math. Control Signals Systems, 17 (2005), 201-216. doi: 10.1007/s00498-005-0152-9. [18] A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, J. Fluid. Mech., 198 (1989), 587-599. doi: 10.1017/S0022112089000261.

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##### References:
 [1] A. Agrachev, D. Barilari and U. Boscain, On the Hausdorff volume in sub-Riemannian geometry, Calc. Var. Partial Differential Equations, 43 (2012), 355-388. doi: 10.1007/s00526-011-0414-y. [2] A. A. Agrachev and Y. L. Sachkov, "Control Theory from the Geometric Viewpoint," Encyclopaedia of Mathematical Sciences, 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004. [3] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. [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), 633-689. doi: 10.1016/S0021-7824(00)00169-0. [5] A. M. Bloch, Nonholonomic mechanics and control, With the collaboration of J. Baillieul, P. Crouch and J. Marsden, With scientific input from P. S. Krishnaprasad, R. M. Murray and D. Zenkov, Interdisciplinary Applied Mathematics, 24, Systems and Control, Springer-Verlag, New York, 2003. doi: 10.1007/b97376. [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), 1398-1430. doi: 10.1137/06065756X. [7] R. W. Brockett, Control theory and singular Riemannian geometry, in "New Directions in Applied Mathematics" (Cleveland, Ohio, 1980), Springer, New York-Berlin, (1982), 11-27. [8] A. E. Bryson, Jr. and Y. C. Ho, Applied optimal control. Optimization, estimation, and control, Revised printing, Hemisphere Publishing Corp. Washington, D. C., distributed by Halsted Press [John Wiley & Sons], New York-London-Sydney, 1975. [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, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309. [11] J.-M. Coron, "Control and Nonlinearity," Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. [12] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. [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), 181-218. doi: 10.1137/1037043. [14] R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, Cambridge, 1985. [15] A. D. Ioffe and V. M. Tihomirov, "Theory of Extremal Problems," Studies in Mathematics and its Applications, 6, North-Holland Publishing Co., Amsterdam-New York, 1979. [16] J. Lohéac, J.-F. Scheid and M. Tucsnak, Controllability and time optimal control for low Reynolds numbers swimmers, Acta Applicandae Mathematicae, 123 (2013), 175-200. doi: 10.1007/s10440-012-9760-9. [17] C. Prieur and E. Trélat, Robust optimal stabilization of the Brockett integrator via a hybrid feedback, Math. Control Signals Systems, 17 (2005), 201-216. doi: 10.1007/s00498-005-0152-9. [18] A. Shapere and F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, J. Fluid. Mech., 198 (1989), 587-599. doi: 10.1017/S0022112089000261.
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