We consider the minimum-time problem for a multi-input control-affine system, where we assume that the controlled vector fields generate a non-involutive distribution of constant dimension, and where we do not assume a priori bounds for the controls. We use Hamiltonian methods to prove that the coercivity of a suitable second variation associated to a Pontryagin singular arc is sufficient to prove its strong-local optimality. We provide an application of the result to a generalization of Dubins problem.
DOI : 10.1051/cocv/2015026
Mots-clés : Control-affine systems, singular extremals, minimum-time problem, sufficient optimality conditions, second variation, Hamiltonian methods
@article{COCV_2016__22_3_786_0,
author = {Chittaro, Francesca and Stefani, Gianna},
title = {Minimum-time strong optimality of a singular arc: {The} multi-input non involutive case},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {786--810},
publisher = {EDP-Sciences},
volume = {22},
number = {3},
year = {2016},
doi = {10.1051/cocv/2015026},
zbl = {1344.49034},
mrnumber = {3527944},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2015026/}
}
TY - JOUR AU - Chittaro, Francesca AU - Stefani, Gianna TI - Minimum-time strong optimality of a singular arc: The multi-input non involutive case JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 786 EP - 810 VL - 22 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015026/ DO - 10.1051/cocv/2015026 LA - en ID - COCV_2016__22_3_786_0 ER -
%0 Journal Article %A Chittaro, Francesca %A Stefani, Gianna %T Minimum-time strong optimality of a singular arc: The multi-input non involutive case %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 786-810 %V 22 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015026/ %R 10.1051/cocv/2015026 %G en %F COCV_2016__22_3_786_0
Chittaro, Francesca; Stefani, Gianna. Minimum-time strong optimality of a singular arc: The multi-input non involutive case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 786-810. doi : 10.1051/cocv/2015026. http://archive.numdam.org/articles/10.1051/cocv/2015026/
A.A. Agrachev and Yu. L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag (2004). | MR | Zbl
, and , An invariant second variation in optimal control. Int. J. Control 71 (1998) 689–715. | DOI | MR | Zbl
, , and , Quadratic order conditions for bang-singular extremals. AIMS J. Numer. Algebra Control Optim. 2 (2012) 511–546. | DOI | MR | Zbl
and , On the relationship between global and local controllability. Math. Syst. Theory 16 (1983) 79–91. | DOI | MR | Zbl
R.M. Bianchini, Good needle-like variations. In vol. 64 of Proceedings of Symposia in Pure Mathematics (1999). | MR | Zbl
R.M. Bianchini, Variational cones and high-order maximum principles. Technical report, Dipartimento di Matematica “Ulisse Dini”, viale Morgagni 67/a, Firenze (1994).
R.M. Bianchini, Variational Approach to Some Optimization Control Problems. In Geometry in Nonlinear Control and Differential Inclusions. Edited by G. Ferreyra R. Gardner H. Hermes and H. Sussmann (1995). | Zbl
R.M. Bianchini and G. Stefani, A High Order Maximum Principle, in Analysis and Control of Linear Systems. Edited by R.E. Saeks C.I. Byrnes, C.F. Martin (1988). | Zbl
and , Impulsive control systems with commutative vector fields. J. Optim. Theory Appl. 71 (1991) 67–84. | DOI | MR | Zbl
and , Impulsive control systems without commutativity assumptions. J. Optim. Theory Appl. 81 (1994) 435–457. | DOI | MR | Zbl
and , Singular extremals in multi-input time-optimal problem: a sufficient condition. Control Cyber. 39 (2010). | MR | Zbl
F.C. Chittaro and G. Stefani, Minimum-Time Strong Optimality of a Singular Arc: extended Dubins Problem. In 52nd IEEE Conference on Decision and Control (2013).
B.D. Craven, Control and Optimization. Chapman & Hall Mathematics Series. Chapman & Hall, London, New York (1995). | MR | Zbl
. Quadratic condition for a weak minimum for singular regimes in optimal control problems. Soviet Math Dokl. 18 (1977). | Zbl
, Jacobi type conditions for singular extremals. Control Cybernet. 37 (2008) 285–306. | MR | Zbl
and , High order necessary conditions for optimality. SIAM J. Control 10 (1972) 127–168. | DOI | MR | Zbl
, The second variation for singular Bolza problems. SIAM J. Control Optim. 4 (1966) 309–325. | DOI | MR | Zbl
and , Fréchet generalized trajectories and minimizers for variational problems of low coercivity. J. Dyn. Contr. Syst. 21 (2015) 351–377. | DOI | MR | Zbl
, Application of the theory of quadratic forms in Hilbert spaces to the calculus of variations. Pac. J. Math. 1 (1951) 525–581. | DOI | MR | Zbl
V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997). | MR | Zbl
V. Jurdjevic and F. Monroy-Pérez, Variational Problems on Lie Groups and Their Homogeneous Spaces: Elastic Curves, Tops, and Constrained Geodesic Problems (2002). | MR | Zbl
, A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control problems. SIAM J. Control Optim. 12 (1974) 43–52. | DOI | MR | Zbl
J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Texts Appl. Math. Springer (1999). | MR | Zbl
E.J. McShane, Unified Integration. Academic Press (1983). | MR | Zbl
and , Bang-singular-bang extremals: sufficient optimality conditions. J. Dyn. Control Syst. 17 (2011) 469–514. | DOI | MR | Zbl
G. Stefani, Minimum-time optimality of a singular arc: second order sufficient conditions. In vol. 1 of 43rd IEEE Conference on Decision and Control (2004).
G. Stefani, Strong Optimality of Singular Trajectories. Geometric Control and nonsmooth analysis. Edited by F. Ancona A. Bressan P. Cannarsa F. Clarke and P. R. Wolenski (2008). | MR | Zbl
and . Constrained regular LQ-control problems. SIAM J. Control Optim. 35 (1997) 876–900. | DOI | MR | Zbl
Cité par Sources :
