Geodesic
Microlocal normal forms for regular fully nonlinear two-dimensional control systems
Informatics and Automation, Differential equations and dynamical systems, Tome 270 (2010), pp. 243-248.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper we deal with fully nonlinear two-dimensional smooth control systems with scalar input $\dot q=\mathbf f(q,u)$, $q\in M$, $u\in U$, where $M$ and $U$ are differentiable smooth manifolds of respective dimensions two and one. For such systems, we provide two microlocal normal forms, i.e., local in the state-input space, using the fundamental necessary condition of optimality for optimal control problems: the Pontryagin maximum principle. One of these normal forms will be constructed around a regular extremal, and the other one will be constructed around an abnormal extremal. These normal forms, which in both cases are parametrized only by one scalar function of three variables, lead to a nice expression for the control curvature of the system. This expression shows that the control curvature, a priori defined for normal extremals, can be smoothly extended to abnormals. 
@article{TRSPY_2010_270_a18,
     author = {Ulysse Serres},
     title = {Microlocal normal forms for regular fully nonlinear two-dimensional control systems},
     journal = {Informatics and Automation},
     pages = {243--248},
     publisher = {mathdoc},
     volume = {270},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a18/}
}
TY  - JOUR
AU  - Ulysse Serres
TI  - Microlocal normal forms for regular fully nonlinear two-dimensional control systems
JO  - Informatics and Automation
PY  - 2010
SP  - 243
EP  - 248
VL  - 270
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a18/
LA  - en
ID  - TRSPY_2010_270_a18
ER  - 
%0 Journal Article
%A Ulysse Serres
%T Microlocal normal forms for regular fully nonlinear two-dimensional control systems
%J Informatics and Automation
%D 2010
%P 243-248
%V 270
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a18/
%G en
%F TRSPY_2010_270_a18
Ulysse Serres. Microlocal normal forms for regular fully nonlinear two-dimensional control systems. Informatics and Automation, Differential equations and dynamical systems, Tome 270 (2010), pp. 243-248. http://geodesic.mathdoc.fr/item/TRSPY_2010_270_a18/

[1] Agrachev A., Zelenko I., “On feedback classification of control-affine systems with one- and two-dimensional inputs”, SIAM J. Control and Optim., 46:4 (2007), 1431–1460 | DOI | MR | Zbl

[2] Agrachev A.A., “Feedback-invariant optimal control theory and differential geometry. II: Jacobi curves for singular extremals”, J. Dyn. and Control Syst., 4:4 (1998), 583–604 | DOI | MR | Zbl

[3] Agrachev A.A., Sachkov Yu.L., Control theory from the geometric viewpoint, Encycl. Math. Sci., 87, Control theory and optimization. II, Springer, Berlin, 2004 | MR | Zbl

[4] Brunovský P., “A classification of linear controllable systems”, Kybernetika (Prague), 6 (1970), 173–188 | MR | Zbl

[5] Gantmacher F.R., The theory of matrices, v. 1, 2, Chelsea Publ., New York, 1959 | MR

[6] Jakubczyk B., “Equivalence and invariants of nonlinear control systems”, Nonlinear controllability and optimal control, Pure and Appl. Math., 133, M. Dekker, New York, 1990, 177–218 | MR | Zbl

[7] Jakubczyk B., “Critical Hamiltonians and feedback invariants”, Geometry of feedback and optimal control, Pure and Appl. Math., 207, M. Dekker, New York, 1998, 219–256 | MR | Zbl

[8] Jakubczyk B., Respondek W., “Feedback classification of analytic control systems in the plane”, Analysis of controlled dynamical systems, Proc. conf. Lyon, 1990, Progr. Syst. Control Theory, 8, Birkhäuser, Boston (MA), 1991, 263–273 | MR | Zbl

[9] Kronecker L., “Algebraische Reduction der Schaaren bilinearer Formen”, Sitzungsber. Preuss. Akad. Wiss. Berlin., 1890, 1225–1237 | Zbl

[10] Pomet J.-B., Kupka I., “Global aspects of feedback equivalence for a parametrized family of systems”, Analysis of controlled dynamical systems, Proc. conf. Lyon, 1990, Progr. Syst. Control Theory, 8, Birkhäuser, Boston (MA), 1991, 337–346 | MR | Zbl

[11] Pomet J.-B., Kupka I.A.K., “On feedback equivalence of a parameterized family of nonlinear systems”, SIAM J. Control and Optim., 33:4 (1995), 1170–1207 | DOI | MR | Zbl

[12] Respondek W., “Feedback classification of nonlinear control systems on $\mathbb R^2$ and $\mathbb R^3$”, Geometry of feedback and optimal control, Pure and Appl. Math., 207, M. Dekker, New York, 1998, 347–381 | MR | Zbl