Control systems on semi-simple Lie groups and their homogeneous spaces
Annales de l'Institut Fourier, Volume 31 (1981) no. 4, p. 151-179

In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In fact, it appears that there is no semi-algebraic necessary and sufficient condition in the bilinear case and that our criterion is in some sense optimal. This will be discussed in a future paper.

Dans cette publication nous considérons la classe de systèmes à commande qui sont induits par l’action d’un groupe de Lie semi-simple sur une variété, et nous établissons un critère suffisant pour q’un tel système puisse être conduit d’un état initial arbitraire à un état final arbitraire lui aussi, par une commande admissible. La classe des systèmes considérés contient, en particulier, essentiellement tous les systèmes bilinéaires. Notre condition est semi-algébrique mais, contrairement à ce qui se passe pour le célèbre critère de Kalman pour les systèmes linéaires, elles n’est pas nécessaire. Apparemment il n’existe pas de condition nécessaire et suffisante semi-algébrique dans notre cas et notre critère est en quelque sorte optimal. Ceci sera étudié dans un papier à paraître.

@article{AIF_1981__31_4_151_0,
     author = {Jurdjevic, Velimir and Kupka, Ivan},
     title = {Control systems on semi-simple Lie groups and their homogeneous spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {31},
     number = {4},
     year = {1981},
     pages = {151-179},
     doi = {10.5802/aif.853},
     zbl = {0453.93011},
     mrnumber = {84a:93014},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1981__31_4_151_0}
}
Jurdjevic, Velimir; Kupka, Ivan. Control systems on semi-simple Lie groups and their homogeneous spaces. Annales de l'Institut Fourier, Volume 31 (1981) no. 4, pp. 151-179. doi : 10.5802/aif.853. http://www.numdam.org/item/AIF_1981__31_4_151_0/

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