Controllability of two-dimensional bilinear systems

• Braga Barros, Carlos José ^[1] ; Gonçalves Filho, Joao Ribeiro ^[1] ; Do Rocío, Osvaldo Germano ^[1] ; San Martín, Luiz A. B. ^[2]
  1. [1] Universidade Estadual de Maringá

     Universidade Estadual de Maringá

     Brasil

     
  2. [2] Universidade Estadual de Campinas

     Universidade Estadual de Campinas

     Brasil

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 15, Nº. 2, 1996, págs. 111-139
• Idioma: inglés
• DOI: 10.22199/S07160917.1996.0002.00002
• Enlaces
  • Texto completo
• Resumen

  • For bilincar control systems x = Ax + uBx, x ∊ R2 , A and B 2 x 2 matrices, necessary and sufficient conditions are given for the controllability on R2 -{0}. The method is through Lie theory, and follows the program outlined by this theory which consists in finding first the connected subgroups of the group Gl(2) of all invertible matrices which are transitive on R2 - {0}, and then look at the subsemigroups of these subgroups which are transitive. A detailed and nearly self contained exposition of the determination of the transitive subgroups is presented. It turns out that they are Gl+(2), Sl(2) and the commutative group of nonzero complex numbers. Controllability is analysed by considering these groups separately. In the case of Sl (2) the controllability is decided with the aid of a result of [15] about semigroups in semi-simple Lie groups. A self contained proof specific for Sl(2) is presented. This case by case analysis recovers the necessary and sufficient conditions given by Lepe and Joó and Tuan (see [10]).

• Referencias bibliográficas
  • Citas [1] Ayala B., V.: Controllability of nilpotent systems. Geometry of nonlinear control and differential inclusions. Banach Center Publications...
  • [2] Boothby, W.: A transitive problem from control theory . .Journal of Differential Equations, 17, pp. 296- 307, (1975).
  • [3] Boothby, W. and Wilson E.N.: Determination of transitivity of bilinear systems. SIAM Journal on Control and Optimization, 17, pp. 212-221,...
  • [4] El Assoudi, R. and J.P. Gauthier: Controllability of right invariant systems on real simple Lie groups of type F4 , G2 , Cn and Bn. Math....
  • [5] Gauthier, .J.P., l. Kupka and G. Sallet: Controllability of right invariant systems on real simple Lie groups. Systems & Control Letters,...
  • [6] Hilgert, .J., K.H. Hofmann and J. Lawson: Controllability of systems on a nilpotent Lie group. Beiträge Algebra Geometrie, 30, pp. 185-190,...
  • [7] Hilgert, J., K.H. Hofmann and J. Lawson: Lie groups, convex cones and semigroups. Oxford University Press ( 1989).
  • [8] Hilgert, .J. and K.-H. Neeb: Lie semigroups and their applications. Lecture Notes in Mathematics, 1552 Springer Verlag ( 1993).
  • [9] Joó, l. and N .M. Tuan: On controllability of bilinear systems I (controllability in finite dimensions). Ann. Univ. Sci. Budapest, 35,...
  • [10] - On controllability of bilinear systems II ( controllability in two dimensions). Ann. Univ. Sci. Budapest, 35, pp. 217-265, (1992).
  • [11] .Jurdjevic, V. and H.J. Sussmann: Control systems on Lie groups. J. of Diff. Eq., 12, pp. 313- 329, ( 1972).
  • [12] .Jurdjevic, V. and l. Kupka: Control systems subordinated to a group action: Accessibility . .J. Diff. Eq., 39, pp. 180- 211, ( 1981 ).
  • [13] - Control systems on semisimple Lie groups and their homogeneous spaces. Ann. Inst. Fourier (Grenoble), 31, pp. 1.51- 179, (1981).
  • [14] Lobry, L.: Controllability of nonlinear systems on compact manifolds. SIAM .J. on Control and Optim., 12, pp. 1-4, (1974).
  • [15] San Martín, L.: lnvariant control sets on flag manifolds. Math. Of Control, Signals and Systems, 6, pp. 41- 61, (1993).
  • [16] - Controllability of families of measure preserving vector fields. Systems & Control Letters, 8, pp. 459- 462, (1987).
  • [17] - On global controllability of discrete-time control systems. Math. of Control, Signals and Systems, 8, pp. 279-297, (1995)
  • [18] San Martín, L.A.B. and P.A. Tonelli: Semigroup actions on homogeneous spaces. Semigroup Forum, 50, pp. 59- 88, ( 1995) .
  • [19] Silva Leite, F. and P.E. Crouch: Controllability on classical Lie groups. Math. of Control, Signals and Systems, 1, pp. 31- 42, (1988).
  • [20] Sontag, E.D.: Mathematical control theory. Springer Verlag (1990).