## Local and global controllability for nonlinear systems.(English)Zbl 0552.93009

Several results concerning local and global controllability of nonlinear systems of the form $$\dot x=f(x)+ug(x)$$, $$x\in {\mathbb{R}}^ n$$, with unbounded control functions are presented. More precisely, the author states two conditions for global controllability in the case $$g(x)=b=const.$$, $$n=2$$. Furthermore, the Hermes-Sussmann condition for local controllability is rederived in the case $$n=2$$. Finally, the global controllability conditions are extended to the case $$g(x)=b=const.$$, n arbitrary. The method is based on the use of sequences of control functions which approximate Dirac impulses.
Reviewer: A.Bacciotti

### MSC:

 93B05 Controllability 93B03 Attainable sets, reachability 93C10 Nonlinear systems in control theory 93C15 Control/observation systems governed by ordinary differential equations
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### References:

 [1] D. Aeyels, Global controllability for nonlinear systems: an extension of the Kalman criterion, forthcoming. [2] Hirsch, M.W.; Smale, S., Differential equations, dynamical systems and linear algebra, (), 305 [3] Hirschorn, R., Global controllability of nonlinear systems, SIAM J. control optim., 14, 700-711, (1976) · Zbl 0341.93006 [4] Sussmann, H.J., Lie brackets and local controllability: a sufficient condition for scalar-input systems, SIAM J. control optim., 21, 5, 686-713, (1983) · Zbl 0523.49026 [5] Hermes, H., Control systems which generate decomposable Lie algebras, J. differential equations, 44, 2, 166-187, (1982) · Zbl 0496.49021 [6] Hunt, L.R., Global controllability of general nonlinear systems, Math. systems theory, 12, 361-370, (1989) · Zbl 0394.93010 [7] Lobry, C., Controlabilité des systèmes non linéaires, Outils et modèles mathématiques pour l’automatique, l’analyse de systèmes et LES traitement du signal, Vol. 1, 187-214, (1981) · Zbl 0476.93015 [8] Kunita, H., On the controllability of nonlinear systems with applications to polynomial systems, Appl. math. optim., 5, 89-99, (1979) · Zbl 0406.93011 [9] Jurdjevic, V., Polynomial control systems, (), 904-906 · Zbl 0746.34028 [10] D. Aeyels, Global controllability for smooth nonlinear systems: a geometric approach. SIAM J. Control Optim. (to appear). · Zbl 0567.93007 [11] Aeyels, D., Controllability for polynomial systems, (), 542-545 [12] Kendig, K., Elementary algebraic geometry, () · Zbl 0364.14001
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