## Summary

A rest solution, p, of a control system is said to possess controlled stability if for each t_{
1>0} a full nbd. of points of p can be steered to p in time t_{
1
} by solutions of the system. Equivalently, if the system acts in a linear space and p is the origin, this states the domain of « null controllability » is open. We consider systems of the form\(X\left( x \right) + \sum\limits_{i = 2}^m {u_i } \left( t \right)Y^i \left( x \right)\) with either each u_{i} measurable with |u_{i}(t)| ⩽*1*, or |u_{i}| ≤*1* inL
_{
1
} or ‖u_{i}‖ ≤*1* in the space of regular countably additive measures, and X, Y^{2}, …, Y^{m} analytic vector fields on an analytic n-manifold. If X(p)=*0* a necessary condition that the rest solution p possess controlled stability is that the dimension, at p, of the Lie algebra generated by X, Y^{2}, …, Y^{m} be n. First order sufficient conditions are well known and can be stated in terms of subsets of the elements of this Lie algebra. This paper provides two higher order sufficiency tests, together with examples of their applications.

## References

- [1]
H. J. Sussmann -V. Jurdjevic,

*Controllability of nonlinear systems*, J. Diff. Eqs.,**12**(1972), pp. 95–116. - [2]
T. Nagano,

*Linear differential systems with singularities and an application of transitive Lie algebras*, J. Math. Soc. Japan,**18**(1966), pp. 398–404. - [3]
A. J. Krener,

*A generalization of Chow's theorem and the Bang-Bang theorem to non-linear control problems*, SIAM J. Control,**12**(1974), pp. 43–52. - [4]
H. Hermes,

*On local and global controllability*, SIAM J. Control,**12**(1974), pp. 252–261. - [5]
H. J. Sussmann -V. Jurdjevic,

*Control systems on Lie groups*, J. Diff. Eqs.,**12**(1972), pp. 313–329. - [6]
R. M. Hirschorn,

*Global controllability of nonlinear systems*(to appear, SIAM J. Control). - [7]
H. Hermes,

*Local controllability and sufficient conditions in singular problems - II*(to appear, SIAM J. Control). - [8]
H. Hermes,

*High order algebraic conditions for controllability*, Proceedings Algebraic Methods in Systems Theory, Udine, Italy. - [9]
H. Hermes,

*High order controlled stability and controllability*, Proceedings International Symposium on Dynamical Systems, Univ. Florida, Gainesville, 1976. - [10]
H. Hermes,

*Local controllability and sufficient conditions in singular problems - I*, J. Diff. Eqs.,**20**(1976), pp. 213–232. - [11]
A. J. Krener,

*The high order maximal principle and its applications to singular extremals*(to appear, SIAM J. Control). - [12]
H. Knobloch,

*Higher order approximations of attainable sets (to appear)*, Proceedings of the International Symposium on Dynamical Systems, Univ. Florida, Gainesville, 1976.

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## Additional information

Entrata in Redazione il 14 giugno 1976.

This research was supported by the National Science Foundation under grant MCS76-04419.

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### Cite this article

Hermes, H. Controlled stability.
*Annali di Matematica* **114, **103 (1977). https://doi.org/10.1007/BF02413781

### Keywords

- Control System
- Vector Field
- Linear Space
- Additive Measure
- Analytic Vector