It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and continuously differentiable stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.
|Original language||English (US)|
|Number of pages||17|
|Journal||Discrete and Continuous Dynamical Systems - Series S|
|State||Published - Dec 2018|
Bibliographical noteFunding Information:
Acknowledgments. The work presented in this paper was carried out while the first author was a postdoctoral fellow at the Institute for Mathematics and its Applications (IMA) during the IMA’s annual program on Control Theory and its Applications. Farhad Jafari also gratefully acknowledges the support and hospitality provided by IMA, where this work was initiated and where he was a visiting professor during the IMA’s annual program on Control Theory and its Applications. The research of Boris Mordukhovich was partly supported by the US National Science Foundation under grant DMS-1512846, by the US Air Force Office of Scientific Research under grant 15RT0462, and by the RUDN University Program 5-100. This research was completed during Mordukhovich’s stay at the University of Padova (February-March 2017), and he gratefully acknowledges fruitful discussions with and the warm hospitality provided by Prof. Giovanni Colombo.
- Brockett's theorem
- Feedback stabilization
- Linear openness
- Metric regularity
- Nonlinear control systems