DCDS-S
Linear openness and feedback stabilization of nonlinear control systems
Rohit Gupta Farhad Jafari Robert J. Kipka Boris S. Mordukhovich
Discrete & Continuous Dynamical Systems - S 2018, 11(6): 1103-1119 doi: 10.3934/dcdss.2018063

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.

keywords: Brockett's theorem feedback stabilization linear openness metric regularity nonlinear control systems
JGM
Neighboring extremal optimal control for mechanical systems on Riemannian manifolds
Anthony M. Bloch Rohit Gupta Ilya V. Kolmanovsky
Journal of Geometric Mechanics 2016, 8(3): 257-272 doi: 10.3934/jgm.2016007
In this paper, we extend neighboring extremal optimal control, which is well established for optimal control problems defined on a Euclidean space (see, e.g., [8]) to the setting of Riemannian manifolds. We further specialize the results to the case of Lie groups. An example along with simulation results is presented.
keywords: optimal control Lie groups neighboring extremal optimal control mechanical systems Riemannian manifolds.

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