User:Saung Tadashi/Brockett's criterion

Brockett's criterion on control theory gives a necessary condition to stabilize a system by continuous time-invariant state feedback. It requires that the mapping (x,u) -> f(x,u) be open at zero...

(see https://books.google.com.br/books?id=-j5rg_0HPksC&lpg=PA165 and https://arxiv.org/pdf/1810.01368.pdf)

Brockett's criterion for continuously differentiable systems
Consider the nonlinear control system described the differential equation

$$\dot{x} = f(x,u), \quad t\ge0, \quad (x,u) \in \mathcal{X} \times \mathcal{U} \subseteq \mathbb{R}^n \times \mathbb{R}^m$$

where f : X x U -> R^n is C1 class and f(0,0) = 0.

If the system is locally asymptotically stabilizable by a stationary C1 feedback law, then it is necessary that f is open at (0,0).

Brockett's criterion for continuous systems
Consider the nonlinear control system described the differential equation

$$\dot{x} = f(x,u), \quad t\ge0, \quad (x,u) \in \mathcal{X} \times \mathcal{U} \subseteq \mathbb{R}^n \times \mathbb{R}^m$$

where f : X x U -> R^n is continuous and f(0,0) = 0.

If the system is locally asymptotically stabilizable by a stationary continuous feedback law, then it is necessary that f is open at (0,0).

Coron's criterion
See.

TODO

 * Add shopping cart figure (see https://ieeexplore.ieee.org/document/4939307)
 * Add region of attraction of Brockett non-holonomic integrator (see 9.4 of https://arxiv.org/pdf/1208.1751.pdf)

Brockett non-holonomic integrator
The Brockett integrator or nonholonomic integrator models a three-wheeled shopping cart

$$\begin{aligned} \dot { x } _ { 1 } & = u _ { 1 } \\ \dot { x } _ { 2 } & = u _ { 2 } \\ \dot { x } _ { 3 } & = u _ { 1 } x _ { 2 } - u _ { 2 } x _ { 1 } \end{aligned}$$

u_1 corresponds to the 'drive' command and u_2 is a steering command.