Sufficient lyapunov-like conditions for stabilization

Stabilization of homogeneous polynomial systems in the plane

“…In [1] it was shown that if the above condition is fulfilled, then the system (1) is stabilizable at the origin by means of a nonlinear feedback law which is smooth for x = 0. The same result was proved independently in [10,11] and [12], where the corresponding stabilizing feedback laws are more explicitly identified. One of the interesting topics in control theory is the stabilization of homogeneous systems affine in control.…”

“…where the state x ∈ R n , the input u ∈ R, f (0) = 0, and f, g are smooth vector fields, the basic stabilization Lyapunov condition provided in [1,10,11] and [12] can be expressed as follows. There exists a positive definite real function V : R n −→ R (i. e., V (0) = 0 and V (x) > 0 for x = 0 near zero) such that for any x = 0 near zero with ∇V.g(x) = 0 it holds ∇V.f (x) < 0.…”

“…In order to make use of this approximation property in the design of locally stabilizing feedbacks for nonlinear systems, the main idea lies in the construction of homogeneous feedback laws that preserve the homogeneity of the resulting closed loop system. These laws can be shown to be locally stabilizing the approximated nonlinear system ( [7,10,11] and [12]). It was shown in [8] that for general controllable homogeneous systems, the existence of a stabilizing feedback does not necessarily imply the existence of a homogeneous stabilizing feedback.…”

Necessary and sufficient Lyapunov-like conditions for robust nonlinear stabilization

“…(iii) The existing feedback design methodology based on partition of unity arguments (see. e.g., [2,33]) does not work because the state space X is infinite-dimensional. (iv ) The feedback design methodology based on Michael's Theorem (see, e.g., [7]) does not work either because simple continuity of the feedback does not suffice or because the hypotheses of Michael's Theorem cannot be verified.…”

Sufficient lie algebraic conditions for sampled-data feedback stabilization

“…In the recent works [22] and [23], the concept of Weak Global Asymptotic Stabilization by Sampled-Data Feedback (SDF-WGAS) is introduced for autonomous systems: This condition constitutes an extension of the well-known "Artstein-Sontag" sufficient condition for asymptotic stabilization of systems (1.2) by means of an almost smooth feedback; (see [3], [19] and [21]). Throughout the paper we adopt the following notations.…”