Local Stabilizability of Nonlinear Control Systems

Abstract

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Feedback Control of Spin Systems

“…The aim of a feedback synthesis is to render a desired state or state trajectory an attractor for the system regardless of the initial condition [32], and the main purpose of this work is to investigate in detail this feedback synthesis from a theoretical viewpoint. The main tools we shall use are Lyapunov-based feedback design for bilinear control systems, adapting a method known as Jurjevic-Quinn condition [7,18] to the case at hand. Standard references for basic material on feedback control are e.g.…”

Stable Barrier-Projection and Barrier-Newton Methods for Linear and Nonlinear Programming

“…All trajectory remains in the set Ω. According to La Salle's Invariance Principle [2] the limit set of the solution is a compact connected set contained in Ω and coincides with the equilibrium point x * , which is unique on Ω. Taking the limit as t → ∞, we obtain from (53) and (54) that Ax * = b, G(x * )v * = 0 n , v * ≥ 0 n , x * ≥ 0 n .…”

Control Lyapunov functions for homogeneous “Jurdjevic-Quinn” systems

“…Hence, from (9), G is homogeneous of degree 0 if and When all the vector fields f i of the system (1) are homogeneous, and when a homogeneous control Lyapunov function V is known for this system, the following result, proved in [26] (see also the textbook [3] for the single-input case), gives a feedback control, simpler than Sontag's one (see Prop. 1.3), that assigns V to be a Lyapunov function for the closed loop system.…”

“…For control systems, the "Lyapunov design" of a stabilizing control law (see [3]), based on Lyapunov direct method, consists in designing a positive definite and radially unbounded function together with a continuous feedback law that assigns this function to be a Lyapunov function for the closed-loop system. Artstein's theorem (see [2,3,24]) points out exactly which relations (control Lyapunov function, small control property, see below) a function has to satisfy in order to allow existence of a feedback law that makes it decrease.…”

“…Artstein's theorem (see [2,3,24]) points out exactly which relations (control Lyapunov function, small control property, see below) a function has to satisfy in order to allow existence of a feedback law that makes it decrease. Definition 1.1 (assignable Lyapunov functions).…”