On the stabilization of quadratic nonlinear systems

Continuous feedback stabilization for a class of affine stochastic nonlinear systems

“…if a v (x) = 0 and Here, we follow the main lines of the proof of Theorem 3.9 from Maniar et al [21]. From the continuity of ϕ, ψ, a v and ∆ v , it is obvious that K is continuous on the open sets:…”

“…The aim of this paper is to relax this restriction, allowing the coefficients of the system (1) as well as the constructed feedback to be only continuous. Inspired by the deterministic case Maniar et al [21] and based on a known SCLF, we first give a necessary condition for the stabilization in probability by a continuous feedback. After that, we present a sufficient stabilization condition which improve the stabilizability condition (H).…”

“…according to the sign of ∆ v (x). To do this, we again follow the main lines of the proof of Theorem 3.9 from Maniar et al [21].…”

Optimality and Passivity of Input-Quadratic Nonlinear Systems

“…By recalling now the decompositions in (8) and (9), it is straightforward to show that equation (16) reduces to (13).…”

“…In particular, most of the proposed results hinge upon the notion and the use of Control Lyapunov functions (CLFs) for the plant, thus partially extending the seminal work of [1] in the general case and [19] in the context of affine nonlinear systems. Specifically, the problem of stabilizing input-quadratic systems via CLFs has been considered in [13], [25] and, in the general case of non-affine models, in [9], in [10] and in [14] by relying on the assumption of Lyapunov stability of the unforced system and on passivity theory of non-affine systems. Moreover, [15] provided a sufficient condition for the existence of a (piece-wise) continuous stabilizing control input for non-affine systems in terms of certain convexity properties of the underlying CLF.…”