Uniform exponential stabilization of nonlinear systems in Banach spaces

Abstract

This paper presents necessary and sufficient conditions for uniform exponential stabilization of a class of nonlinear systems in Banach state spaces. The stabilization assumptions are formulated in terms of integral estimates involving the control operator and the state of the uncontrolled version of the system at hand. An explicit estimate of the convergence speed is given. Applications to feedback stabilization of affine control systems are given. Illustrative examples are further provided.

Robust stabilization for affine control systems in Banach spaces

“…Then there exists 𝜌 1 > 0 such that the feedback control (18) exponentially stabilizes (1) for any 0 < 𝜌 < 𝜌 1 .…”

“…In this section, we discuss the robustness of the stabilizing control (18) under some classes of reasonable perturbations of the parameters of system (1), namely, the dynamic A and the operator of control B. Here, we will consider perturbations having the same nature as the operator subject to the perturbation.…”

“…[13]) (see also Khlebnikov [16] for the case of finite dimensional). Recently, the question of feedback stabilization of system () in a reflexive Banach state space has been tackled in Benzaza and Ouzahra [17] and the case of general Banach state space (not necessary reflexive) in Ouzahra [18]. The case of unbounded bilinear systems (i.e., B is an unbounded linear operator) has been considered in Ouzahra [19].…”

Feedback stabilization of linear and bilinear unbounded systems in Banach space

“…The problem of feedback stabilization of some classes of linear and nonlinear systems has been investigated in case of bounded and unbounded control operators in [2,3,4,5,6,7,8,9]. Feedback stabilization of the bilinear system (3) has been investigated in the case of a bounded control operator by numerous authors using various control approaches, such as quadratic control laws, sliding mode control, piecewise constant feedback and optimal control laws (see [10,31] and the references therein). Recently, the question of stabilization of bilinear systems with unbounded control operator has been treated in [12,18,19,30].…”

“…In [30], the exponential stabilizability of bilinear systems has been considered for Miyadera's control operator, and the stabilizing control is a switching one which leads to a closed-loop system like (2) evolving in a reflexive state space. More recently, the case of nonreflexive state space was considered in the context of bounded control operator [31]. In this paper, we deal with a wide class of linear/bilinear systems evolving on a nonreflexive state space with unbounded control operators, including control operators of type Weiss-Staffans, Miyadera-Voigt or Desch-Schappacher.…”