Abstract
In this paper, we consider the feedback stabilization of linear systems in a Hilbert state space. The paper proposes a class of nonlinear controls that guarantee exponential sta- bility for linear systems. Applications to stabilization with saturating controls are provided. Also the robustness of constrained stabilizing controls is analyzed.
Highlights
IntroductionWhere the state space is a Hilbert H with inner product ·, · and corresponding norm . , the Hilbert space U with norm · U is the space of control and u(t) ∈ U is a control subject to the constraint u(t) U ≤ umax, umax > 0
In this paper, we consider the following linear system : dz(t) dt = Az(t) +Bu(t), z(0) = z0, (1)where the state space is a Hilbert H with inner product ·, · and corresponding norm . , the Hilbert space U with norm · U is the space of control and u(t) ∈ U is a control subject to the constraint u(t) U ≤ umax, umax > 0
Where the state space is a Hilbert H with inner product ·, · and corresponding norm . , the Hilbert space U with norm · U is the space of control and u(t) ∈ U is a control subject to the constraint u(t) U ≤ umax, umax > 0
Summary
Where the state space is a Hilbert H with inner product ·, · and corresponding norm . , the Hilbert space U with norm · U is the space of control and u(t) ∈ U is a control subject to the constraint u(t) U ≤ umax, umax > 0. The radial projection onto the unit ball enables us to define the following bounded control : u1(t) =. This control guarantees weak and strong stabilization for a class of linear systems under the approximate controllability assumption : B∗S(t)y = 0, ∀t ≥ 0 ⇒ y = 0 (see [13, 14]). The purpose of this paper is to give necessary and sufficient conditions for exponential stabilization of an autonomous nonlinear systems. The plan of the paper is as follows : in the second section, we give necessary and sufficient conditions for exponential stability of an autonomous nonlinear system. The third section is devoted to problems of stabilization of the linear system (1) using bounded controls. An illustrating example is given in the fifth section
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