[Paper Review] Linearization and exponential stability

Hans Zwart|arXiv (Cornell University)|Apr 14, 2014

Control and Stability of Dynamical Systems1 references4 citations

TL;DR

This paper establishes sufficient conditions under which exponential stability of the linearization of a nonlinear infinite-dimensional system implies local exponential stability of the original system. It proves that if the nonlinear term has a zero Fréchet derivative at equilibrium and certain operator conditions on the linear part are satisfied, then local exponential stability holds—highlighting that Gateaux differentiability is insufficient, as shown by a counterexample with finite escape time.

ABSTRACT

We give sufficient conditions such that the exponential stability of the linearization of a non-linear system implies that the non-linear system is (locally) exponentially stable. One of these conditions is that the non-linear system is Fréchet differential at the equilibrium, if it is only Gateaux differentiable, then we show by means of an example that the result does not hold.

Motivation & Objective

• To determine sufficient conditions under which exponential stability of the linearization implies local exponential stability for nonlinear infinite-dimensional systems.
• To clarify the role of Fréchet versus Gateaux differentiability in stability analysis of nonlinear systems in Hilbert spaces.
• To demonstrate via a counterexample that Gateaux differentiability alone is insufficient to guarantee stability, even when linearization is exponentially stable.
• To provide a Lyapunov-based proof framework using bounded, self-adjoint, invertible operators to establish local exponential stability.

Proposed method

• Uses a Lyapunov function $ V(x) = \langle x, P_2 x \rangle $, where $ P_2 = 2P + \frac{1}{\omega}Q $, with $ P $ derived from the exponentially stable semigroup generated by $ A $.
• Imposes condition (2) involving a self-adjoint, boundedly invertible operator $ Q $, ensuring $ \langle x, QAx \rangle + \langle Ax, Qx \rangle \leq \omega \|x\|^2 $.
• Applies the Lyapunov derivative to show $ \dot{V}(x) \leq -\|x\|^2 + 2\|P_2\|\|x\|\|f(x)\| $, leveraging the zero Fréchet derivative of $ f $ at 0.
• Establishes local exponential stability by showing $ \dot{V}(x) \leq -\frac{1}{2M_1}V(x) $ for $ V(x) \leq \delta $, using equivalence of norms via $ P_2 $.
• Constructs a counterexample in $ \ell^2(\mathbb{N}) $ with $ f(x)_n = 3\sqrt[n]{|x_n|}x_n $, showing Gateaux differentiability does not suffice.
• Demonstrates finite escape time and instability for initial states arbitrarily close to zero, despite exponentially stable linearization.

Experimental results

Research questions

• RQ1 Under what conditions does exponential stability of the linearization imply local exponential stability for nonlinear infinite-dimensional systems?
• RQ2 Is Gateaux differentiability of the nonlinear term sufficient to ensure stability when the linearization is exponentially stable?
• RQ3 Can a nonlinear system be unstable at equilibrium even if its linearization is exponentially stable, when only Gateaux differentiability holds?
• RQ4 What role does the Fréchet derivative condition play in ensuring stability via Lyapunov methods in Hilbert spaces?
• RQ5 How can a Lyapunov function be constructed using bounded, self-adjoint, invertible operators to prove local exponential stability?

Key findings

• Exponential stability of the linearization implies local exponential stability of the nonlinear system if the nonlinear term has a zero Fréchet derivative at the origin and condition (2) holds.
• The counterexample shows that Gateaux differentiability with zero derivative is insufficient; the system can be unstable even when linearization is exponentially stable.
• The counterexample system has finite escape time for initial conditions near zero, indicating strong instability despite stable linearization.
• The nonlinear term $ f(x)_n = 3\sqrt[n]{|x_n|}x_n $ is locally Lipschitz but not Fréchet differentiable at zero.
• The origin is unstable because for any $ \delta > 0 $, there exists an initial state with $ \|x(0)\| < \delta $ that diverges due to instability in a single mode.
• The system's right-hand side is uniformly Lipschitz continuous only after modification, but the original example is not uniformly Lipschitz, leading to finite escape time.