[Paper Review] On the controllability of some steady states in the case of nonlinear discrete dynamical systems with control

Éva Kaslik, A.M. Balint+2 more|ArXiv.org|Nov 11, 2003

Chaos control and synchronization4 references3 citations

TL;DR

This paper establishes that in nonlinear discrete dynamical systems with control, two asymptotically stable steady states connected by an analytic path can be gradually transferred into one another through a sequence of successful control parameter maneuvers. The key result relies on the analyticity of the Lyapunov function's domain of attraction, ensuring that transitions are feasible as long as each intermediate state lies within the domain of attraction of the next.

ABSTRACT

The main objective of this paper is to show that two asymptotically stable steady states which belong to an analytic path of asymptotically stable steady states can be gradually transferred one to the other by successive changes of the control parameters.

Motivation & Objective

• To investigate the controllability of asymptotically stable steady states in nonlinear discrete dynamical systems with control parameters.
• To extend results from continuous systems to discrete-time systems by leveraging analyticity and Lyapunov function theory.
• To establish conditions under which a sequence of control parameter changes can transfer the system from one stable steady state to another.
• To characterize the domain of attraction of asymptotically stable fixed points using the natural domain of analyticity of a Lyapunov function.
• To demonstrate that successful maneuvers are possible if and only if the initial state lies within the domain of attraction of the target steady state.

Proposed method

• Utilizes an analytic path of steady states, defined as a function φ: D₁ ⊂ D → Ω satisfying φ(α) = f(φ(α), α) for all α ∈ D₁.
• Applies a transformation y_k = x_k - φ(α) to shift the steady state to the origin, enabling analysis of stability around y=0.
• Relies on a theorem from [4] stating that the domain of attraction of an asymptotically stable fixed point coincides with the natural domain of analyticity of a solution V to the functional equation V(g(y)) - V(y) = -||y||² with V(0)=0.
• Establishes that the maneuver α′ → α″ is successful if and only if φ(α′) ∈ DA(φ(α''))
• Uses the implicit function theorem to prove the existence of a maximal analytic path φ(α) through a given stable steady state (x⁰, α⁰) under the condition ||∂ₓf(x⁰, α⁰)|| < 1.
• Applies the inverse function theorem to ensure the path φ(α) is analytic and unique in a neighborhood of α⁰.

Experimental results

Research questions

• RQ1 Can two asymptotically stable steady states connected by an analytic path be transferred into one another in a nonlinear discrete dynamical system with control?
• RQ2 What conditions ensure that a control parameter change α′ → α″ results in a successful transition from one steady state to another?
• RQ3 How is the domain of attraction of a stable fixed point related to the analyticity domain of a Lyapunov function in discrete systems?
• RQ4 Under what conditions does a sequence of control parameter changes allow navigation from one stable steady state to another via intermediate states?
• RQ5 Can the controllability result from continuous systems be extended to discrete-time systems using analyticity and Lyapunov function theory?

Key findings

• An analytic path of asymptotically stable steady states exists locally around any given stable steady state (x⁰, α⁰) if the spectral norm of the Jacobian ∂ₓf(x⁰, α⁰) is less than 1.
• The domain of attraction of a stable fixed point x⁰ = φ(α) coincides with the natural domain of analyticity of a Lyapunov function V satisfying V(g(y)) - V(y) = -||y||².
• A maneuver α′ → α″ is successful if and only if the initial state φ(α′) lies within the domain of attraction of the target state φ(α'').
• In Example 1, the maneuver from α=1 to α=3 fails because the target state is unstable, but transitions via intermediate stable states are possible.
• In Example 2, a finite sequence of successful maneuvers transfers the system from φ₁(0)=0 to φ₁(2)=2, with each step satisfying the domain-of-attraction condition: 0 ∈ DA(φ₁(0.7)), φ₁(0.7) ∈ DA(φ₁(1.4)), and φ₁(1.4) ∈ DA(φ₁(2)).
• In Example 3, a four-step maneuver from α=−1 to α=1 successfully transfers the system from (−1,−1) to (1,1), with each intermediate state lying within the unit ball around the next target.