Stability Preservation in Linear Dynamical Systems Under Nonlinear Perturbations: A Comprehensive Analysis and Conditions for Robust Stability

Question

Barrier of the Process: The statement we are investigating concerns the stability of a point under a perturbation. Specifically, we want to determine whether a point that is Lyapunov stable for the linear system $\dot{x}=Ax$ remains stable when perturbed by a term $O(\|x\|^2)$ in the form $\dot{x}=v(x)=Ax+O(\|x\|^2)$, assuming that the function $v(x)$ belongs to $C^k(U)$ and that we are dealing with a system in $U \subset \mathbb{R}^n$ where $n \geq 2$.

Analysis of the Barrier: As previously discussed, the statement is not necessarily true. We demonstrated this with the example:

\begin{align*} \dot{x} &= -\frac{y}{x} + x^3 \\ \dot{y} &= x + y^3 \end{align*}

In this example, the origin is Lyapunov stable for the linearized system $\dot{x}=Ax$. However, when we introduce the nonlinear term $O(\|x\|^2)$ in $\dot{x}=v(x)$, the origin is no longer Lyapunov stable. This raises the question of finding a suitable Lyapunov function $g'(x,y)$ for the perturbed system that can capture the stability or instability of the origin.

Related Question: Given that we've observed that adding nonlinear terms to the system can change its stability, a related question arises:

Question: Under what conditions can we guarantee the preservation of stability (i.e., Lyapunov stability) when introducing perturbations, such as $O(\|x\|^2)$ terms, to a system of the form $\dot{x}=Ax$? Are there specific properties of the perturbation function $v(x)$ that ensure stability is maintained?

This question delves into the broader topic of stability analysis in the presence of perturbations and seeks to identify conditions under which the stability of a system remains robust despite the addition of nonlinear terms.

Answer 1

The local (at the origin) stability of a linear system perturbed by nonlinearities is preserved if the Jacobian of the nonlinear term evaluated at the origin is zero and if the linear part does not have marginally stable modes.

This equivalent to the following mathematical notation

$$ \dot{x} = v(x), \tag{1} $$

with $v(0)=0$ and

$$ \left.\frac{\partial\,v(x)}{\partial x}\right|_{x=0} = A \tag{2} $$

and $A$ a Hurwitz matrix (i.e. all eigenvalues of $A$ have a negative real part). Which is also equivalent to stating

$$ v(x) = A\,x + f(x), \tag{3} $$

with $A$ a Hurwitz matrix and $f(x)$ such that $f(0)=0$ and

$$ \left.\frac{\partial\,f(x)}{\partial x}\right|_{x=0} = 0. \tag{4} $$

Though, it has to be noted that this is a sufficient but not necessary condition to ensure local stability of the nonlinear system.

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A Lyapunov function $V(x)$ that would show local asymptotic stability of $(1)$ can be found using any positive definite matrix $Q$ and matrix $P$ satisfying the following Lyapunov equation

$$ A^\top P + P\,A = -Q. \tag{5} $$

The corresponding Lyapunov function can be obtained as

$$ V(x) = x^\top P\,x. \tag{6} $$

Local asymptotic stability of $(1)$ using $(6)$ can be shown by taking the derivative of $(5)$, which yields

\begin{align} \dot{V}(x) &= \dot{x}^\top P\,x + x^\top P\,\dot{x}, \\ &= v(x)^\top P\,x + x^\top P\,v(x), \\ &= x^\top A^\top P\,x + f(x)^\top P\,x + x^\top P\,A\,x + x^\top P\,f(x), \\ &= -x^\top Q\,x + f(x)^\top P\,x + x^\top P\,f(x). \end{align}

Due to $(4)$ is follows that $f(x)^\top P\,x + x^\top P\,f(x)$ is $O(\|x\|^3)$ or higher, while $x^\top Q\,x$ is $O(\|x\|^2)$. Therefore, near the origin the term $-x^\top Q\,x$ dominates $\dot{V}(x)$ and thus implies asymptotic stability.

Answer 2

The system under consideration is given by:

$$ \dot{x} = Ax $$

When perturbed by a term $O(|x|^2)$in the form:

$$ \dot{x} = v(x) = Ax + O(\|x\|^2) $$

with (v(x)) belonging to $C^k(U)$ and $U \subset \mathbb{R}^n$ where $n \geq 2$.

The stability analysis involves finding a Lyapunov function $V(x)$ that satisfies the stability conditions.

The example system is given by:

$$ \dot{x}\dot{y} = -yx + x^3 = x + y^3 $$

To investigate stability, a Lyapunov function $g'(x, y)$ for the perturbed system needs to be determined.

The broader question is expressed as:

Under what conditions can we guarantee the preservation of stability (i.e., Lyapunov stability) when introducing perturbations, such as $O(\|x\|^2)$ terms, to a system of the form $ \dot{x} = Ax$

The related analysis involves frequency equations. These equations capture the behavior of perturbations and are often expressed in terms of the system matrix $A$ and additional terms related to the perturbation function.