### Asymptotic stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov. In simple terms, if all solutions of the dynamical system that start out near an equilibrium point $x\_e$ stay near $x\_e$ forever, then $x\_e$ is **Lyapunov stable**. More strongly, if $x\_e$ is Lyapunov stable and all solutions that start out near $x\_e$ converge to $x\_e$, then $x\_e$ is asymptotically stable. The notion of **exponential stability** guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs.

## Contents

## History

Lyapunov stability is named after Aleksandr Lyapunov, a Russian mathematician who published his book *The General Problem of Stability of Motion* in 1892.^{[1]} Lyapunov was the first to consider the modifications necessary in nonlinear systems to the linear theory of stability based on linearizing near a point of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. Interest in it started suddenly during the Cold War (1953–1962) period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature.^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}
More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.^{[7]}

## Definition for continuous-time systems

Consider an autonomous nonlinear dynamical system

- $\backslash dot\{x\}\; =\; f(x(t)),\; \backslash ;\backslash ;\backslash ;\backslash ;\; x(0)\; =\; x\_0$,

where $x(t)\; \backslash in\; \backslash mathcal\{D\}\; \backslash subseteq\; \backslash mathbb\{R\}^n$ denotes the system state vector, $\backslash mathcal\{D\}$ an open set containing the origin, and $f:\; \backslash mathcal\{D\}\; \backslash rightarrow\; \backslash mathbb\{R\}^n$ continuous on $\backslash mathcal\{D\}$. Suppose $f$ has an equilibrium $x\_e$.

- The equilibrium of the above system is said to be
**Lyapunov stable**, if, for every $\backslash epsilon\; >\; 0$, there exists a $\backslash delta\; =\; \backslash delta(\backslash epsilon)\; >\; 0$ such that, if $\backslash |x(0)-x\_e\backslash |\; <\; \backslash delta$, then $\backslash |x(t)-x\_e\backslash |\; <\; \backslash epsilon$, for every $t\; \backslash geq\; 0$. - The equilibrium of the above system is said to be
**asymptotically stable**if it is Lyapunov stable and if there exists $\backslash delta\; >\; 0$ such that if $\backslash |x(0)-x\_e\; \backslash |<\; \backslash delta$, then $\backslash lim\_\{t\; \backslash rightarrow\; \backslash infty\}\; \backslash |x(t)-x\_e\backslash |\; =\; 0$. - The equilibrium of the above system is said to be
**exponentially stable**if it is asymptotically stable and if there exist $\backslash alpha,\; \backslash beta,\; \backslash delta\; >0$ such that if $\backslash |x(0)-x\_e\backslash |\; <\; \backslash delta$, then $\backslash |x(t)-x\_e\backslash |\; \backslash leq\; \backslash alpha\backslash |x(0)-x\_e\backslash |e^\{-\backslash beta\; t\}$, for $t\; \backslash geq\; 0$.

Conceptually, the meanings of the above terms are the following:

- Lyapunov stability of an equilibrium means that solutions starting "close enough" to the equilibrium (within a distance $\backslash delta$ from it) remain "close enough" forever (within a distance $\backslash epsilon$ from it). Note that this must be true for
*any*$\backslash epsilon$ that one may want to choose. - Asymptotic stability means that solutions that start close enough not only remain close enough but also eventually converge to the equilibrium.
- Exponential stability means that solutions not only converge, but in fact converge faster than or at least as fast as a particular known rate $\backslash alpha\backslash |x(0)-x\_e\backslash |e^\{-\backslash beta\; t\}$.

The trajectory *x* is (locally) *attractive* if

- $\backslash |y(t)-x(t)\backslash |\; \backslash rightarrow\; 0$

for $t\; \backslash rightarrow\; \backslash infty$ for all trajectories that start close enough, and *globally attractive* if this property holds for all trajectories.

That is, if *x* belongs to the interior of its stable manifold. It is *asymptotically stable* if it is both attractive and stable. (There are counterexamples showing that attractivity does not imply asymptotic stability. Such examples are easy to create using homoclinic connections.)

### Lyapunov's second method for stability

Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability.^{[1]} The first method developed the solution in a series which was then proved convergent within limits. The second method, which is almost universally used nowadays, makes use of a *Lyapunov function V(x)* which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system having a point of equilibrium at x=0. Consider a function $V(x):\; \backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ such that

- $V(x)\; \backslash ge\; 0$ with equality if and only if $x=0$ (positive definite)
- $\backslash dot\{V\}(x)\; =\; \backslash frac\{d\}\{dt\}V(x)\; \backslash le\; 0$ with equality if and only if $x=0$ (negative definite).

Then *V(x)* is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov. (Note that $V(0)=0$ is required; otherwise for example $V(x)\; =\; 1/(1+|x|)$ would "prove" that $\backslash dot\; x(t)\; =\; x$ is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global asymptotic stability.)

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering the energy of such a system. If the system loses energy over time and the energy is never restored then eventually the system must grind to a stop and reach some final resting state. This final state is called the attractor. However, finding a function that gives the precise energy of a physical system can be difficult, and for abstract mathematical systems, economic systems or biological systems, the concept of energy may not be applicable.

Lyapunov's realization was that stability can be proven without requiring knowledge of the true physical energy, provided a Lyapunov function can be found to satisfy the above constraints.

## Definition for discrete-time systems

The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts.

Let (*X*, *d*) be a metric space and *f* : *X* → *X* a continuous function. A point *x* in *X* is said to be **Lyapunov stable**, if,

- $\backslash forall\; \backslash epsilon>0\; \backslash \; \backslash exists\; \backslash delta>0\; \backslash \; \backslash forall\; y\backslash in\; X\; \backslash \; \backslash left\; [d(x,y)<\backslash delta\; \backslash Rightarrow\; \backslash forall\; n\; \backslash in\; \backslash mathbf\{N\}\; \backslash \; d\backslash left\; (f^n(x),f^n(y)\; \backslash right\; )<\backslash epsilon\; \backslash right\; ].$

We say that *x* is **asymptotically stable** if it belongs to the interior of its stable set, *i.e.* if,

- $\backslash exists\; \backslash delta>0\; \backslash left\; [\; d(x,y)<\backslash delta\; \backslash Rightarrow\; \backslash lim\_\{n\backslash to\backslash infty\}\; d\; \backslash left(f^n(x),f^n(y)\; \backslash right)=0\backslash right\; ].$

## Stability for linear state space models

A linear state space model

- $\backslash dot\{\backslash textbf\{x\}\}\; =\; A\backslash textbf\{x\}$,

where $A$ is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of $A$ are negative. This condition is equivalent to the following one:

- $A^\{T\}M\; +\; MA\; +\; N\; =\; 0$

has a solution where $N\; =\; N^\{T\}\; >\; 0$ and $M\; =\; M^\{T\}\; >\; 0$ (positive definite matrices). (The relevant Lyapunov function is $V(x)\; =\; x^TMx$. )

Correspondingly, a time-discrete linear state space model

- $\{\backslash textbf\{x\}\_\{t+1\}\}\; =\; A\backslash textbf\{x\}\_t$

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of $A$ have a modulus smaller than one.

This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices $\backslash \{A\_1,\; \backslash dots,\; A\_m\backslash \}$)

- $\{\backslash textbf\{x\}\_\{t+1\}\}\; =\; A\_\{i\_t\}\backslash textbf\{x\}\_t,\backslash quad\; A\_\{i\_t\}\; \backslash in\; \backslash \{A\_1,\; \backslash dots,\; A\_m\backslash \}$

is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set $\backslash \{A\_1,\; \backslash dots,\; A\_m\backslash \}$ is smaller than one.

## Stability for systems with inputs

A system with inputs (or controls) has the form

- $\backslash dot\{\backslash textbf\{x\}\}\; =\; \backslash textbf\{f(x,u)\}$

where the (generally time-dependent) input u(t) may be viewed as a *control*, *external input*,
*stimulus*, *disturbance*, or *forcing function*. The study of such systems is the subject
of control theory and applied in control engineering. For systems with inputs, one must
quantify the effect of inputs on the stability of the system. The main two approaches to this
analysis are BIBO stability (for linear systems) and input-to-state (ISS) stability (for nonlinear systems)

## Example

Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:

- $\backslash ddot\{y\}\; +\; y\; -\backslash varepsilon\; \backslash left(\; \backslash frac\{\backslash dot\{y\}^\{3\}\}\{3\}\; -\; \backslash dot\{y\}\backslash right)\; =\; 0.$

The equilibrium is at :$\backslash ddot\{y\}\; =\; y\; =\; 0.$

Here is a good example of an unsuccessful try to find a Lyapunov function that proves stability:

Let

- $x\_\{1\}\; =\; y\; ,\; x\_\{2\}\; =\; \backslash dot\{y\}$

so that the corresponding system is

- $\backslash dot\{x\_\{2\}\}\; =\; -x\_\{1\}\; +\; \backslash varepsilon\; \backslash left(\; \backslash frac^\{3\}\}\{3\}\; -\; \{x\_\{2\}\}\backslash right).$

Let us choose as a Lyapunov function

- $V\; =\; \backslash frac\; \{1\}\{2\}\; \backslash left(x\_\{1\}^\{2\}+x\_\{2\}^\{2\}\; \backslash right)$

which is clearly positive definite. Its derivative is

- $\backslash begin\{align\}$

\dot{V} &= x_{1} \dot x_{1} +x_{2} \dot x_{2}\\ &= x_{1} x_{2} - x_{1} x_{2}+\varepsilon \left(\frac{x_{2}^4}{3} -{x_{2}^2}\right)\\ &= -\varepsilon \left({x_{2}^2} - \frac{x_{2}^4}{3}\right). \end{align}

It seems that if the parameter $\backslash varepsilon$ is positive, stability is asymptotic for $x\_\{2\}^\{2\}\; <\; 3.$ But this is wrong, since $\backslash dot\{V\}$ does not depend on $x\_1$, and will be 0 everywhere on the $x\_1$ axis.

## Barbalat's lemma and stability of time-varying systems

Assume that f is function of time only.

- Having $\backslash dot\{f\}(t)\; \backslash to\; 0$ does not imply that $f(t)$ has a limit at $t\backslash to\backslash infty$. For example, $f(t)=\backslash sin(\backslash ln(t)),\backslash ;\; t>0$.
- Having $f(t)$ approaching a limit as $t\; \backslash to\; \backslash infty$ does not imply that $\backslash dot\{f\}(t)\; \backslash to\; 0$. For example, $f(t)=\backslash sin(t^2)/t,\backslash ;\; t>0$.
- Having $f(t)$ lower bounded and decreasing ($\backslash dot\{f\}\backslash le\; 0$) implies it converges to a limit. But it does not say whether or not $\backslash dot\{f\}\backslash to\; 0$ as $t\; \backslash to\; \backslash infty$.

(Barbalat's Lemma says:

- If $f(t)$ has a finite limit as $t\; \backslash to\; \backslash infty$ and if $\backslash dot\{f\}$ is uniformly continuous (or $\backslash ddot\{f\}$ is bounded), then $\backslash dot\{f\}(t)\; \backslash to\; 0$ as $t\; \backslash to\backslash infty$.

Usually, it is difficult to analyze the *asymptotic* stability of time-varying systems because it is very difficult to find Lyapunov functions with a *negative definite* derivative.

We know that in case of autonomous (time-invariant) systems, if $\backslash dot\{V\}$ is negative semi-definite (NSD), then also, it is possible to know the asymptotic behaviour by invoking invariant-set theorems. However, this flexibility is not available for *time-varying* systems.
This is where "Barbalat's lemma" comes into picture. It says:

- IF $V(x,t)$ satisfies following conditions:
- $V(x,t)$ is lower bounded

- $\backslash dot\{V\}(x,t)$ is negative semi-definite (NSD)

- $\backslash dot\{V\}(x,t)$ is uniformly continuous in time (satisfied if $\backslash ddot\{V\}$ is finite)

- $V(x,t)$ is lower bounded
- then $\backslash dot\{V\}(x,t)\backslash to\; 0$ as $t\; \backslash to\; \backslash infty$.

The following example is taken from page 125 of Slotine and Li's book *Applied Nonlinear Control*.

Consider a non-autonomous system

- $\backslash dot\{e\}=-e\; +\; g\backslash cdot\; w(t)$
- $\backslash dot\{g\}=-e\; \backslash cdot\; w(t).$

This is non-autonomous because the input $w$ is a function of time. Assume that the input $w(t)$ is bounded.

Taking $V=e^2+g^2$ gives $\backslash dot\{V\}=-2e^2\; \backslash le\; 0.$

This says that $V(t)<=V(0)$ by first two conditions and hence $e$ and $g$ are bounded. But it does not say anything about the convergence of $e$ to zero. Moreover, the invariant set theorem cannot be applied, because the dynamics is non-autonomous.

Using Barbalat's lemma:

- $\backslash ddot\{V\}=\; -4e(-e+g\backslash cdot\; w)$.

This is bounded because $e$, $g$ and $w$ are bounded. This implies $\backslash dot\{V\}\; \backslash to\; 0$ as $t\backslash to\backslash infty$ and hence $e\; \backslash to\; 0$. This proves that the error converges.

## References

## Further reading

- Jean-Jacques E. Slotine and Weiping Li,
*Applied Nonlinear Control*, Prentice Hall, NJ, 1991 - Parks P.C: "A. M. Lyapunov's stability theory - 100 years on",
*IMA Journal of Mathematical Control & Information*1992 9 275-303

## External links

- http://www.mne.ksu.edu/research/laboratories/non-linear-controls-lab (login required)

*This article incorporates material from asymptotically stable on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*