Lyapunov stability

Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892.[1] A. M. Lyapunov was a pioneer in successful endeavors to develop a global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of his suicide in 1918 . For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanician Nikolay Gur'yevich Chetaev working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev[2] was so significant that many mathematicians, physicists and engineers consider him Lyapunov's direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability.

The interest in it suddenly skyrocketed during the Cold War 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.[3][4][5][6][7] 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.[8]

Consider an autonomous nonlinear dynamical system

${\displaystyle {\dot {x}}=f(x(t)),\;\;\;\;x(0)=x_{0}}$,

where ${\displaystyle x(t)\in {\mathcal {D}}\subseteq \mathbb {R} ^{n}}$ denotes the system state vector, ${\displaystyle {\mathcal {D}}}$ an open set containing the origin, and ${\displaystyle f:{\mathcal {D}}\rightarrow \mathbb {R} ^{n}}$ is a continuous vector field on ${\displaystyle {\mathcal {D}}}$. Suppose ${\displaystyle f}$ has an equilibrium at ${\displaystyle x_{e}}$ so that ${\displaystyle f(x_{e})=0}$ then

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

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

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

The trajectory ${\displaystyle x(t)=\phi (t)}$ is (locally) attractive if

${\displaystyle \|x(t)-\phi (t)\|\rightarrow 0}$ as ${\displaystyle t\rightarrow \infty }$

for all trajectories ${\displaystyle x(t)}$ that start close enough to ${\displaystyle \phi (t)}$, 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 examples showing that attractivity does not imply asymptotic stability.[9][10][11] Such examples are easy to create using homoclinic connections.)

If the Jacobian of the dynamical system at an equilibrium happens to be a stability matrix (i.e., if the real part of each eigenvalue is strictly negative), then the equilibrium is asymptotically stable.

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,

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

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

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

A linear state space model

${\displaystyle {\dot {\textbf {x}}}=A{\textbf {x}}}$,

where ${\displaystyle A}$ is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of ${\displaystyle A}$ are negative. This condition is equivalent to the following one:[12]

${\displaystyle A^{\textsf {T}}M+MA}$

is negative definite for some positive definite matrix ${\displaystyle M=M^{\textsf {T}}}$. (The relevant Lyapunov function is ${\displaystyle V(x)=x^{\textsf {T}}Mx}$.)

Correspondingly, a time-discrete linear state space model

${\displaystyle {\textbf {x}}_{t+1}=A{\textbf {x}}_{t}}$

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of ${\displaystyle A}$ 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 ${\displaystyle \{A_{1},\dots ,A_{m}\}}$)

${\displaystyle {{\textbf {x}}_{t+1}}=A_{i_{t}}{\textbf {x}}_{t},\quad A_{i_{t}}\in \{A_{1},\dots ,A_{m}\}}$

is asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set ${\displaystyle \{A_{1},\dots ,A_{m}\}}$ is smaller than one.

A system with inputs (or controls) has the form

${\displaystyle {\dot {\textbf {x}}}={\textbf {f}}({\textbf {x}},{\textbf {u}})}$

where the (generally time-dependent) input u(t) may be viewed as a control, external input, stimulus, disturbance, or forcing function. It has been shown [13] that near to a point of equilibrium which is Lyapunov stable the system remains stable under small disturbances. For larger input disturbances 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 stability (ISS) (for nonlinear systems)

This example shows a system where a Lyapunov function can be used to prove Lyapunov stability but cannot show asymptotic stability. Consider the following equation, based on the Van der Pol oscillator equation with the friction term changed:

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

Let

${\displaystyle x_{1}=y,x_{2}={\dot {y}}}$

so that the corresponding system is

${\displaystyle {\begin{aligned}&{\dot {x}}_{1}=x_{2},\\&{\dot {x}}_{2}=-x_{1}+\varepsilon \left({\frac {x_{2}^{3}}{3}}-{x_{2}}\right).\end{aligned}}}$

The origin ${\displaystyle x_{1}=0,\ x_{2}=0}$ is the only equilibrium point. Let us choose as a Lyapunov function

${\displaystyle V={\frac {1}{2}}\left(x_{1}^{2}+x_{2}^{2}\right)}$

which is clearly positive definite. Its derivative is

${\displaystyle {\dot {V}}=x_{1}{\dot {x}}_{1}+x_{2}{\dot {x}}_{2}=x_{1}x_{2}-x_{1}x_{2}+\varepsilon {\frac {x_{2}^{4}}{3}}-\varepsilon {x_{2}^{2}}=\varepsilon {\frac {x_{2}^{4}}{3}}-\varepsilon {x_{2}^{2}}.}$

It seems that if the parameter ${\displaystyle \varepsilon }$ is positive, stability is asymptotic for ${\displaystyle x_{2}^{2}<3.}$ But this is wrong, since ${\displaystyle {\dot {V}}}$ does not depend on ${\displaystyle x_{1}}$, and will be 0 everywhere on the ${\displaystyle x_{1}}$ axis. The equilibrium is Lyapunov stable but not asymptotically stable.

Assume that f is a function of time only.

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

Barbalat's Lemma says:

If ${\displaystyle f(t)}$ has a finite limit as ${\displaystyle t\to \infty }$ and if ${\displaystyle {\dot {f}}}$ is uniformly continuous (or ${\displaystyle {\ddot {f}}}$ is bounded), then ${\displaystyle {\dot {f}}(t)\to 0}$ as ${\displaystyle t\to \infty }$.[14]

An alternative version is as follows:

Let ${\displaystyle p\in [1,\infty )}$ and ${\displaystyle q\in (1,\infty ]}$. If ${\displaystyle f\in L^{p}(0,\infty )}$ and ${\displaystyle {\dot {f}}\in L^{q}(0,\infty )}$, then ${\displaystyle f(t)\to 0}$ as ${\displaystyle t\to \infty .}$ [15]

In the following form the Lemma is true also in the vector valued case:

Let ${\displaystyle f(t)}$ be a uniformly continuous function with values in a Banach space ${\displaystyle E}$ and assume that ${\displaystyle \textstyle \int _{0}^{t}f(\tau )\mathrm {d} \tau }$ has a finite limit as ${\displaystyle t\to \infty }$. Then ${\displaystyle f(t)\to 0}$ as ${\displaystyle t\to \infty }$.[16]

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

Consider a non-autonomous system

${\displaystyle {\dot {e}}=-e+g\cdot w(t)}$

${\displaystyle {\dot {g}}=-e\cdot w(t).}$

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

Taking ${\displaystyle V=e^{2}+g^{2}}$ gives ${\displaystyle {\dot {V}}=-2e^{2}\leq 0.}$

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

Using Barbalat's lemma:

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

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

• Lyapunov function
• LaSalle's invariance principle
• Lyapunov–Malkin theorem
• Markus–Yamabe conjecture
• Libration point orbit
• Hartman–Grobman theorem
• Perturbation theory

• Bhatia, Nam Parshad; Szegő, Giorgio P. (2002). Stability theory of dynamical systems. Springer. ISBN 978-3-540-42748-3.
• Chervin, Robert (1971). Lyapunov Stability and Feedback Control of Two-Stream Plasma Systems (PhD). Columbia University.
• Gandolfo, Giancarlo (1996). Economic Dynamics (Third ed.). Berlin: Springer. pp. 407–428. ISBN 978-3-540-60988-9.
• Parks, P. C. (1992). "A. M. Lyapunov's stability theory—100 years on". IMA Journal of Mathematical Control & Information. 9 (4): 275–303. doi:10.1093/imamci/9.4.275.
• Slotine, Jean-Jacques E.; Weiping Li (1991). Applied Nonlinear Control. NJ: Prentice Hall.
• Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
• Wiggins, S. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos (2nd ed.). New York: Springer Verlag. ISBN 978-0-387-00177-7.

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