This article'slead sectionmay need to be rewritten. Please review thelead guide and helpimprove the lead of this article if you can.(December 2021) (Learn how and when to remove this message)

Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Various types ofstability may be discussed for the solutions ofdifferential equations ordifference equations describingdynamical 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 ofAleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point${\displaystyle x_e}$ stay near${\displaystyle x_e}$ forever, then${\displaystyle x_e}$ isLyapunov stable. More strongly, if${\displaystyle x_e}$ is Lyapunov stable and all solutions that start out near${\displaystyle x_e}$ converge to${\displaystyle x_e}$, then${\displaystyle x_e}$ is said to beasymptotically stable (seeasymptotic analysis). The notion ofexponential 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 asstructural 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.

Lyapunov stability is named afterAleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesisThe General Problem of Stability of Motion atKharkov University (nowVN Karazin Kharkiv National 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.^[2] For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanicianNikolay 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^[3] 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 theCold War period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospaceguidance 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.^{[4][5][6][7][8]}More recently the concept of theLyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection withchaos theory. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.^[9]

Consider anautonomous 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 thesystem state vector,${\displaystyle \mathcal{D}}$ an open set containing the origin, and${\displaystyle f:\mathcal{D}\to {\mathbb{R}}^n}$ 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 beLyapunov stable if for every${\displaystyle ϵ>0}$ there exists a${\displaystyle \delta >0}$ such that if then for every${\displaystyle t\ge 0}$ we have.
2. The equilibrium of the above system is said to beasymptotically stable if it is Lyapunov stable and there exists${\displaystyle \delta >0}$ such that if then${\displaystyle \underset{t\to \mathrm{\infty }}{lim}\Vert x(t)-x_e\Vert =0}$.
3. The equilibrium of the above system is said to beexponentially stable if it is asymptotically stable and there exist${\displaystyle \alpha >0,\beta >0,\delta >0}$ such that if then${\displaystyle \Vert x(t)-x_e\Vert \le \alpha \Vert x(0)-x_e\Vert e^{-\beta t}}$ for all${\displaystyle t\ge 0}$.

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${\displaystyle ϵ}$ from it). Note that this must be true forany${\displaystyle ϵ}$ 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${\displaystyle \alpha \Vert x(0)-x_e\Vert e^{-\beta t}}$.

The trajectory${\displaystyle \varphi (t)}$ is (locally)attractive if

${\displaystyle \Vert x(t)-\varphi (t)\Vert \to 0}$ as${\displaystyle t\to \mathrm{\infty }}$

for all trajectories${\displaystyle x(t)}$ that start close enough to${\displaystyle \varphi (t)}$, andglobally attractive if this property holds for all trajectories.

That is, ifx belongs to the interior of itsstable manifold, it isasymptotically stable if it is both attractive and stable. (There are examples showing that attractivity does not imply asymptotic stability.^[10][11][12] Such examples are easy to create usinghomoclinic connections.)

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

Instead of considering stability only near an equilibrium point (a constant solution${\displaystyle x(t)=x_e}$), one can formulate similar definitions of stability near an arbitrary solution${\displaystyle x(t)=\varphi (t)}$. However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations". Define${\displaystyle y=x-\varphi (t)}$, obeying the differential equation:

${\displaystyle \dot{y}=f(t,y+\varphi (t))-\dot{\varphi }(t)=g(t,y)}$.

This is no longer an autonomous system, but it has a guaranteed equilibrium point at${\displaystyle y=0}$ whose stability is equivalent to the stability of the original solution${\displaystyle x(t)=\varphi (t)}$.

Lyapunov, in his original 1892 work, proposed twomethods 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 now referred to as theLyapunov stability criterion or the Direct Method, makes use of aLyapunov function V(x) which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system${\displaystyle \dot{x}=f(x)}$ having a point of equilibrium at${\displaystyle x=0}$. Consider a function${\displaystyle V:{\mathbb{R}}^n\to \mathbb{R}}$ such that

• ${\displaystyle V(x)=0}$ if and only if${\displaystyle x=0}$
• ${\displaystyle V(x)>0}$ if and only if${\displaystyle x\ne 0}$
• ${\displaystyle \dot{V}(x)=\frac{d}{dt}V(x)=\sum _{i=1}^n\frac{\mathrm{\partial }V}{\mathrm{\partial }{x}_i}{f}_i(x)=\mathrm{\nabla }V\cdot f(x)\le 0}$ for all values of${\displaystyle x\ne 0}$ . Note: for asymptotic stability, for${\displaystyle x\ne 0}$ is required.

ThenV(x) is called aLyapunov function and the system is stable in the sense of Lyapunov. (Note that${\displaystyle V(0)=0}$ is required; otherwise for example${\displaystyle V(x)=1/(1+|x|)}$ would "prove" that${\displaystyle \dot{x}(t)=x}$ is locally stable.) An additional condition called "properness" or "radial unboundedness" is required in order to conclude global stability. Global asymptotic stability (GAS) follows similarly.

It is easier to visualize this method of analysis by thinking of a physical system (e.g. vibrating spring and mass) and considering theenergy 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 theattractor. 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 aLyapunov function can be found to satisfy the above constraints.

The definition fordiscrete-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 ametric space andf :X →X acontinuous function. A pointx inX is said to beLyapunov stable, if,

We say thatx isasymptotically stable if it belongs to the interior of itsstable set,i.e. if,

A linearstate space model

${\displaystyle \dot{\mathbf{\text{x}}}=A\mathbf{\text{x}}}$,

where${\displaystyle A}$ is a finite matrix, is asymptotically stable (in fact,exponentially stable) if all real parts of theeigenvalues of${\displaystyle A}$ are negative. This condition is equivalent to the following one:^[13]

${\displaystyle A^{\mathsf{\text{T}}}M+MA}$

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

Correspondingly, a time-discrete linearstate space model

${\displaystyle {\mathbf{\text{x}}}_{t+1}=A{\mathbf{\text{x}}}_t}$

is asymptotically stable (in fact, exponentially stable) if all the eigenvalues of${\displaystyle A}$ have amodulus smaller than one.

This latter condition has been generalized to switched systems: a linear switched discrete time system (ruled by a set of matrices${\displaystyle \{A_1,\dots ,A_m\}}$)

${\displaystyle {\mathbf{\text{x}}}_{t+1}=A_{i_t}{\mathbf{\text{x}}}_t,A_{i_t}\in \{A_1,\dots ,A_m\}}$

is asymptotically stable (in fact, exponentially stable) if thejoint spectral radius of the set${\displaystyle \{A_1,\dots ,A_m\}}$ is smaller than one.

A system with inputs (or controls) has the form

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

where the (generally time-dependent) input u(t) may be viewed as acontrol,external input,stimulus,disturbance, orforcing function. It has been shown^[14] 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 ofcontrol theory and applied incontrol 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 areBIBO stability (forlinear systems) andinput-to-state stability (ISS) (fornonlinear 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 theVan der Pol oscillator equation with the friction term changed:

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

Let

${\displaystyle x_1=y,x_2=\dot{y}}$

so that the corresponding system is

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 clearlypositive definite. Its derivative is

${\displaystyle \dot{V}=x_1{\dot{x}}_1+x_2{\dot{x}}_2=x_1{x}_2-x_1{x}_2+\epsilon \frac{x_2^4}{3}-\epsilon x_2^2=\epsilon \frac{x_2^4}{3}-\epsilon x_2^2.}$

It seems that if the parameter${\displaystyle \epsilon }$ is positive, stability is asymptotic for 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.

It may be difficult to find a Lyapunov function with a negative definite derivative as required by the Lyapunov stability criterion, however a function${\displaystyle V}$ with${\displaystyle \dot{V}}$ that is only negative semi-definite may be available. In autonomous systems,the invariant set theorem can be applied to prove asymptotic stability, but this theorem is not applicable when the dynamics are a function of time.^[15]

Instead, Barbalat's lemma allows for Lyapunov-like analysis of these non-autonomous systems. The lemma is motivated by the following observations. Assuming 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${\displaystyle t\to \mathrm{\infty }}$. For example,${\displaystyle f(t)=\sin (\ln (t)),t>0}$.
• Having${\displaystyle f(t)}$ approaching a limit as${\displaystyle t\to \mathrm{\infty }}$ 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 (${\displaystyle \dot{f}\le 0}$) implies it converges to a limit. But it does not say whether or not${\displaystyle \dot{f}\to 0}$ as${\displaystyle t\to \mathrm{\infty }}$.

Barbalat'sLemma says:

If${\displaystyle f(t)}$ has a finite limit as${\displaystyle t\to \mathrm{\infty }}$ and if${\displaystyle \dot{f}}$ is uniformly continuous (a sufficient condition for uniform continuity is that${\displaystyle \ddot{f}}$ is bounded), then${\displaystyle \dot{f}(t)\to 0}$ as${\displaystyle t\to \mathrm{\infty }}$.^[16]

An alternative version is as follows:

Let${\displaystyle p\in [1,\mathrm{\infty })}$ and${\displaystyle q\in (1,\mathrm{\infty }]}$. If${\displaystyle f\in L^p(0,\mathrm{\infty })}$ and${\displaystyle \dot{f}\in L^q(0,\mathrm{\infty })}$, then${\displaystyle f(t)\to 0}$ as${\displaystyle t\to \mathrm{\infty }.}$^[17]

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${\displaystyle {\int }_0^{t}f(\tau )\mathrm{d}\tau }$ has a finite limit as${\displaystyle t\to \mathrm{\infty }}$. Then${\displaystyle f(t)\to 0}$ as${\displaystyle t\to \mathrm{\infty }}$.^[18]

The following example is taken from page 125 of Slotine and Li's bookApplied Nonlinear Control.^[15]

Consider anon-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${\displaystyle \dot{V}=-2e^2\le 0.}$

This says that${\displaystyle V(t)\le V(0)}$ 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, as${\displaystyle \dot{V}}$ is only negative semi-definite (note${\displaystyle g}$ can be non-zero when${\displaystyle \dot{V}}$=0) and the dynamics are 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${\displaystyle t\to \mathrm{\infty }}$ 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
• Stability 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.
• Shcherbakov, Pavel S. (1992), "Alexander Mikhailovitch Lyapunov: On the centenary of his doctoral dissertation on stability of motion",Automatica,28 (5):865–871,doi:10.1016/0005-1098(92)90140-B
• 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.

This article incorporates material from asymptotically stable onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

• Stability theory
• Dynamical systems
• Lagrangian mechanics
• Three-body orbits

• CS1 Russian-language sources (ru)
• CS1: long volume value
• Articles with short description
• Short description matches Wikidata
• Wikipedia introduction cleanup from December 2021
• All pages needing cleanup
• Articles covered by WikiProject Wikify from December 2021
• All articles covered by WikiProject Wikify
• Wikipedia articles incorporating text from PlanetMath