Various types of

stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...

may be discussed for the solutions of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s or difference equations describing

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

s. 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

Aleksandr Lyapunov Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Lia ...

. In simple terms, if the solutions 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

asymptotically stable Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. T ...

. 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 Input-to-state stability (ISS)Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002.restricted three-body problem which do not exhibit asymptotic stability.

History

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. Lyapunov, A. M. ''The General Problem of the Stability of Motion'' (In Russian), Doctoral dissertation, Univ. Kharkov 1892 English translations: (1) ''Stability of Motion'', Academic Press, New-York & London, 1966 (2) ''The General Problem of the Stability of Motion'', (A. T. Fuller trans.) Taylor & Francis, London 1992. Included is a biography by Smirnov and an extensive bibliography of Lyapunov's work. 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 the Russian revolution of 1917 . 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 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. English tr. Princeton 1961 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.

Definition for continuous-time systems

Consider an

autonomous In developmental psychology and moral, political, and bioethical philosophy, autonomy, from , ''autonomos'', from αὐτο- ''auto-'' "self" and νόμος ''nomos'', "law", hence when combined understood to mean "one who gives oneself one's ow ...

nonlinear dynamical system :

\dot = f(x(t)), \;\;\;\; x(0) = x_0

, where

x(t) \in \mathcal \subseteq \mathbb^n

denotes the system state vector,

\mathcal

an open set containing the origin, and

f: \mathcal \rightarrow \mathbb^n

is a continuous vector field on

\mathcal

. Suppose

f

has an equilibrium at

x_e

so that

f(x_e)=0

then # This equilibrium is said to be Lyapunov stable, if, for every

\epsilon > 0

, there exists a

\delta > 0

such that, if

\, x(0)-x_e\,  < \delta

, then for every

t \geq 0

we have

\, x(t)-x_e\,  < \epsilon

. # The equilibrium of the above system is said to be asymptotically stable if it is Lyapunov stable and there exists

\delta > 0

such that if

\, x(0)-x_e \, < \delta

, then

\lim_ \, x(t)-x_e\,  = 0

. # The equilibrium of the above system is said to be exponentially stable if it is asymptotically stable and there exist

\alpha >0, \beta >0, \delta >0

such that if

\, x(0)-x_e\,  < \delta

, then

\, x(t)-x_e\,  \leq \alpha\, x(0)-x_e\, e^

, for all

t \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

\delta

from it) remain "close enough" forever (within a distance

\epsilon

from it). Note that this must be true for ''any''

\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

\alpha\, x(0)-x_e\, e^

. The trajectory ''

x(t) = \phi(t)

'' is (locally) ''attractive'' if :

\, x(t)-\phi(t)\,  \rightarrow 0

t \rightarrow \infty

for all trajectories

x(t)

that start close enough to

\phi(t)

, and ''globally attractive'' if this property holds for all trajectories. That is, if ''x'' belongs to the interior of its

stable manifold In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repello ...

, it is ''asymptotically stable'' if it is both attractive and stable. (There are examples showing that attractivity does not imply asymptotic stability. 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.

System of deviations

Instead of considering stability only near an equilibrium point (a constant solution

x(t)=x_e

), one can formulate similar definitions of stability near an arbitrary solution

x(t) = \phi(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

y = x - \phi(t)

, obeying the differential equation: :

\dot = f(t, y + \phi(t)) - \dot(t) = g(t, y)

. This is no longer an autonomous system, but it has a guaranteed equilibrium point at

y=0

whose stability is equivalent to the stability of the original solution

x(t) = \phi(t)

Lyapunov's second method for stability

Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. 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 the Lyapunov stability criterion or the Direct Method, 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

\dot = f(x)

having a point of equilibrium at

x=0

. Consider a function

V : \mathbb^n \rightarrow \mathbb

such that *

V(x)=0

if and only if

x=0

V(x)>0

if and only if

x \ne 0

\dot(x) = \fracV(x) = \sum_^n\fracf_i(x) = \nabla V \cdot f(x) \le 0

for all values of

x\ne 0

. Note: for asymptotic stability,

\dot(x)<0

for

x \ne 0

is required. Then ''V(x)'' is called a

Lyapunov function In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s s ...

and the system is stable in the sense of Lyapunov. (Note that

V(0)=0

is required; otherwise for example

V(x) = 1/(1+, x, )

would "prove" that

\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 the

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...

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 In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...

. 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

can be found to satisfy the above constraints.

Definition for discrete-time systems

The definition for

discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...

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 In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

and ''f'' : ''X'' → ''X'' a continuous function. A point ''x'' in ''X'' is said to be Lyapunov stable, if, :

\forall \epsilon>0 \  \exists \delta>0 \  \forall y\in X \ \left (x,y)<\delta \Rightarrow \forall n \in \mathbf \  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, :

\exists \delta>0 \left d(x,y)<\delta \Rightarrow \lim_ d \left(f^n(x),f^n(y) \right)=0\right

Stability for linear state space models

A linear

state space A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the to ...

model :

\dot = A\textbf

, where

A

is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

s of

A

are negative. This condition is equivalent to the following one: :

A^\textsfM + MA

is negative definite for some positive definite matrix

M = M^\textsf

. (The relevant Lyapunov function is

V(x) = x^\textsfMx

.) Correspondingly, a time-discrete linear

model :

\textbf_ = A\textbf_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

\

) :

= A_\textbf_t,\quad A_ \in \

is asymptotically stable (in fact, exponentially stable) if the

joint spectral radius In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topi ...

of the set

\

is smaller than one.

Stability for systems with inputs

A system with inputs (or controls) has the form :

\dot = \textbf(\textbf, \textbf)

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 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 Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...

and applied in

control engineering Control engineering or control systems engineering is an engineering discipline that deals with control systems, applying control theory to design equipment and systems with desired behaviors in control environments. The discipline of controls o ...

. 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 In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the ...

(for

linear system In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstractio ...

s) and

input-to-state stability Input-to-state stability (ISS)Eduardo D. Sontag. Mathematical Control Theory: Finite-Dimensional Systems. Springer-Verlag, London, 1998Hassan K. Khalil. Nonlinear Systems. Prentice Hall, 2002.nonlinear systems)

Example

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 In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation: :-\mu(1-x^2)+x= 0, where ''x'' is the position coordinate—which is a f ...

equation with the friction term changed: :

\ddot + y -\varepsilon \left( \frac - \dot\right) = 0.

Let :

x_ = y , x_ = \dot

so that the corresponding system is :

\begin
&\dot_ =  x_, \\
&\dot_ = -x_ + \varepsilon \left( \frac - \right).
\end

The origin

x_1= 0,\ x_2=0

is the only equilibrium point. Let us choose as a Lyapunov function :

V = \frac  \left(x_^+x_^ \right)

which is clearly positive definite. Its derivative is :

\dot = x_ \dot x_ + x_ \dot x_
= x_ x_ - x_ x_+\varepsilon
\frac - \varepsilon 
=   \varepsilon \frac -\varepsilon .

It seems that if the parameter

\varepsilon

is positive, stability is asymptotic for

x_^ < 3.

But this is wrong, since

\dot

does not depend on

x_1

, and will be 0 everywhere on the

x_1

axis. The equilibrium is Lyapunov stable but not asymptotically stable.

Barbalat's lemma and stability of time-varying systems

Assume that f is a function of time only. * Having

\dot(t) \to 0

does not imply that

f(t)

has a limit at

t\to\infty

. For example,

f(t)=\sin(\ln(t)),\; t>0

. * Having

f(t)

approaching a limit as

t \to \infty

does not imply that

\dot(t) \to 0

. For example,

f(t)=\sin\left(t^2\right)/t,\; t>0

. * Having

f(t)

lower bounded and decreasing (

\dot\le 0

) implies it converges to a limit. But it does not say whether or not

\dot\to 0

t \to \infty

. Barbalat's Lemma says: :If

f(t)

has a finite limit as

t \to \infty

and if

\dot

is uniformly continuous (or

\ddot

is bounded), then

\dot(t) \to 0

t \to\infty

. An alternative version is as follows: :Let

p\in,\infty)

and

q\in (1,\infty/math>. If f \in L^p(0,\infty) and \in L^q(0,\infty), then f(t)\to 0 as t\to \infty. In the following form the Lemma is true also in the vector valued case:

:Let f(t) be a uniformly continuous function with values in a Banach space E and assume that \textstyle\int_0^t f(\tau)\mathrm \tau has a finite limit as t\to \infty . Then f(t)\to 0 as t\to \infty . B. Farkas et al., Variations on Barbălat's Lemma, Amer. Math. Monthly (2016) 128, no. 8, 825-830, DOI: 10.4169/amer.math.monthly.123.8.825, p. 826. The following example is taken from page 125 of Slotine and Li's book ''Applied Nonlinear Control''.

Consider a non-autonomous system : \dot=-e + g\cdot w(t) : \dot=-e \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 \dot=-2e^2 \le 0. This says that V(t)\leq 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:

: \ddot= -4e(-e+g\cdot w) .

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

History

Definition for continuous-time systems

System of deviations

Lyapunov's second method for stability

Definition for discrete-time systems

Stability for linear state space models

Stability for systems with inputs

Example

Barbalat's lemma and stability of time-varying systems

See also

References

Further reading