Asymptotic Stability
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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, ...
s or
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s 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 i ...
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 is
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 In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
which do not exhibit asymptotic stability.


History

Lyapunov stability is named after
Aleksandr Mikhailovich 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, Liap ...
, 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 Nikolay Gur'yevich Chetaev (23 November 1902 – 17 October 1959) was a Russian Soviet mechanician and mathematician. He was born in Karaduli, Laishevskiy uyezd, Kazan province, Russian Empire (now Tatarstan of Russian Federation) and died in Mos ...
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 The Cold War is a term commonly used to refer to a period of geopolitical tension between the United States and the Soviet Union and their respective allies, the Western Bloc and the Eastern Bloc. The term '' cold war'' is used because t ...
period when the so-called "Second Method of Lyapunov" (see below) was found to be applicable to the stability of aerospace
guidance system A guidance system is a virtual or physical device, or a group of devices implementing a controlling the movement of a ship, aircraft, missile, rocket, satellite, or any other moving object. Guidance is the process of calculating the changes in po ...
s 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 In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with ini ...
(related to Lyapunov's First Method of discussing stability) has received wide interest in connection with
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
. 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 as 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 se ...
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 heat a ...
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
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 se ...
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 settin ...
and ''f'' : ''X'' → ''X'' a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
. 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 toy ...
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 b ...
s of A are negative. This condition is equivalent to the following one: :A^\textsfM + MA is negative definite for some
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
matrix M = M^\textsf. (The relevant Lyapunov function is V(x) = x^\textsfMx.) Correspondingly, a time-discrete 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 toy ...
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 abstraction o ...
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 system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
s)


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 dynamical system, dynamics, the Van der Pol oscillator is a Conservative force, non-conservative oscillator with nonlinearity, non-linear Damping ratio, damping. It evolves in time according to the second-order differential equation: :-\mu(1-x ...
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 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
. 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 as 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 as 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.


See also

*
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 se ...
*
LaSalle's invariance principle LaSalle's invariance principle (also known as the invariance principle, Barbashin-Krasovskii-LaSalle principle, or Krasovskii-LaSalle principle) is a criterion for the asymptotic stability of an autonomous (possibly nonlinear) dynamical system. Gl ...
* Lyapunov–Malkin theorem *
Markus–Yamabe conjecture In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks ...
* Libration point orbit *
Hartman–Grobman theorem In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that lineari ...
*
Perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...


References


Further reading

* * * * * * * {{Authority control Stability theory Dynamical systems Lagrangian mechanics Three-body orbits