Lyapunov stability

of a point relative to a family of mappings

$$ \tag{1 } \{ f _ {t} \} _ {t \in G ^ {+} } : E \rightarrow E $$

of a certain space $ E $

Equicontinuity of this family of mappings at the point (here $ G ^ {+} $ is the set of non-negative numbers in $ G $; for example, the real numbers $ G = \mathbf R $ or the integers $ G = \mathbf Z $). Lyapunov stability of a point relative to the family of mappings

is equivalent to the continuity at this point of the mapping $ x \mapsto x ( \cdot ) $ of a neighbourhood of this point into the set of functions $ x ( \cdot ) $ defined by the formula $ x ( t) = f _ {t} ( x) $, equipped with the topology of uniform convergence on $ G ^ {+} $. Lyapunov stability of a point relative to a mapping is defined as Lyapunov stability relative to the family of non-negative powers of this mapping. Lyapunov stability of a point relative to a dynamical system $ f ^ { t } $ is Lyapunov stability of this point relative to the family $ \{ f ^ { t } \} _ {t \in G ^ {+} } $. Lyapunov stability of the solution $ x _ {0} ( \cdot ) $ of an equation $ x ( t + 1 ) = g _ {t} ( t) $ given on $ t _ {0} + \mathbf Z ^ {+} $ is Lyapunov stability of the point $ x _ {0} ( t _ {0} ) $ relative to the family of mappings $ \{ f _ {t} = g _ {t _ {0} + t } \dots g _ {t _ {0} } \} _ {t \in \mathbf Z ^ {+} } $.

Lyapunov stability of the solution $ x _ {0} ( \cdot ) $ of a differential equation $ \dot{x} = f ( x , t ) $ given on $ t _ {0} + \mathbf R ^ {+} $ is Lyapunov stability of the point $ x _ {0} ( t _ {0} ) $ relative to the family of mappings $ \{ X ( t _ {0} + t , t _ {0} ) \} _ {t \in \mathbf R ^ {+} } $, where $ X ( \theta , \tau ) $ is the Cauchy operator of this equation. Lyapunov stability of the solution $ y ( \cdot ) $ of a differential equation

$$ y ^ {(} m) = g ( y , \dot{y} \dots y ^ {(} m- 1) , t ) $$

of order $ m $, given on $ t _ {0} + \mathbf R ^ {+} $, is Lyapunov stability of the solution $ x ( \cdot ) = ( y ( \cdot ) , \dot{y} ( \cdot ) \dots y ^ {(} m- 1) ( \cdot ) ) $ of the corresponding first-order differential equation $ \dot{x} = f ( x , t ) $, given on $ t _ {0} + \mathbf R ^ {+} $, where

$$ x = ( x _ {1} \dots x _ {m} ) , $$

$$ f ( x , t ) = ( x _ {2} \dots x _ {m} , g ( x _ {1} \dots x _ {m} , t ) ) . $$

The definitions 1–7 given below are some concrete instances of the above and related definitions.

1) Let a differential equation $ \dot{x} = f ( x , t ) $ be given, where $ x $ lies in an $ n $- dimensional normed space $ E $. A solution $ x _ {0} ( \cdot ) : t _ {0} + \mathbf R ^ {+} \rightarrow E $ of this equation is called Lyapunov stable if for every $ \epsilon > 0 $ there exists a $ \delta > 0 $ such that for every $ x \in E $ satisfying the inequality $ | x - x _ {0} ( t _ {0} ) | < \delta $, the solution $ x ( \cdot ) $ of the Cauchy problem

$$ \dot{x} = f ( x , t ) ,\ \ x ( t _ {0} ) = x $$

is unique, defined on $ t _ {0} + \mathbf R ^ {+} $, and for each $ t \in t _ {0} + \mathbf R ^ {+} $ satisfies the inequality $ | x ( t) - x _ {0} ( t) | < \epsilon $. If, in addition, one can find a $ \delta _ {0} > 0 $ such that for every solution $ x ( \cdot ) $ of the equation $ \dot{x} = f ( x , t ) $ whose initial value satisfies the inequality

$$ | x ( t _ {0} ) - x _ {0} ( t _ {0} ) | < \delta _ {0} , $$

the equation

$$ \lim\limits _ {t \rightarrow + \infty } | x ( t) - x _ {0} ( t) | = 0 $$

holds (respectively, the inequality

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ \frac{1}{t} \mathop{\rm ln} | x ( t) - x _ {0} ( t) | < 0 $$

holds; here and elsewhere one puts $ \mathop{\rm ln} 0 = \infty $), then the solution $ x _ {0} ( \cdot ) $ is called asymptotically (respectively, exponentially) stable.

A solution of the equation

$$ \tag{2 } \dot{x} = f ( x , t ) , $$

where $ x \in \mathbf R ^ {n} $ or $ x \in \mathbf C ^ {n} $, is called Lyapunov stable (asymptotically, exponentially stable) if it becomes such after equipping the space $ \mathbf R ^ {n} $( or $ \mathbf C ^ {n} $) with a norm. This property of the solution does not depend on the choice of the norm.

2) Let a mapping $ f : S \rightarrow S $ be given, where $ ( S , d ) $ is a metric space. The point $ x _ {0} \in S $ is called Lyapunov stable relative to the mapping $ f $ if for every $ \epsilon > 0 $ there exists a $ \delta > 0 $ such that for any $ x \in S $ satisfying the inequality $ d ( x , x _ {0} ) < \delta $, the inequality

$$ d ( f ^ { t } x , f ^ { t } x _ {0} ) < \epsilon $$

holds for each $ t \in \mathbf N $. If, moreover, one can find a $ \delta _ {0} > 0 $ such that for each $ x \in S $ satisfying $ d ( x , x _ {0} ) < \delta _ {0} $ one has the equation

$$ \lim\limits _ {t \rightarrow + \infty } \ d ( f ^ { t } x , f ^ { t } x _ {0} ) = 0 $$

(the inequality

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ \frac{1}{t} \mathop{\rm ln} d ( f ^ { t } x , f ^ { t } x _ {0} ) < 0 , $$

respectively), then the point $ x _ {0} $ is called asymptotically (respectively, exponentially) stable relative to $ f $.

Let $ f $ be a mapping from a compact topological space $ S $ into itself. A point $ x _ {0} \in S $ is called Lyapunov stable (asymptotically stable) relative to $ f $ if it becomes such after equipping $ S $ with a metric. This property of the point does not depend on the choice of the metric.

If $ S $ is a compact differentiable manifold, then a point $ x _ {0} \in S $ is called exponentially stable relative to a mapping $ f : S \rightarrow S $ if it becomes such after equipping $ S $ with a certain Riemannian metric. This property of the point does not depend on the choice of the Riemannian metric.

3) Suppose that a differential equation (2) is given, where $ x $ lies in a topological vector space $ E $. A solution $ x _ {0} ( \cdot ) : t _ {0} + \mathbf R ^ {+} \rightarrow E $ of this equation is called Lyapunov stable if for each neighbourhood of zero $ U \subset E $ there is a neighbourhood $ V $ of $ x _ {0} ( t _ {0} ) $ in $ E $ such that for every $ x \in E $ the solution $ x ( \cdot ) $ of the Cauchy problem (2), $ x ( t _ {0} ) = x $, is unique, defined on $ t _ {0} + \mathbf R ^ {+} $ and satisfies the relation $ x ( t)- x _ {0} ( t) \in U $ for all $ t \in t _ {0} + \mathbf R ^ {+} $. If, in addition, one can find a neighbourhood $ V _ {0} \subset E $ of the point $ x _ {0} ( t _ {0} ) $ such that for every solution $ x ( \cdot ) $ of (2) satisfying $ x ( t _ {0} ) \in V _ {0} $ one has the equation

$$ \lim\limits _ {t \rightarrow + \infty } ( x ( t) - x _ {0} ( t) ) = 0 $$

(respectively,

$$ \lim\limits _ {t \rightarrow + \infty } \ e ^ {\alpha t } ( x ( t) - x _ {0} ( t) ) = 0 $$

for a certain $ \alpha > 0 $), then the solution $ x _ {0} ( \cdot ) $ is called asymptotically (respectively, exponentially) stable. If $ E $ is a normed space, then this definition may be formulated as in 1 above, if as norm $ | \cdot | $ one takes any norm compatible with the topology on $ E $.

4) Let a differential equation (2) be given on a Riemannian manifold $ U $( for which a Euclidean or a Hilbert space can serve as a model) or, in a more general situation, on a Finsler manifold $ U $( for which a normed space can serve as a model); the distance function in $ U $ is denoted by $ d ( \cdot , \cdot ) $. A solution $ x _ {0} ( \cdot ) : t _ {0} + \mathbf R ^ {+} \rightarrow U $ of this equation is called Lyapunov stable if for each $ \epsilon > 0 $ one can find a $ \delta > 0 $ such that for each $ x \in U $ satisfying $ d ( x , x _ {0} ( t _ {0} ) ) < \delta $, the solution $ x ( \cdot ) $ of the Cauchy problem (2), $ x ( t _ {0} ) = x $, is unique, defined for $ t _ {0} + \mathbf R ^ {+} $ and satisfies the inequality $ d ( x ( t) , x _ {0} ( t) ) < \epsilon $ for all $ t \in t _ {0} + \mathbf R ^ {+} $. If, in addition, one can find a $ \delta _ {0} > 0 $ such that for every solution $ x _ {0} ( \cdot ) $ of (2) whose initial value satisfies the inequality $ d ( x ( t _ {0} ) , x _ {0} ( t _ {0} ) ) < \delta _ {0} $ one has the equation

$$ \lim\limits _ {t \rightarrow + \infty } d ( x ( t) , x _ {0} ( t) ) = 0 $$

(the inequality

$$ \overline{\lim\limits}\; _ {t \rightarrow + \infty } \ \frac{1}{t} \mathop{\rm ln} d ( x ( t) , x _ {0} ( t) ) < 0 , $$

respectively), then the solution $ x _ {0} ( \cdot ) $ is called asymptotically (respectively, exponentially) stable.

Suppose that the differential equation (2) is given on a compact differentiable manifold $ V ^ {n} $. A solution of this equation is called Lyapunov stable (asymptotically, exponentially stable) if it becomes such when the manifold $ V ^ {n} $ is equipped with some Riemannian metric. This property of the solution does not depend on the choice of the Riemannian metric.

5) Let $ E $ be a uniform space. Let

$$ f _ {t} : U \rightarrow E ,\ \ t \in G ^ {+} \ \ ( G = \mathbf R \textrm{ or } = \mathbf Z ) , $$

be a mapping defined on an open set $ U \subset E $. A point $ x _ {0} \in U $ is called Lyapunov stable relative to the family of mappings $ \{ f _ {t} \} _ {t \in G ^ {+} } $ if for every entourage $ W $ there exists a neighbourhood $ V $ of $ x _ {0} $ such that the set of all $ x \in U $ satisfying $ ( f _ {t} x , f _ {t} x _ {0} ) \in W $ for all $ t \in G ^ {+} $, is a neighbourhood of $ x _ {0} $. If, in addition, there exists a neighbourhood $ V _ {0} $ of $ x _ {0} $ such that for every $ x \in V _ {0} $ and every entourage $ W $ one can find a $ t ( x , w ) \in G ^ {+} $ such that $ ( f _ {t} x , f _ {t} x _ {0} ) \in W $ for all $ t \in t ( x , w ) + \mathbf R ^ {+} $, then the point $ x _ {0} $ is called asymptotically stable.

If $ E $ is a compact topological space and $ f _ {t} : U \rightarrow E $, $ t \in G ^ {+} $, is a mapping given on some open set $ U \subset E $, then the point $ x _ {0} \in U $ is called Lyapunov stable (asymptotically stable) relative to the family of mappings $ \{ f _ {t} \} _ {t \in G ^ {+} } $ if it becomes such after the space $ E $ is equipped with the unique uniform structure that is compatible with the topology on $ E $.

6) Let $ E $ be a topological space and $ U $ an open subspace in it. Let $ f _ {t} : U \rightarrow E $, $ t \in G ^ {+} $, where $ G $ is $ \mathbf R $ or $ \mathbf Z $, be a mapping having $ x _ {0} $ as fixed point. The fixed point $ x _ {0} $ is called Lyapunov stable relative to the family of mappings $ \{ f _ {t} \} _ {t \in G ^ {+} } $ if for every neighbourhood $ V $ of $ x _ {0} $ there exists a neighbourhood $ W $ of the same point such that $ f _ {t} W \subset V $ for all $ t \in G ^ {+} $. If, in addition, there exists a neighbourhood $ V _ {0} $ of $ x _ {0} $ such that $ \lim\limits _ {t \rightarrow + \infty } f _ {t} x = x _ {0} $ for every $ x \in V _ {0} $, then the point $ x _ {0} $ is called asymptotically stable relative to the family of mappings $ \{ f _ {t} \} _ {t \in G ^ {+} } $.

7) Lyapunov stability (asymptotic, exponential stability) of a solution $ y _ {0} ( \cdot ) $ of an equation of arbitrary order, $ y ^ {(} m) = g ( y , \dot{y} \dots y ^ {(} m- 1) , t ) $, is understood to mean Lyapunov stability (respectively asymptotic, exponential stability) of the solution $ x _ {0} ( \cdot ) = ( y _ {0} ( \cdot ) , \dot{y} _ {0} ( \cdot ) \dots y _ {0} ^ {(} m- 1) ( \cdot ) ) $ of the corresponding first-order equation (2), where $ x = ( x _ {1} \dots x _ {m} ) $, $ f ( x , t ) = ( x _ {2} \dots x _ {m} , g ( x _ {1} \dots x _ {m} , t ) ) $.

Definitions 1, 2, 4, 6, 7 include stable motions of systems with a finite number of degrees of freedom (where the equations on manifolds arise naturally when considering mechanical systems with a constraint). Definitions 2–7 include stable motions in the mechanics of continuous media and in other parts of physics, stable solutions of operator equations, functional-differential equations (in particular, equations with retarded arguments) and other equations.

Study of the stability of an equilibrium position of an autonomous system.

Let $ \dot{x} = f ( x) $ be an autonomous differential equation defined in a neighbourhood of a point $ x _ {0} \in \mathbf R ^ {n} $, where the function $ f ( \cdot ) $ is continuously differentiable and vanishes at this point. If the real parts of all eigen values of the derivative $ d f _ {x _ {0} } $ are negative, then the fixed point $ x _ {0} $ of $ \dot{x} = f ( x) $ is exponentially stable (Lyapunov's theorem on stability in a first approximation); to facilitate the verification of the condition in this theorem one applies criteria for stability. If under these conditions at least one of the eigen values of the derivative $ d f _ {x _ {0} } $ has positive real part (this condition may be checked without finding the eigen values themselves, cf. Stability criterion), then the fixed point of the differential equation $ \dot{x} = f ( x) $ is unstable.

Example. The equation of the oscillation of a pendulum with friction is

$$ \dot{y} dot + a \dot{y} + b \sin y = 0 ,\ \ a , b > 0 . $$

The lower equilibrium position $ y = \dot{y} = 0 $ is exponentially stable, since the roots of the characteristic equation $ \lambda ^ {2} + a \lambda + b = 0 $ of the variational equation (cf. Variational equations) have negative real parts. The upper equilibrium position $ y = \pi , \dot{y} = 0 $ is unstable, since the characteristic equation $ \lambda ^ {2} + a \lambda - b = 0 $ of the variational equation $ \dot{y} dot + a \dot{y} - b y = 0 $ has a positive root. This instability takes place even in the absence of friction $ ( a = 0 ) $. The lower equilibrium position of a pendulum without friction is one of the so-called critical cases, when all eigen values of the derivative $ d f _ {x _ {0} } $ are contained in the left complex half-plane, and at least one of them lies on the imaginary axis.

For the study of stability in critical cases, A.M. Lyapunov proposed the so-called second method for studying stability (cf. Lyapunov function). For a pendulum without friction,

$$ \dot{y} dot + b \sin y = 0 ,\ b > 0 , $$

the lower equilibrium position is Lyapunov stable, since there exists a Lyapunov function

$$ V ( y , \dot{y} ) = \frac{1}{2} \dot{y} ^ {2} + b ( 1 - \cos y ) $$

— the total energy of the pendulum; the condition of non-positivity for the derivative of this function is a consequence of the law of conservation of energy.

A fixed point $ x _ {0} $ of a differentiable mapping $ f : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ is exponentially stable relative to $ f $ if all eigen values of the derivative $ d f _ {x _ {0} } $ are less than 1 in modulus, and it is unstable if at least one of them has modulus $ > 1 $.

The study of the stability of periodic points of differentiable mappings reduces to the study of stability of fixed points relative to the powers of these mappings. Periodic solutions of autonomous differential equations are not asymptotically stable (cf. Orbit stability; Andronov–Witt theorem).

It should not be believed that exponential stability of the null solution of the variational equation of the autonomous differential equation $ \dot{x} = f ( x) $ along a solution $ x ( \cdot ) $ implies stability of the solution. This is shown by Perron's example (cf. [2], [3]):

$$ \tag{3 } \left . \begin{array}{c} \dot{u} = - a u , \\ \dot{v} = ( {\sin \mathop{\rm ln} } t + {\cos \mathop{\rm ln} } t - 2 a ) v + u ^ {2} ; \\ \end{array} \right \} $$

for $ a > 1 / 2 $ the null solution of the system of variational equations,

$$ \tag{4 } \left . \begin{array}{c} \dot{u} = - a u , \\ \dot{v} = ( {\sin \mathop{\rm ln} } t + {\cos \mathop{\rm ln} } t - 2 a ) v, \\ \end{array} \right \} $$

of the system (3) (along the null solution) is exponentially stable (the Lyapunov characteristic exponents of the system (4) are $ - a , 1 - 2 a $, cf. Lyapunov characteristic exponent), but for $ a \in ( 1 / 2 , ( 2 + e ^ {- \pi } ) / 4 ) $ the null solution of the system (3) is unstable. However, stability in the first approximation is typical, in a sense explained below.

Let $ S $ be the set of diffeomorphisms $ f $ of a Euclidean space $ E ^ {n} $ onto itself having uniformly continuous derivatives that satisfy the inequality

$$ \sup _ {x \in E ^ {n} } \ \max \{ \| d f _ {x} \| , \| ( d f _ {x} ) ^ {-} 1 \| \} \ < + \infty . $$

For each diffeomorphism $ j \in S $ denote by $ S _ {j} $ the set of all $ f \in S $ satisfying the inequality

$$ \sup _ {x \in E ^ {n} } | f x - j x | < + \infty ; $$

one endows $ S _ {j} $ with the distance function

$$ d ( f , g ) = \ \sup _ {x \in E ^ {n} } ( | f x - g x | + \| d f _ {x} - d g _ {x} \| ) . $$

For each $ j \in S $ there is in $ S _ {j} \times E ^ {n} $ an everywhere-dense set $ D _ {j} $ of type $ G _ \delta $ with the following property: If $ ( f , x ) \in D _ {j} $ is such that for every $ g \in T _ {x} E ^ {n} $ the inequality

$$ \overline{\lim\limits}\; _ {m \rightarrow + \infty } \ \frac{1}{m} \mathop{\rm ln} | d f ^ { m } g | < 0 $$

holds, then there is a neighbourhood $ U $ of $ ( f , x ) $ in $ S _ {j} \times E ^ {n} $ such that for every $ ( g , y ) \in U $ the point $ y $ is exponentially stable relative to the diffeomorphism $ g $.

For a dynamical system given on a compact differentiable manifold, an analogous theorem can be formulated more simply and as a differential-topologically invariant statement. Let $ V ^ {n} $ be a closed differentiable manifold. The set $ S $ of all diffeomorphisms $ f $ of class $ C ^ {1} $ mapping $ V ^ {n} $ to $ V ^ {n} $ can be equipped with the $ C ^ {1} $- topology. In the space $ S \times V ^ {n} $ there is an everywhere-dense set $ D $ of type $ G _ \delta $ with the following property: If for an $ ( f , x ) \in D $ the inequality

$$ \lim\limits _ {m \rightarrow + \infty } \ \frac{1}{m} \mathop{\rm ln} | d f ^ {m} g | < 0 $$

holds for all $ g \in T _ {x} V ^ {n} $, then there is a neighbourhood $ U $ of $ ( f , x ) $ in $ S \times V ^ {n} $ such that for each $ ( g , y ) \in U $ the point $ y $ is exponentially stable relative to the diffeomorphism $ g $.

The concepts of Lyapunov stability, asymptotic stability and exponential stability were introduced by Lyapunov [1] in order to develop methods for studying stability in the sense of these definitions (cf. Lyapunov stability theory).

References

[1]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2]	O. Perron, "Ueber Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen" Math. Z. , 29 (1928) pp. 129–160
[3]	R.E. Bellman, "Stability theory of differential equations" , Dover, reprint (1969)

Comments

For stability questions of differential equations with discontinuous right-hand sides cf. [a6].

References

[a1]	J.P. Lasalle, S. Lefschetz, "Stability by Lyapunov's direct method with applications" , Acad. Press (1961)
[a2]	M.W. Hirsch, S. Smale, "Differential equations, dynamical systems, and linear algebra" , Acad. Press (1974)
[a3]	W. Hahn, "Stability of motion" , Springer (1967) pp. 422
[a4]	N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36
[a5]	M.I. Rabinovich, D.I. Trubetskov, "Oscillations and waves in linear and nonlinear systems" , Kluwer (1989) pp. Chapt. 6–7 (Translated from Russian)
[a6]	A.F. Filippov, "Differential equations with discontinuous righthand sides" , Kluwer (1989) pp. §15 (Translated from Russian)