Lyapunov Stability
Zhao Zhao
  2022-09-28 22:39:06 2022-09-28 22:39    

Lyapunov Stability

Autonomous System

Consider the autonomous system where is a locally Lipschitz map from a domain into and there is at least one equilibrium point , that is .

Goal: Stability analysis of the equilibrium point .

Without loss of generality, we consider that . If it is not, then consider the change of variables . Then

Definition:

The equilibrium point is

  • stable if, for each , there is such that

  • unstable if it is not stable

  • asymptotically stable if it is stable and can be chosen such that

A function is said to be

  • positive definite if and
  • positive semidefinite if and
  • negative definite (resp. negative semi definite) if is definite positive (resp. definite semi positive).

In particular, for (quadratic form), where is a real symmetric matrix, is positive (semi)definite if and only if all the eigenvalues of are positive (nonnegative), which is true if all leading principal minors (顺序主子式) of are positive (all principal minors of are nonnegative). (阶矩阵 为正定矩阵的充要条件是A的所有顺序主子式 , 为实对称矩阵)

Example： The leading principal minors of are Therefore, if .

Lyapunov's stability theorem

Theorem: Lyapunov's stability theorem

Let be an equilibrium point for and be a domain containing . Let be a continuously differentiable function such that

Then, is stable. Moreover, if

then is asymptotically stable.

Examples: Consider the Pendulum example without friction Assume the following energy function Clearly, and . Thus the origin is stable. Since , we can also conclude that the origin is not asymptotically stable.

Examples: Pendulum equation, but this time with friction Consider

Then, which is negative semidefinite. why? because for irrespective of the value of . We can only conclude that the origin is stable! (use Corollary of La Salle's Theorem)

Theorem: Globally Asymptotically Stability (GAS)

Let be an equilibrium point for and be a domain containing . Let be a continuously differentiable function such that

Then, is globally asymptotically stable (GAS).

La Salle's Theorem

Theorem 4.11 - La Salle's Theorem

Let

  • be a compact positively invariant set.
  • be a continuously differentiable function such that in .
  • be the largest invariant set in .

Then every solution starting in approaches as .

Note that is not needed to be positive definite.

When is the origin?

Corollary 4.1

Let be an equilibrium point of . Let be a positive definite function containing the origin such that in . Let and suppose that no solution can stay identically in , other than the trivial solution . Then, the origin is asymptotically stable.

Corollary 4.2

Let be an equilibrium point of . Let be a , radially unbounded, positive definite function such that for all . Let and suppose that no solution can stay identically in , other than the trivial solution . Then, the origin is global asymptotically stable (GAS).

Example: Consider Clearly and as . i.e, negative semidefinite. Is it GAS?

Let be a solution that belongs identically to : Therefore, the only solution that can stay identically in is the trivial solution . Thus, is GAS.

Comparison Functions

Definition:

A continuous function is said to belong to class if it is strictly increasing and . It is of class if and as .

Definition:

A continuous function is said to belong to class if, fore each fixed , the mapping belong to class with respect to and, for each fixed , the mapping is decreasing with respect to and as .

Examples:

Class : Class : Class :

Lemma 4.4

Consider the scalar autonomous differential equation where is a locally Lipschitz class function defined on . Then, for all , the solution is unique and defined for all . Moreover, where is defined on .

  1. the solution is given by and we can define

  2. the solution is given by and we can dene

Nonautonomous Systems

考虑如下系统 如上式所示的 中显含时间 的系统就是非自治系统（nonautonomous system），也称时变系统（time varying system). 如果不显含时间 t ,即 则系统被称为自治系统（autonomous system），也称为时不变系统（time invariant system）。

注1：决定自治和非自治的是 是否“显含时间 ”。注意这里的描述是“显含”而不是“不含”。事实上， 本身就是时间的函数，即 ，也就是说系统(1)的完整描述是 ，系统（2）的完整描述是 。 中包含的时间 对于系统来说不是显含的，而是隐含的，所以 中所包含的时间 与判断系统是否自治无关。

注2：如果 是线性的，那么系统（1）就是线性非自治系统（线性时变系统），系统（2）就是线性自治系统（线性时不变系统）

Consider the nonautonomous system Definition: The origin is an equilibrium point at if

The equilibrium point x = 0 of the nonautonomous system is

  • Stable if

  • Uniformly Stable (US) if is independent of

  • Asymptotically Stable if it is stable and as

  • Uniformly Asymptotically Stable (UAS) if is independent of and the convergence is uniformly in , that is

  • Globally Uniformly Asymptotic Stable (GUAS) if can be chosen to satisfies and

The following lemma make the above definitions more clear.

Lemma 4.5

The equilibrium point of the nonautonomous system is

  • US there exist and (independent of ) such that

  • UAS there exist and (independent of ) such that

  • GUAS the inequality in UAS holds for all , that is

Definition

The equilibrium point is exponentially stable if there exists positive constants and such that and globally exponentially stable if this inequality is satisfied

Theorem 4.8:

Let be an equilibrium point that belongs to . Let be a function such that and where are continuous positive functions on . Then, is uniformly stable.

Theorem 4.9 Same assumptions as Theorem 4.8 but with where is a continuous positive definite function on . Then, is UAS.

Moreover, if and are such that Then for some .

If in addition and is radially unbounded then is GUAS.

Definition:

  • is said to be positive semidefinite if
  • is said to be positive definite if , for some positive definite function
  • is said to be unbounded if is radially unbounded.

Theorem 4.10:

Let be an equilibrium point for the nonautonomous system that belongs to some . Let be a function such that where and a are positive constants. Then is exponentially stable. If the assumption holds globally. Then is GES.

Proof: Hence, Examples 1 : Consider Then, and , thus is GUAS. Note we cannot conclude exponential because is not the same .

Example 2 Consider Note that Thus is positive definite and radially unbounded. Therefore, , and are positive definite quadratic functions () and so we conclude that is GES.

Boundedness and Ultimate Boundedness

Until now we have used Lyapunov theory to study the behavior of the system about the equilibrium point. What happens when the system does not have any equilibrium point? We will see that Lyapunov analysis can be used to show boundedness of the solution of the state equation.

Example: There are no equilibrium points!

Nevertheless, with Note that , which means that the set with is an invariant set because . Hence, the solutions are uniformly bounded.

Moreover for some in the set which shows that will reach in finite time and the solution enter the set .

Thus, we can conclude that the solution is uniformly ultimately bounded with ultimate bound .

Definition:

The solutions of are

  • Uniformly Bounded (UB) if there exists a , independent of such that

  • Globally Uniformly Bounded (GUB) if can be arbitrarily large

  • Uniformly Ultimately Bounded (UUB) with ultimate bound , if there exists (independent of ) such that

  • Globally Uniformly Ultimately Bounded (GUUB) if can be arbitrarily large.

Theorem 4.18 (Global)

Let be a function such that where and . Then, there exists a and such that and

Example - Mass-spring system and assuming certain numerical values we get Choosing Then Therefore which shows that the solutions are GUUB. To compute the ultimate bound, we have to find .

Let which ensures that

But Therefore