28. The main Lyapunov theorems.

Basic theorem of Lyapunov

Let V(x,t) be a non-negative function with derivative V’ along the trajectories of the system.

1.If V(x,t) is locally positive definite and V’(x,t) ≤0 locally in x and for all t, then the origin of the system is locally stable(in the sense of Lyapunov)

2.If V(x,t) is locally positive definite and decrescent,and V’(x,t) ≤0 locally in x and for all t, then the origin of the system is uniformly locally stable(in the sense of Lyapunov)

3. If V(x,t) is locally positive definite and decrescent, and -V’(x,t) ≤0 locally positive definite, then the origin of the system is uniformly locally asymptotically stable

4.If V(x,t) is locally positive definite and decrescent, and -V’(x,t) ≤0 locally positive definite, then the origin of the system is uniformly locally asymptotically stable

Theorem. Exponential stability theorem

x*=0 is an exponentially stable equilibrium point of x’=f(x,t) if and only if there exists an e>0 and a function V(x,t) which satisfies

29.Chetaev theorem n n n

Theorem.[adapted from Chetaev 1934].Suppose E € R is open, 0 € E, f;E→R and V: E→R are continuously differentiable, f(0)=0, V(0)=0. Suppose U an open neighborhood of 0 with compact closure F=U€E such that the restriction of V’=(DV)f to F∩V pow-1(0,∞) is strictly positive. If for every open neighborhood W€R pow n of 0 the set W∩W pow -1 (0,∞) is nonempty , then the origin is an unstable equilibrium of x’=f(x).

Proof:

1.Assume above hypotheses

2.Let W be an open neighborhood of 0€R pow n

3There exists z€W∩V pow -1(0,∞).Define a=V(z)>0

4.The set K=F∩ V pow -1[a,∞) is compact, and it is nonempty since z€K

5Using the contiuity of V and V’ there exist m=minV’(x) and M=maxV(x)

6 Since K=F∩ V pow -1[a,∞) € F∩ V pow -1[a,∞) it follows that m>0

7 Let I € [0,∞) be the maximum interval of existence for empty set:I→E such that empty(0)=z

8 Define the set T={t€ I: for all s€[0,t], empty set(s)=K }

9.For all t€ T d/dt(V*empty set)(t)=(V’*empty set)(t)>m

10Hence for all t€ T V(empty set(t))=V(empty set(0))+∫V(empty(s))ds≥a+mt

11 Since V is bounded above by M,T is also bounded above.Let t0 the least bounded of T.

12 Since K €E,I [0,t0] and empty(t0) is well-defined. Moreover t0 lies in the interior of I.

13 Since for all t€[0,t0),empty(t0) €K and for every b>0 there exist t€I,t0<t<t0+b such that it follows that empty(t0) lies on the boundary of the set K.

14 The boundary of K is contained in the union ӘK€V pow -1(a)U (F\U)

15 Since V(empty(t0))>a+m*t0>a it follows that empty(t0) € not V(a)pow -1

16 Therefore empty(t0) €(F\U) and in particular empty(t0) € not U

17 Since W was arbitrary this shows that the origin is an unstable equilibrium point of f.

30) Studying stability by Lyapunov function.

Consider a dynamical system which satisfies

= f(x, t) x( ) = x ∈

We will assume that f(x, t) satisfies the standard conditions for the existence and uniqueness of solutions. Such conditions are, for instance, that f(x, t) is Lipschitz continuous with respect to x, uniformly in t, and piecewise continuous in t. A point ∈ is an equilibrium point of if f( , t) ≡ 0. Intuitively and somewhat crudely speaking, we say an equilibrium point is locally stable if all solutions which start near (meaning that the initial conditions are in a neighborhood of ) remain near for all time. The equilibrium point is said to be locally asymptotically stable if is locally stable and, furthermore, all solutions starting near tend towards as t → ∞. By shifting the origin of the system, we may assume that the equilibrium point of interest occurs at x= 0. If multiple equilibrium points exist, we will need to study the stability of each by appropriately shifting the origin.

The equilibrium point = 0 is stable (in the sense of Lyapunov)

at t = if for any e > 0 there exists a δ( , e) > 0 such that

||x(t0)|| < δ =⇒ ||x(t)|| < e, ∀t ≥ .

Lyapunov stability is a very mild requirement on equilibrium points.