Transformation of solutions and system.

The transformation of the kth derivative of a function in on variable is a

follows:

Y (k) =1/ k![ dky/dxk (x)] x = x 0, (1)

and the inverse transformation is defind by,

y (x) =S ∞k =0 Y (k)(x − x 0) k, (2)

The following theorems that can be deduced from equations (1) and (2) are

given below:

Theorem 1. If y (x) = y 1(x) ± y 2(x),then Y (k) = Y 1(k) ± Y 2(k).

Theorem 2. If y (x) = cy 1(x), then Y (k) = cY 1(k),where c is a constant.

Theorem 3. If y (x) = dny 1(x)/ dxn, then Y (k) = (k + n)! / k! Y 1(k + n).

Theorem 4. If y (x) = y 1(x) y 2(x), then Y (k) =S k,k 1=0 Y 1(k 1) Y 2(k − k 1).

Theorem 5. If y (x) = xn, then Y (k) = δ (k−n) where δ (k−n) = 1 k = n

0 k ¹ n

Theorem 6. If y (x) = eλx, then Y (k) = λk

k!,where λ is a constant.

Theorem 7. If y (x) = sin(ωx + α), then Y (k) = ωk

k! sin(kπ/ 2 + α), where ω and α constants.

Theorem 8. If y (x) = cos(ωx + α), then Y (k) = ωk/k! cos(kπ/ 2 + α), where ω and α constants.

 

Consider the following system of liner differential equations

 

y_ 1(x) = y 1(x) + y 2(x) (1)

y_ 2(x) = −y 1(x) + y 2(x)

 

with the conditions

y 1(0) = 0

y 2(0) = 1 (2)

By using Theorems 1,2 and 3 choosing x 0 = 0,equations (1) and (2) are

transformed as follows:

(k + 1) Y 1(k + 1) − Y 1(k) − Y 2(k) = 0

(k + 1) Y 2(k + 1) + Y 1(k) − Y 2(k) = 0

Y 1(0) = 0, Y 2(0) = 1,

consequently, we find

Y 1(1) = 1, Y 2(1) = 1

Y 1(2) = 1, Y 2(2) = 0

Y 1(3) = 1/3, Y 2(3) = − 1/3

Y 1(4) = 0, Y 2(3) = − 1/6

Y 1(5) = − 1/30, Y 2(5) = − 1/30.

...

...

Therefore, from(4),the solution of equation(11) is given by

y 1(x) = x + x 2 + 1/3 x 3 – 1/30 x 5 ± O (x 6),

y 2(x) = 1+ x – 1/3 x 3 – 1/6 x 4 ± O (x 5).

 

Basic concepts. Stability by Lyapunov. Geometric means.

Definition1. Stability in the sense of Lyapunov

The equilibrium point x*=of is stable(in the sense of Lyapunov) at t=t0 if for any e>0 there exists a b(t0;e)>0 such that ||x(t0)||<b => ||x(t)||<e, any t>t0.

Definition2.Asymptotic stability

An equilibrium point x*=0 of is asymptotically stable at t=t0 if

1.x*=0 is stable, and

2.x*=0 is locally attractive; i.e., there exists b(t0) such that ||x(t0)||<b =>limx(t)=0

Definition3. Exponential stability, rate of convergence

The equilibrium point x*=0 is an exponentially stable equilibrium point of if there exist constants m,a>0 and e>0 such that ||x(t)||≤m*exp(-a(t-t0))||x(t0)|| for all ||x(t0)|| ≤e and t≥t0. The largest constant a which may be utilized in ||x(t)||≤m*exp(-a(t-t0))||x(t0)|| is called the rate of convergence.

Exponential stability is a strong form of stability; in partial, it implies uniform, asymptotic stability. Exponential convergence is important in applications because it can be shown to be robust to perturbations and is essential for the consideration of more advanced control algorithms,

Orbital stability

The orbital stability differs from the Lyapunov stabilities in that it concerns with the stability of a system output (or state) trajectory under small external perturbations. n

Let f(x) be a p-periodic solution, p>0, of the autonomous system x(t)=f(x), x(t0)=x0?۠۠۠R and let Г represent the closed orbit of f(x) in the state space, namely, Г={y|y=f(x0),0≤t<p}

If, for any e>0, there exists a constant b=b(e)>0 such that d(x0, Г):=inf||x0-y||<b the solution of the system, f(x) satisfies d(f(x), Г) < e, for all t≥t0 then this p-periodic solution trajectory, f(x) is said to be orbitally stable.

 

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 nn 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.