26) 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=x0, (1)

and the inverse transformation is defind by,

y(x) =∞k=0Y (k)(x − x0)k, (2)

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

given below:

Theorem 1.If y(x) = y1(x) ± y2(x),then Y (k) = Y1(k) ± Y2(k).

Theorem 2.If y(x) = cy1(x), then Y (k) = cY1(k) ,where c is a constant.

Theorem 3.If y(x) = dny1(x)/dxn , then Y (k) = (k+n)! / k! Y1(k + n).

Theorem 4.If y(x) = y1(x)y2(x), then Y (k) =k,k1=0 Y1(k1)Y2(k − k1).

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

0 k  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) = y1(x) + y2(x) (1)

y_2(x) = −y1(x) + y2(x)

with the conditions

y 1(0) = 0

y2(0) = 1 (2)

By using Theorems 1,2 and 3 choosing x0 = 0,equations (1) and (2) are

transformed as follows:

(k + 1)Y1(k + 1) − Y1(k) − Y2(k) = 0

(k + 1)Y2(k + 1) + Y1(k) − Y2(k) = 0

Y1(0) = 0, Y2(0) = 1,

consequently, we find

Y1(1) = 1, Y2(1) = 1

Y1(2) = 1, Y2(2) = 0

Y1(3) = 1/3, Y2(3) = −1/3

Y1(4) = 0, Y2(3) = −1/6

Y1(5) = −1/30, Y2(5) = −1/30 .

...

...

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

y1(x) = x + x2 + 1/3x3 – 1/30x5 ± O(x6),

y2(x) = 1+x – 1/3x3 – 1/6x4 ± O(x5).

27. 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.