Sufficient conditions for the existence of Laplace Transform of a function:

The function f(t) must satisfy the following conditions for the existence of the Laplace Transform

1. The function f(t) must be piecewise continuous in any limited intervals like 0 < a ≤ t ≤ b.
2. The function f(t) is of exponential order like 

F (t) ={t2, 0 < t < 5 or 2t, t > 5 is a piece – wise continuous for t > 0.

1. F (t) = 1/t is not a piece – wise continuous in any intervals conditioning t = 0
2. The function f(t) is said to be of exponential order ‘a’ if e e-atf(t) = 0 = finite quantity. F(t) = t2 , sin at, eat are of exponential order and continuous.
3. F (t) = et2 is not of exponential order and its Laplace transform doesn’t exist.

Different Methods of Laplace Transform:

Laplace transform is an important part of control system engineering. To study and analyze the control system we have many different Laplace and inverse Laplace transform functions. The different types of methods that are used by control system engineers are

• Initial Value Theorem (IVT): 

• • It is the 1st basic property of Laplace transform.
  • It is used to find the initial value at a time of a given function to transform without working hard to find f (t).
    • Conditions for Initial Value Theorem:
      • The function f (t) and its derivative should be Laplace transformable.
      • If time t approaches 0+ then f (t) should exist.
• If f (t) and f (s) are a pair. i.e. f (t)🡨 L 🡪f (s), then Initial Value Theorem is given by 

ft= sFs=f(0+) 

• Final Value Theorem (FVT):

• The final value theorem is used to find the solutions without knowing its function which inverse Laplace transforms.
• If both the functions are Laplace Transformable and g (f(s)) have no point on the axis then

 gfs=  f(t)

Properties of Laplace Transform:

• Linear Property:

If L (f (t)) = f(s) and L (g (t)) = g(s) then 

L [c1 f (t) + c2 g (t)] = c1 (L (f (t))) + c2 (L (f (t))) = c1 f(s) + c2 f(s).

• First Shifting Theorem: