Stability
The concept of stability is the idea that any bounded input into a function
yields a bounded output. If one puts an unbounded input in and it yields
an unbounded output this is fine since the definition says the input must
be bounded. By this definition if every bounded input yields a bounded output
the conditions are satisfied. However by definition a stronger condition is
that the integration or summation of the function is also bounded.
Examples If you have the unit step function u(t) it is not stable because its integration
is infinite. The dirac delta function is a bit trickier. This is of course the
continuous time dirac delta function. When one integrates this function from negative
infinity to infinity it yields one. However the dirac delta function at zero
is infinite. This seems to contradict the defintion, however since it satisfies
the stronger condition it is correct.
y(t) = e^(-8t)*u(t) is stable
It integrates to e^(-8t)/(-8) evaluated from 0 to infinity
Therefore it yields 0 - - 1/8 = 1/8 which is less than infinity.
