Laplace transform of a signal multiplied to a time function

Question

Laplace transform of multiplication of two time functions can be calculated easily using standard formulae. But I have a problem in which signal $x(t)$ is multiplied to a time function. I have to do control analysis of the problem. Feedback loop diagram

Here, $z(t)=cos(\Omega t) x(t)$.

For control analysis, I need to get relationship between $Z(s)$ and $X(s)$.

As per defination of Laplace transform, $X(s)=\int_{0}^{\infty} x(t) e^{-s t} dt$, it can be derived

$Z(s) = \int_{0}^{\infty} x(t) e^{j \Omega t} e^{-s t} dt = X(s-j \Omega)$ But, this relationship cannot be used for control analysis like root locus and many other things.

Can someone help me for getting relationship between $X(s)$ and $Z(s)$? How can convolution theorem be used in this regard?

Answer 1

Multiplication in the time-domain corresponds to convolution in the Laplace domain.