For a proper understanding of the behaviour of a system, one has to represent the system in the form of mathematical relations. The voltage equation in algebraic form or a differential equation of an electrical circuit are a few such mathematical relations.
The Laplace transformation offers a very simple and elegant method of solving linear or linearised differential equations which result from the mathematical modelling of a process or a system. The Laplace transform also allows us to develop input-process-output models which are very useful for the study of control systems. It is because of the above reasons that the Laplace transform is used extensively in control engineering.