Stochastic differential equations Laplace transforms

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

The following section contains the particularization of the integral Laplace transformation for the case of the stochastic model given by the assembly of relations (4.146)-(4.147). This particularization illustrates how the Laplace transformation is used to solve partial differential equations. We start by applying the integral Laplace operator to all the terms of relation (4.146) the result is in ... 

We make now, similarly as is common with the different integral transforms, a correspondence table between the stochastic variable and the associated characteristic function. Note, there are several integral transforms. The most well-known integral transformation might be the Fourier transform. Further, we emphasize the Laplace transform, the Mellin transform, and the Hilbert transform. These transformations are useful for the solution of various differential equations, in communications technology, all ranges of the frequency analysis, also for optical problems and much other more. We designate the stochastic variable with X. The associated characteristic function should be... 

The mathematical treatment of stochastic models of bicomponential reactions is rather difficult. The reactions X Yand X Y Z were investigated by Renyi (1953) using Laplace transformation. The method of the generating function does not operate very well in the general case, since it leads to higher-order partial differential equations. In principle chemical... 

---