Variational embedding method and application to distributed systems control

The main aim of the present paper is to show the variational embedding method as a powerful tool for solving control problems associated with distributed systems. It is shown that, variational embedding subsumes many approximation methods that are available in the literature for solving constitutive equations of distributed systems. The well-known method of weighted residuals and its ramifications are shown to be the appropriate specializations of the embedding method. The method thus offers a vantage point from which it is conceptually easy to see both the plurality and unity of these methods. In the process of illustrating the hierarchy of the method, the attendant dilemma associated with the method of weighted residuals is successfully resolved. The variational embedding method yields a unified approach for obtaining lumped models for distributed parameter systems. The application is demonstrated by considering examples of engineering importance, and it is shown how lumped models of non-linear distributed systems could be constructed by methods other than the conventional discretization methods.