Liapunov functionals for systems with time delay and propagation

Starting from some recently analyzed controlled models with propagation i.e. described by hyperbolic partial differential equations, it is shown that a Liapunov functional may be attached in a natural way by using development of the energy integral. As known from the general theory of the hyperbolic partial differential equations, the energy integral is a useful tool for obtaining uniqueness theorems. On the other hand, existence theorems may be obtained either starting from difference schemes (the approach of S.K. Godunov) but also by associating some functional differential equations (the approach of A.D. Myshkis, K.L. Cooke and their followers). Due to the one to one correspondence between the two mathematical objects, the counterpart of the Liapunov functional deduced from the energy integral is obtained for the functional differential equations. Using these mathematical tools some process applications are discussed: flexible beam (manipulator), flexible overhead crane and to applications of controlling the oil drilling plants. The outcome of the analysis is given by control structures and stability criteria.