This paper discusses the mathematical description of propagation and growth of wave-type phenomena which may occur in many physical systems, such as magneto-hydrodynamic system, fluid flow system, electromagnetic field system. This work also discusses the statement of a maximum principle for this wave generating system, and the problems associated with approximating this distributed system by discretization into lumped systems for the sake of computations. A one-dimensional version of this system with a quadratic cost function has been taken, and the method of gradients and quasilinearization techniques has been applied to derive the optimal control function