Hamiltonian Control Systems

In this paper it is intended to elaborate a framework in which we can incorporate external forces in the systems prescription with emphasis on Hamiltonian systems with external forces and on the consequences of external forces. An appealing tool for this case is the language of symplectic geometry. Definitions of Hamiltonian systems with external forces are given and it is shown how they fit very naturally into the framework. It is also shown that forces are basic variables and that they have to be included in the definitions of mechanical systems. Hamiltonian systems with dissipation. The structural properties of these systems were discussed, in particular the existence of Casimir functions and their implications for stability. (4) has shown the geometric property and structure of the Hamilton--Jacobi equation arising from nonlinear control theory are investigated using symplectic geometry. (5) made an analysis on Hamiltonian systems and his results revealed a systematic geometric frame for generalized controlled Hamiltonian systems. The pseudo-Poisson manifold and the ω-manifold are proposed as the state space of the generalized controlled Hamiltonian systems were established. However the above results did not consider the external forces as basic variables. In this paper it is therefore intended to show that external forces should. It will be shown that it is necessary to maintain forces as basic variables and those have to be included in the definition of mechanical systems.