In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

Abstract. We investigate necessary and sufficient conditions under which a general nonlinear affine control system with outputs can be written as a gradient control system corresponding to some pseudoRiemannian metric defined on the state space. The results rely on a suitable notion of compatibility of the system with respect to a given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role in the discussion.Key words. gradient control systems, symmetric product, prolongation and gradient extension of a nonlinear system, externally equivalent systems. AMS subject classifications. 93C10, 93B29, 53B05, 93B151. Introduction. A physically motivated class of nonlinear systems are gradient control systems, see [4,9,21,22,23,24,25] and the references quoted therein. These systems are described in the following way: they are nonlinear affine control systems, which are endowed with a pseudo-Riemannian metric on the state space manifold. The drift vector field of the system is the gradient vector field associated with an internal potential function with respect to the pseudo-Riemannian metric, and the input vector fields are the gradient vector fields associated with the output functions of the system. Examples of gradient control systems include nonlinear electrical RLC networks, and dissipative systems where the inertial effects are neglected. In the case of RL or RC networks, the pseudo-Riemannian metric is positive-definite, and thus is a usual Riemannian metric, while for general RLC networks the metric is indefinite. We refer to [4,9,23,24] for more background on the modeling of nonlinear networks as gradient systems.Another relevant class of nonlinear systems is the family formed by the Hamiltonian control systems. In this case, the state space manifold is equipped with a symplectic form. The drift vector field and the input vector fields are the Hamiltonian vector fields associated, respectively, to an internal energy function and the output functions of the system with respect to the symplectic form. Hamiltonian equations are of central importance in the modeling of physical systems as they are the starting point to describe the dynamics of a very large class of phenomena, including mechanical, electrical and electromagnetic systems.Apart from their physical and engineering importance, gradient and Hamiltonian systems also possess very peculiar mathematical properties. For instance, a linear input-state-output system is a Hamiltonian control system [5] (respectively a gradient control system [25]) if and only if its impulse response matrix W (t) satisfies