Networks of Hamiltonian systems and feedback

Motivated by the design of complex networks transmitting pulse-like signals, we investigate and develop a network theory for Hamiltonian systems. The main topics of this paper are articulated around the following result: under a few natural assumptions, a network of Hamiltonian systems with energy-conserving interactions will itself be Hamiltonian for the symplectic form ω_{c l} = ω - G^* (K (I - C K)^{- 1}) where ω is the symplectic form of the individual components of the network, K encodes the network structure and C, which we call the symplectic gain matrix, is related to the non-involutivity of the control vector fields. A further analysis of this form will reveal how certain interactions result in effectively putting constraints on the system, giving a new result in the flavor of the Dirac theory of constrained Hamiltonian systems but of a control-theoretic origin. Throughout the paper, we use the Toda lattice to illustrate the results and find a new approach to the description of the periodic Toda lattice as a Hamiltonian system on the space of Jacobi matrices.