Geometric pseudospectral method for spatial integration of dynamical systems

Abstract

A reduction method that preserves geometric structure and energetic properties of non-linear distributed parameter systems is presented. It is stated as a general pseudospectral method using approximation spaces generated by polynomials bases. It applies to Hamiltonian formulations of distributed parameter systems that may be derived for hyperbolic systems (wave equation, beam model, shallow water model) as well as for parabolic ones (heat or diffusion equations, reaction–diffusion models). It is defined in order to preserve the geometric symplectic interconnection structure (Stokes–Dirac structure) of the infinite-dimensional model by performing exact differentiation and by a suitable choice of port variables. This leads to a reduced port-controlled Hamiltonian finite-dimensional system of ordinary differential equations. Moreover, the stored and dissipated power in the reduced model are approximations of the distributed ones. The method thus allows the direct use of thermodynamics phenomenological constitutive equations for the design of passivity-based or energy-shaping control techniques.