Seminar Details
2012-10-02 (14.00) : Potential realization of nonlinear systems
At Euler building
Organized by Mathematical Engineering
Speaker :
Martin GUAY
( Queen's University, Canada)
Abstract :
This presentation considers the problem of representing a sufficiently smooth nonlinear dynamical as a structured potential-driven system. The problem is related to recent nonlinear control contributions where feedback controllers are designed for systems modeled as structured potential-driven systems, for example gradient systems, generalized Hamiltonian systems and systems given in Brayton--Moser form. In recent years, matching conditions, expressed as partial differential equations, were developed to transform general nonlinear system in potential-driven forms. Departing from this approach, the present work proposes a geometric decomposition technique to re-express a given vector field into a desired potential-driven form. The proposed method is based on a decomposition of a differential one-form associated that encodes the divergence of the given vector fields into its exact and anti-exact components, and into its co-exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems, and generalized Hamiltonian systems are given to illustrate the proposed approach, as well as the three-dimensional rigid body problem.