GEOMETRIC DECOMPOSITION AND POTENTIAL-BASED REPRESENTATION OF NONLINEAR SYSTEMS M. GUAY * , N. HUDON † , AND K. H ¨ OFFNER ‡ Abstract. This paper considers the problem of representing a sufficiently smooth nonlinear dy- namical as a potential-driven system. The problem is related to recent contributions where feedback controllers are designed for gradient systems, generalized Hamiltonian systems and systems given in Brayton–Moser form.The approach proposed in the present note is based on a decomposition of a differential one-form associated to the given vector field into its exact and anti-exact components, and into its coexact and anti-coexact components. In particular, a dual operator to the standard homotopy operator, which inverts locally the exterior differential operator, is defined and used in the present paper to invert the codifferential operator. A study of Brayton–Moser representation is presented to illustrate the proposed approach. Key words. Nonlinear systems, systems representation, (dual) homotopy decomposition. AMS subject classifications. 93A30, 93B27 1. Introduction. Analysis and control design using on physically-based non- linear representations, such as gradient systems [2], generalized Hamiltonian systems [1], and systems given by Brayton–Moser equations [5], are now central to nonlinear control theory and practice. For applications where the concept of free energy is ill- defined, such representations are not available a priori and the problem of deriving a potential-based representation proved difficult to be solved. In the present note, we consider a nonlinear dynamical system of the form ˙ x = f (x),x ∈ R n , (1.1) where f ∈C k , with k ≥ 2 and we assume that the origin is an isolated equilibrium for f (x). The general problem considered in the present paper is to represent the system (1.1) as ˙ x = −Q(x) ∂P T (x) ∂x , (1.2) where P (x) is a potential for the system and Q(x) is a structure matrix. The approach proposed in the present paper seeks to refine the technique pro- posed originally in [6], which was based on the application of a homotopy operator (see Section 3) on a differential one-form associated to the system (1.1), to find a potential for a given system using the Poincar´ e lemma [10, 5, 4]. In particular, we characterize further the structure of the dynamics by proposing a dual to the homo- topy operator (Section 4). This operator locally inverts the codifferential operator [8]. In particular, the proposed approach can be related to the representation of smooth nonlinear dynamics as the sum of a gradient system and (n − 1) Hamiltonian systems, as presented for example in [9] using Hodge theory. * Corresponding author, martin.guay@chee.queensu.ca. Department of Chemical Engineering, Queen’s University, Kingston, ON, Canada, K7L 3N6. † School of Chemical Engineering, The University of New South Wales, UNSW Sydney, NSW 2052, Australia. ‡ Process Systems Engineering Laboratory, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. 02139. 1