Feedback equivalence of input–output contact systems

Abstract

Control contact systems represent controlled (or open) irreversible processes which allow to represent simultaneously the energy conservation and the irreversible creation of entropy. Such systems systematically arise in models established in Chemical Engineering. The differential-geometric of these systems is a contact form in the same manner as the symplectic 2-form is associated to Hamiltonian models of mechanics. In this paper we study the feedback preserving the geometric structure of controlled contact systems and renders the closed-loop system again a contact system. It is shown that only a constant control preserves the canonical contact form, hence a state feedback necessarily changes the closed-loop contact form. For strict contact systems, arising from the modelling of thermodynamic systems, a class of state feedback that shapes the closed-loop contact form and contact Hamiltonian function is proposed. The state feedback is given by the composition of an arbitrary function and the control contact Hamiltonian function. The similarity with structure preserving feedback of input-output Hamiltonian systems leads to the definition of inputoutput contact systems and to the characterization of the feedback equivalence of input-output contact systems. An irreversible thermodynamic process, namely the heat exchanger, is used to illustrate the results.

Thermodynamics and evolutionary biology through optimal control

“…Let us start by reviewing briefly (and extend in its scope) the lift of controlled dynamical systems proposed recently in [3] (see also [6][7][8][9][10][11][12]). We consider a dynamical system of interest, represented by the equationsẋ…”

“…Provided (1) and (10) describe the same system, we can equate the corresponding Hamiltonians h ψ in (8) and H in (13) and infer that, if u(t) is chosen so that h ψ is minimized at every t as in (15), then the lifted system (4)-(5) is equivalent to Pontryagin's equations (14) for the cost functional…”

“…The aim of this work is to regard evolution, either of a thermodynamic system or of a population of individuals, as an adaptive process in which the system dynamically controls some parameters in order to evolve towards the optimum state allowed by the constraints. To do so, first we review and generalize a specific instance for the lift of irreversible thermodynamic systems proposed in [3,[6][7][8][9][10][11][12] and prove that such lift is equivalent to an optimal control in the sense of Pontryagin [34].…”

Contact Hamiltonian Dynamics: The Concept and Its Use

“…Several other examples, concerning also the generalization to open systems (called control contact systems) can be found in [28] and also in subsequent works, e.g., [29][30][31][32][33][34]. Further (and different) approaches to the use of contact Hamiltonian dynamics in irreversible thermodynamics can be found in [12,15,16,[35][36][37].…”

Geometry of Thermodynamic Processes

“…Starting from Gibbs’ fundamental relation, contact geometry has been recognized since the 1970s as an appropriate framework for the geometric formulation of thermodynamics; see in particular [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. More recently, the interest in contact-geometric descriptions has been growing, from different points of view and with different motivations; see, e.g., [ 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ].…”