## A Thermodynamic Automaton with Four Temperatures and Three Controls

**Guido Cosenza***Dipartimento di Scienze Fisiche, Universita di Napoli,**Mostra d'Oltremare, Padiglione 19, I-80125 Napoli, Italy*

**Anthony Massobrio**

**Luigi Sertorio***Present address: Service de Chimie Physique, ULB, Bruxelles, Belgium.*

*Dipartimento di Fisica Teorica dell'Universita di Torino, INFN,**Via P. Giuria 1, I-10125 Torino, Italy*

#### Abstract

A Thermodynamic Automaton is a finite set of subsystems interacting according to the laws of non-equilibrium thermodynamics. The thermal flows between subsystems, determining the evolution in the state space, are modulated by bilinear controls that map the state space onto a finite set. The physical motivation for introducing Thermodynamical Automata is the problem of obtaining a thermostatic ambient with zero energy consumption. The mathematical context of Thermodynamic Automata is the theory of finite-dimensional dynamical systems: the general distributed parameter problem is reduced to a lumped parameter problem *via* compartmental modeling. Corresponding to various degrees of complexity of the system, various theoretical models can be developed; a compartmental reduction into four components and three control functions is studied here and is compared with Optimal Control Theory.