Contact Geometry in Optimal Control of Thermodynamic Processes for Gases
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Contact Geometry in Optimal Control of Thermodynamic Processes for Gases A. G. Kushnera,b,*, V. V. Lychaginc,**, and M. D. Roop a,c,*** Presented by Academician of the RAS S.N. Vassilyev February 18, 2020 Received March 27, 2020; revised April 14, 2020; accepted June 6, 2020
Abstract—We solve an optimal control problem for thermodynamic processes in an ideal gas. The thermodynamic state is given by a Legendrian manifold in a contact space. Pontryagin’s maximum principle is used to find an optimal trajectory (thermodynamic process) on this manifold that maximizes the work of the gas. In the case of ideal gases, it is shown that the corresponding Hamiltonian system is completely integrable and its quadrature-based solution is given. Keywords: contact geometry, thermodynamics, optimal control, Hamiltonian systems, integrability DOI: 10.1134/S1064562420040109
Optimal control is important for solving a number of practical problems in gas dynamics and gas flows in porous media when a thermodynamic process involving the medium can be controlled by applying external disturbances. Of course, the case to be considered the first is when, among all processes, there is one maximizing the work of the gas. In contrast to [1], where control methods for nonequilibrium thermodynamic processes are proposed, we consider only equilibrium thermodynamics. Our approach is based on a geometric thermodynamic formulation that goes back to [2–4] and, as was shown in [5], is closely related to the theory of measurements. Specifically, the thermodynamic state of the gas is represented as a Legendrian submanifold in a thermodynamic contact space and the nonnegativity of the variance of measurements of extensive thermodynamic variables leads to Riemannian structures on this manifold [5]. As a result, Pontryagin’s maximum principle can be applied to the problem of finding a curve on the Legendrian manifold that maximizes the work functional. In the case of a Faculty
of Physics, Lomonosov Moscow State University, Moscow, 119991 Russia b Moscow Pedagogical State University, Moscow, 119435 Russia c Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 117997 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
ideal gases, this formulation of the problem leads to a Liouville integrable Hamiltonian system, so the optimal control problem admits an exact solution. 1. THERMODYNAMICS Consider a contact space (R5, θ) with coordinates ( s , e , v , p, T ), which denote specific entropy, internal energy, volume, pressure, and temperature, respectively. Suppose that a contact structure on (R5, θ) is specified by the differential 1-form
θ = ds − T −1de − pT −1d v . By the thermodynamic state, we mean a maximum integral manifold L of the form θ, i.e., a Legendrian manifold, which means in terms of physics that the first law of thermodynamics holds on L. Using the extensive variables (e , v ) as coordinates on L and applying the condition θ L = 0 , we see that L can b
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