Theory of Non-equilibrium Thermodynamics in the Optimal Control Processes Part II: Physical Approaches to the Pontryagin’s Maximum Principle

The theory of non-equilibrium thermodynamics in the optimal control processes is established by using the Pontryagin’s theory of optimal control and the physical meaning of the the Pontryagin’s maximum principle is revealed. For this purpose, we strat with the system of nonlinear differential equations ̇⃗ x = f⃗ (x⃗, u⃗, t) where x⃗ is an n-dimensional vector of state variables and u⃗ an r-dimensional vector of control variables in the admissible control region U. We will show that when we would like to obtain the maximality of the function J = ∫ t1 t0 L(x⃗, u⃗, t)dt, where t0, t1 are given initial and final times, then there is a Hamiltonian H ≡ ∑ni=1 ψi fi(x⃗, u⃗, t)+L(x⃗, u⃗, t), called the Pontryagin’s Hamiltonian. This provides the Hamilton equations of motion for ψ⃗ and xi: dxi dt = ∂H ∂ψi , dψi dt = − ∂H ∂xi ,with the optimality condition: ∂H ∂ui = 0, and the maximality condition: ∂2H ∂ui ≤ 0. Thereby the Pontryagin’s maximum principle is given by H ≤ maxu⃗∈U H ≡ M = 0. We will apply this theory to the non-equilibrium thermodynamics in the optimal control processes, regarding L(x⃗, u⃗, t) as the Onsager-Prigogine’s variational function such that L(x⃗, u⃗, t) = Ṡ (x⃗, u⃗, t) − 2Φ(x⃗, u⃗, t). Thus, we will show that the optimality condition for the Hamiltonian provides the concepts of the optimal dissipation of energy and the optimal production of entropy: ∑n i=1 ψi(t) ∂ fi(x⃗(t),u⃗(t),t) ∂us + ∂[Ṡ (x⃗,u⃗,t)−2Φ(x⃗,u⃗,t)] ∂us = 0, where the units of ψi are taken so as to adjust with the problem situation. And we show that as long as we use suitable choice for the units of the vector ψ⃗, the Pontryagin’s maximum principle assures that the system realizes the energy conservation so that the power vanishes(i.e.,W ≡ dE dt = 0) when the system approaches the optimal state. The proofs and the details will be discussed in this paper.