Theory of Non-equilibrium Thermodynamics in the Optimal Control Processes Part I: Mathematical Formulation

The theory of non-equilibrium thermodynamics in the optimal control processes is established, by using the Pontryagin’s theory of optimal control and the Bellman’s theory of dynamic programming, for the first time. For this theory, we study the systems of nonlinear differential equations ̇⃗ x = f⃗ (x⃗, u⃗, t) of the state variables x⃗ under the condition that the control variables u⃗ be optimal. It is the natural generalization of Onsager-Prigogine’s theory of the irreversible processes to the theory of the optimal control processes. It is shown that the principle of the least dissipation of energy by Onsager and the principle of the least production of entropy should be generalized to the principle of the optimal dissipation of energy and the principle of the optimal production of entropy, respectively. In this theory, it is shown that for the system of the optimal control processes, the Pontryagin’s Hamiltonian plays the same role of the Hamiltonian in classical mechanics, while the Bellman’s optimality equation plays the same role of the Hamilton-Jacobi equation, respectively. The main result is summarized as follows: Define the Pontryagin’s Hamiltonian H(ψ⃗(t), x⃗(t), u⃗(t)) ≡ ψ⃗(t) · f⃗ (x⃗, u⃗, t) + T [Ṡ (x⃗, u⃗) − Φ(x⃗, u⃗)], where Ṡ (x⃗, u⃗) means the entropy production and Φ(x⃗, u⃗) the dissipation function of energy, respectively. Then, the following Pontryagin’s Hamilton dynamics must be satisfied: