This paper introduces a control-theoretic formulation of nonequilibrium thermodynamic systems with Gibbs states of Gaussian distributions, i.e., thermodynamic systems characterized by Gaussian probability densities. The distinct features of the paper are three-fold. First, the dynamics of a thermodynamic system is studied by transferring the original state variable from the non-Euclidean and nonlinear manifold state space to a Lie group, and further to the dual space of a Lie algebra, which is endowed with vector space structures. Consequently, the resulting reduced dynamics significantly reduces the high nonlinearity appearing in the original non-Euclidean state space of a thermal process. Second, the obtained equations of motion interestingly indicate that thermodynamics can be naturally viewed as a generalization of rigid body motions, and this bridges control theory, thermodynamics, information theory, and rigid body dynamics. Third, the optimality conditions for the energy-minimum optimal control problem of probability densities are derived via geometric Pontryagin’s principle by regarding the reduced dynamics as dynamical constraints, and a unified control algorithm is developed. Finally, the proposed approach is applied to three different scenarios including two benchmark examples to demonstrate the applicability and effectiveness. The purpose of the paper is to provide a deeper understanding of thermodynamics from a control perspective, and also to draw intrinsic connections between control theory, thermodynamics, information theory, and rigid body dynamics.

中文翻译：

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非平衡热力学的简化动力学和几何最优控制：高斯情况

本文介绍了具有高斯分布吉布斯态的非平衡热力学系统的控制理论公式，即以高斯概率密度为特征的热力学系统。本文的显着特征有三个方面。首先，通过将原始状态变量从非欧非线性流形状态空间转移到李群，并进一步转移到具有向量空间结构的李代数对偶空间来研究热力学系统的动力学。因此，由此产生的动力学降低显着降低了热过程的原始非欧几里得状态空间中出现的高非线性。其次，所获得的运动方程有趣地表明，热力学可以自然地被视为刚体运动的推广，并且它连接了控制理论、热力学、信息论和刚体动力学。第三，以约化动力学为动力学约束，根据几何庞特里亚金原理推导了概率密度能量最小最优控制问题的最优性条件，并提出了统一的控制算法。最后，将所提出的方法应用于包括两个基准示例在内的三种不同场景，以证明其适用性和有效性。本文的目的是从控制角度更深入地理解热力学，并绘制控制理论、热力学、信息论和刚体动力学之间的内在联系。