一国碳减排的最小费用研究

本文研究一国碳排放量的最优控制问题. 假设这个国家的总碳排放量由国内生产总值（GDP）和人口决定，而GDP满足几何布朗运动模型，国内人口数量满足Logistic模型. 国家采取一些策略降低国内的碳排放量，这些措施也会产生相应的成本. 这个国家需要在降低碳排放量的同时，使得控制过程中的总费用最小. 利用随机控制理论的相关结论，可以对这一问题进行建模，并得到相应的Hamilton-Jacobi-Bellman（HJB）方程. 由此得出的半线性方程可以通过Cole-Hopf变换变为线性并得出显式解，从而得出相应的最小成本和最优控制策略的表达式. 我们对解进行数值计算，得出了值函数与不同参数的关系图.

Minimization of the national cost due to carbon emission

In this paper, an optimal control problem based on carbon emission is analyzed. The carbon emission of a country follows a process deduced from the Gross Domestic Product (GDP), which follows a geometric Brownian motion, and the population of the country, which satisfies the Logistic model. A policy is utilized to reduce the CO₂ emission of the country, resulting in some related costs. The country aims at reducing the carbon emission while minimizing the relevant costs contemporarily. The problem can be solved in the framework of stochastic control and Hamilton-Jacobi-Bellman (HJB) equation. Via Cole-Hopf transformation, the quasi-linear equation can be linearized and solved explicitly, and the optimal policy and minimal cost can be derived afterwards. Some numerical results are provided, and the relationships between the value functions and different parameters are analyzed. Furthermore, we analysed the case in which an upper bound is put on the control policy. Since the equation cannot be solved explicitly, a numerical solution is presented, showing the properties of the value function and the optimal policy.