The authors demonstrated an optimal stochastic control algorithm to obtain desirable cancer treatment based on the Gompertz model. Two external forces as two time-dependent functions are presented to manipulate the growth and death rates in the drift term of the Gompertz model. These input signals represent the effect of external treatment agents to decrease tumour growth rate and increase tumour death rate, respectively. Entropy and variance of cancerous cells are simultaneously controlled based on the Gompertz model. They have introduced a constrained optimisation problem whose cost function is the variance of a cancerous cells population. The defined entropy is based on the probability density function of affected cells was used as a constraint for the cost function. Analysing growth and death rates of cancerous cells, it is found that the logarithmic control signal reduces the growth rate, while the hyperbolic tangent-like control function increases the death rate of tumour growth. The two optimal control signals were calculated by converting the constrained optimisation problem into an unconstrained optimisation problem and by using the real-coded genetic algorithm. Mathematical justifications are implemented to elucidate the existence and uniqueness of the solution for the optimal control problem.

中文翻译：

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基于Fokker-Planck方程的肿瘤生长过程的最佳最小方差熵控制。

作者展示了一种基于Gompertz模型的最优随机控制算法，以获得理想的癌症治疗方法。在Gompertz模型的漂移项中，提出了两个作为两个时间相关函数的外力来控制增长率和死亡率。这些输入信号分别代表外部治疗剂降低肿瘤生长速率和增加肿瘤死亡率的作用。根据Gompertz模型同时控制癌细胞的熵和方差。他们提出了一个约束优化问题，其代价函数是癌细胞群的方差。定义的熵基于受影响单元的概率密度函数作为成本函数的约束。分析癌细胞的生长和死亡率，发现对数控制信号降低了生长速率，而双曲线正切样控制功能增加了肿瘤生长的死亡率。通过将约束优化问题转换为无约束优化问题并使用实编码遗传算法来计算两个最优控制信号。实施数学上的解释以阐明最优控制问题的解决方案的存在性和唯一性。