This paper investigates the finite-time optimal control problems for positive linear systems with a time-varying control input. Cost functional of the system state is constructed. Under some assumptions on the designed parameters and cost functionals, the optimization problem with piecewise-constant matrix functions is first proved to be log–log convex and can be solved via geometric programming. Then, the log–log convex result is further extended to the optimization problem with continuous functions. Finally, an optimal control problem is investigated to verify the effectiveness of the proposed optimization framework.

中文翻译：

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正线性系统的最优控制问题的对数-对数凸性

本文研究了具有时变控制输入的正线性系统的有限时间最优控制问题。构造系统状态的成本泛函。在对设计参数和成本泛函的一些假设下，分段常数矩阵函数的优化问题首先被证明是对数-对数凸的，并且可以通过几何规划求解。然后，对数-对数凸结果进一步扩展到具有连续函数的优化问题。最后，研究了一个最优控制问题，以验证所提出的优化框架的有效性。