Theory of Probability &Its Applications, Volume 67, Issue 1, Page 28-43, May 2022.
We consider an optimal linear-quadratic control problem for a control system where the matrices corresponding to the state in the controlled process equation and the cost functional are absolutely integrable over an infinite time interval. The integral quadratic performance index includes two mutually inversely proportional time-weighting functions. It is shown that a well-known linear stable feedback law turns out to be optimal with respect to criteria from the class of the extended long-run averages. The results are applied to studying a control system under time-varying dynamic scaling of its parameters.

中文翻译：

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成本中具有反比例时间加权的最优随机线性二次控制

概率理论及其应用，第 67 卷，第 1 期，第 28-43 页，2022 年 5 月。
我们考虑控制系统的最优线性二次控制问题，其中与受控过程方程中的状态和成本泛函相对应的矩阵为在无限的时间间隔内绝对可积。积分二次性能指标包括两个相互成反比的时间加权函数。结果表明，一个众所周知的线性稳定反馈定律对于扩展的长期平均值类的标准是最优的。将结果应用于研究其参数随时间变化的动态标度下的控制系统。