Asymptotic expansion of the solution to an optimal control problem for a linear autonomous system with a terminal convex quality index depending on slow and fast variables

A. R. Danilin, O. O. Kovrizhnykh
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Abstract

In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.
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一类终端凸质量指标依赖于慢变量和快变量的线性自治系统最优控制问题解的渐近展开
本文研究了一类具有光滑几何约束的球形分段连续控制中具有慢变量和快变量的线性自治系统在有限时间区间的最优控制问题。我们考虑了一个依赖于慢速和快速变量的终端凸质量指标。我们证明了当小参数趋于零时确定最优控制的向量的极限关系。我们改进了具有终端质量指标的间接控制问题的极限关系,该问题是两个严格凸连续可微函数的值和,其中第一个函数只依赖于慢变量,第二个函数只依赖于快变量,并且在零处有最小值。在此过程中,我们证明了决定向量的第一个分量收敛于极限问题的决定向量,而第二个分量趋于零。在有阻力介质中质点系统的间接控制问题中,我们得到了决定向量的小参数幂的完全渐近性。
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