论文标题
终端随机线性互补约束的最佳控制问题
An Optimal Control Problem with Terminal Stochastic Linear Complementarity Constraints
论文作者
论文摘要
在本文中，我们研究了使用终端随机线性互补约束（SLCC）的最佳控制问题，并使用弛豫，样本平均近似（SAA）和隐式Euler时间步变方案进行离散近似。我们在SLCC中随机矩阵的期望是Z-Matrix或适当的矩阵的条件下，显示了最佳控制问题的可行解决方案和最佳解决方案的存在。此外，我们证明，离散近似产生的解决方案序列会收敛到原始最佳控制问题的解决方案1，概率为1，为$ε\ downarrow 0 $，$ν\ to \ infty $和$ h \ histrow 0 $，其中$ε$是放松参数，$ν$是样本尺寸和$ h $。我们还提供了SAA最佳价值和时间步长方法的误差界的渐近学。数值示例用于说明最佳解决方案的存在，离散方案和误差估计。
In this paper, we investigate an optimal control problem with terminal stochastic linear complementarity constraints (SLCC), and its discrete approximation using the relaxation, the sample average approximation (SAA) and the implicit Euler time-stepping scheme. We show the existence of feasible solutions and optimal solutions to the optimal control problem and its discrete approximation under the conditions that the expectation of the stochastic matrix in the SLCC is a Z-matrix or an adequate matrix. Moreover, we prove that the solution sequence generated by the discrete approximation converges to a solution of the original optimal control problem with probability 1 as $ε\downarrow 0$, $ν\to \infty $ and $h\downarrow 0$, where $ε$ is the relaxation parameter, $ν$ is the sample size and $h$ is the mesh size. We also provide asymptotics of the SAA optimal value and error bounds of the time-stepping method. A numerical example is used to illustrate the existence of optimal solutions, the discretization scheme and error estimation.
