无约束椭圆型最优控制问题Schwarz交替法的收敛性分析
Convergence analysis of the Schwarz alternating method for unconstrained elliptic optimal control problems
摘要：
本文分析了求解无约束椭圆型最优控制问题的Schwarz交替法。我们讨论了该方法的收敛性在连续的情况下，然后应用到有限差分离散的情况下的参数。在这两种情况下，我们证明了Schwarz交替方法是收敛的，如果其对应的椭圆型方程是收敛的。同时，椭圆型方程方法在最大范数下的收敛速度也给出了最优控制问题方法在连续情况下或离散情况下在真误差价值函数或向量的最大范数下的收敛速度的一致上界（关于正则化参数$α$）.我们的数值结果证实了我们的理论结果，并表明，随着$α$减少到零，该方法将收敛得更快。并对这一现象进行了阐述。
英文摘要：
In this paper we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems. We discuss the convergence properties of the method in the continuous case first and then apply the arguments to the finite difference discretization case. In both cases, we prove that the Schwarz alternating method is convergent if its counterpart for an elliptic equation is convergent. Meanwhile, the convergence rate of the method for the elliptic equation under the maximum norm also gives a uniform upper bound (with respect to the regularization parameter $α$) of the convergence rate of the method for the optimal control problem under the maximum norm of proper error merit functions in the continuous case or vectors in the discrete case. Our numerical results corroborate our theoretical results and show that with $α$ decreasing to zero, the method will converge faster. We also give some exposition of this phenomenon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
