We consider a parabolic optimal control problem with an initial measure control. The cost functional consists of a tracking term corresponding to the observation of the state at final time. Instead of a regularization term in the cost functional, we follow an approach by E. Casas and K. Kunisch and consider a bound on the measure norm of the initial control. The variational discretization of the problem together with the optimality conditions induce maximal discrete sparsity of the initial control, i.e. Dirac measures in space. We present numerical experiments to illustrate our approach.
Variational discretization and semi-smooth Newton methods; implementation, convergence and globalization in pde constrained optimization with control constraints
An optimal order error estimate for the variational discretization of optimal control problems in the presence of pointwise control and state constraints
Variational discretization of a control-constrained parabolic bang-bang optimal control problem
研究了跟踪泛函中不含Tikhonov项的控制约束抛物型最优控制问题。对于数值处理，我们使用其Tikhonov正则化的变分离散：对于状态和伴随方程，我们在时间上应用[Daniels et al 2015]的Petrov-Galerkin格式，在空间上应用通常的协调有限元。我们证明了离散正则化问题和极限问题之间的误差的先验估计。由于这些估计是不稳健的，如果正则化参数趋于零，我们建立强大的估计，这-取决于问题的规律性-提高以前的。在bang-bang解的特殊情况下，这些估计得到了进一步的改进。一个数值例子证实了我们的分析结果。
The Popov's Algorithm with Optimal Bounded Stepsize for Generalized Monotone Variational Inequalities
Faculty of Applied Sciences, HCMC University of Technology and Engineering（胡志明市技术与工程学院应用科学学院）；Faculty of Basic Sciences, Van Lang University（文郎大学基础科学学院）；School of Mathematical Sciences, University of Southampton（南安普顿大学数学科学学院）
