SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1057-1082, January 2021.
Controlling the shape and position of moving and pinned droplets on a solid surface is an important feature often found in microfluidic applications. In this work, we consider a well investigated phase field model including contact line dynamics as the state system for an (open-loop) optimal control problem. Here the spatially and temporally changeable contact angles between droplet and solid are considered as the control variables. We consider a suitable, energy stable, time discrete version of the state equation in our optimal control problem. We discuss regularity of the solution to the time discrete state equation and its continuity and differentiability properties. Furthermore, we show existence of solutions and state first order optimality conditions to the optimal control problem. We illustrate our results by actively pushing a droplet uphill against gravity in an optimal way.

中文翻译：

---

利用接触角分布的滑动液滴最优控制

SIAM控制与优化杂志，第59卷，第2期，第1057-1082页，2021年1月。
控制在固体表面上移动和固定的液滴的形状和位置是微流体应用中经常发现的重要特征。在这项工作中，我们考虑了一个经过充分研究的相场模型，其中包括接触线动力学作为（开环）最优控制问题的状态系统。在此，液滴和固体之间的在空间和时间上可改变的接触角被认为是控制变量。我们在最优控制问题中考虑了状态方程的合适的，能量稳定的，时间离散的版本。我们讨论了时间离散状态方程解的正则性及其连续性和微分性质。此外，我们显示了最优控制问题的解和状态一阶最优条件的存在。