A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.

中文翻译：

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带钉扎电介质电润湿的有限水平模型预测控制

具有接触线钉扎的时间离散空间连续电介质电润湿 (EWOD) 模型被认为是最佳控制框架中的状态系统。固定模型基于互补条件。除了描述速度、压力和电压的物理变量之外，固-液-气界面，即接触线，作为随时间演变的几何变量而出现。由于互补条件，由此产生的自由边界问题的最优控制是函数空间中具有平衡约束（MPEC）的数学程序。为了应对几何变量，提出了一种有限层模型预测控制方法。通过应用正则化程序，利用 PDE 约束优化技术，推导出双平稳性条件，然后传递到正则化参数中的限制。此外，通过遵循用于推导双重平稳性条件的理论极限参数，开发了基于函数空间的数值程序。通过几个例子展示了算法的性能；包括重心匹配和轨迹跟踪。