We develop a framework and numerical method for controlling the full space-time tube of a geometrically driven flow. We consider an optimal control problem for the mean curvature flow of a curve or surface with a volume constraint, where the control parameter acts as a forcing term in the motion law. The control of the trajectory of the flow is achieved by minimizing an appropriate tracking-type cost functional. The gradient of the cost functional is obtained via a formal sensitivity analysis of the space-time tube generated by the mean curvature flow. We show that the perturbation of the tube may be described by a transverse field satisfying a parabolic equation on the tube. We propose a numerical algorithm to approximate the optimal control and show several results in two and three dimensions demonstrating the efficiency of the approach.

我们开发了用于控制几何驱动流的全时空管的框架和数值方法。我们考虑具有体积约束的曲线或曲面的平均曲率流的最优控制问题，其中控制参数在运动定律中充当强迫项。通过最小化适当的跟踪型成本函数来实现对流动轨迹的控制。成本函数的梯度是通过对平均曲率流生成的时空管的形式灵敏度分析来获得的。我们表明，可以通过满足管上抛物线方程的横向场来描述管的摄动。我们提出了一种数值算法来逼近最优控制，并在二维和三维中显示了一些结果，证明了该方法的有效性。