AI 中文总结
针对流形上由几何变分方程描述的PDE约束最优控制问题，提出拉格朗日-牛顿方法，通过局部分裂技术实现，并证明其与牛顿法等价，数值示例验证了有效性。
AI 中文摘要
在偏微分方程约束优化中，控制量被施加于偏微分方程（PDE）内部，例如作为力场，以实现PDE解的期望构型。目标通常是找到控制成本与受控状态偏离期望构型之间的最优平衡。我们在一个设定中考虑此类问题，其中PDE由几何变分方程给出，且状态是到非线性流形的映射。我们针对该几何最优控制问题定义了一个拉格朗日函数，并导出了一阶最优性条件。此外，我们提出了一种用于其数值求解的拉格朗日-牛顿方法。我们引入了一种使用局部分裂技术的可实现算法，并证明了其等价于对拉格朗日函数导数应用牛顿法。该理论框架通过数值示例加以说明，包括弹性测地线和最小能量曲面在力场作用下的最优控制，以及不可伸长杆和壳体的控制。
英文摘要
In PDE-constrained optimization a control is applied within a partial differential equation (PDE), e.g., as a force field, to achieve a desired configuration of the PDE solution. The objective is usually to find an optimal balance between control cost and deviation of the controlled state from the desired configuration. We consider this problem class in a setting, where the PDE is given by a geometric variational equation and the state is a mapping into a nonlinear manifold. We define a Lagrangian function corresponding to this geometric optimal control problem and derive first-order optimality conditions. Moreover, we present a Lagrange-Newton method for their numerical solution. We introduce an implementable algorithm using a local splitting technique and show its equivalence to Newton's method applied to the derivative of the Lagrangian. The theoretical framework is illustrated by numerical examples, including the optimal control of elastic geodesics and surfaces of minimal energy by force fields and the control of inextensible rods and shells.
