标题： 非线性最优控制中的线性结构 显示英文标题

标题： Linear structures in nonlinear optimal control

摘要： 我们研究仿射动力系统的最优控制，即在控制上是线性的，但在状态上是非线性的。 控制任务是使系统状态尽可能紧密地跟随预定的期望轨迹，这一任务也被称为最优轨迹跟踪。 为了获得最优控制的良好解，必须在代价泛函中包含一个系数为$\varepsilon$的正则化项。 假设$\varepsilon$很小，我们将仿射最优控制问题重新解释为奇异摄动微分方程。 进行奇异摄动展开，推导出任意期望轨迹最优跟踪的近似方法。 对于$\varepsilon=0$，状态轨迹可能变得不连续，而控制可能发散。 另一方面，解析处理变得精确。 我们确定导致线性演化方程的条件。 这些结果导致一类非线性轨迹跟踪问题的精确解析解。 该类包括但不限于一维空间中的机械控制系统以及作用于激活剂上的 FitzHugh-Nagumo 模型。 显示英文摘要

摘要： We investigate optimal control of dynamical systems which are affine, i.e., linear in control, but nonlinear in state. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible, a task also known as optimal trajectory tracking. To obtain well-behaved solutions to optimal control, a regularization term with coefficient $\varepsilon$ must be included in the cost functional. Assuming $\varepsilon$ to be small, we reinterpret affine optimal control problems as singularly perturbed differential equations. Performing a singular perturbation expansion, approximations for the optimal tracking of arbitrary desired trajectories are derived. For $\varepsilon=0$, the state trajectory may become discontinuous, and the control may diverge. On the other hand, the analytical treatment becomes exact. We identify the conditions leading to linear evolution equations. These result in exact analytical solutions for an entire class of nonlinear trajectory tracking problems. The class comprises, among others, mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model with a control acting on the activator.