论文标题
抽象的非线性灵敏度和收费公路分析以及半连接抛物线PDE的应用
Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs
论文作者
论文摘要
我们分析了非线性最佳控制问题对动力学和初始数据的扰动的一阶最佳最佳条件产生的极端方程的灵敏度。为此，我们提出了一种带有缩放空间的抽象隐式函数方法。我们将把这种抽象方法应用于由半线性PDE管辖的问题。在这种情况下，我们证明了指数收费的结果，并表明极端方程动力学的扰动，例如离散化错误会及时呈指数衰减。后者可以在模型预测控制器中用于非常有效的离散化方案，其中仅需要准确地计算解决方案的一部分。我们通过在二维结构域上具有非线性热方程的两个示例来展示理论结果。
We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.
