论文标题
Lipschitz-Nonlinelarear的最佳控制中的收费公路
Turnpike in Lipschitz-nonlinear optimal control
论文作者
论文摘要
当运行目标是稳定的控制状态对的基础系统时，我们为非线性最佳控制问题提供了收费公路属性的新证明。我们的策略结合了通过可控性和引导性参数的准转换控制的构建，并且不依赖于分析最优性系统或线性化技术。反过来，这使我们能够解决具有有限维度的控制型系统的几个最佳控制问题，该系统具有全球Lipschitz（可能是非平滑）的非线性，而初始数据或运行目标上没有任何较小的条件。这些结果是由通过深层残留神经网络在机器学习中的应用进行的，这些神经网络可能适合我们的环境。我们表明，我们的方法也适用于受控的PDE，例如具有全球Lipschitz非线性的半线性波和热方程，而没有任何较小的假设。
We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by applications in machine learning through deep residual neural networks, which may be fit within our setting. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
