This paper introduces harmonic control Lyapunov barrier functions (harmonic
CLBF) that aid in constrained control problems such as reach-avoid problems.
Harmonic CLBFs exploit the maximum principle that harmonic functions satisfy to
encode the properties of control Lyapunov barrier functions (CLBFs). As a
result, they can be initiated at the start of an experiment rather than trained
based on sample trajectories. The control inputs are selected to maximize the
inner product of the system dynamics with the steepest descent direction of the
harmonic CLBF. Numerical results are presented with four different systems
under different reach-avoid environments. Harmonic CLBFs show a significantly
low risk of entering unsafe regions and a high probability of entering the goal
region.

本文介绍了谐波控制 Lyapunov 屏障函数 (harmonic CLBF)，以解决受限控制问题，如到达 - 避免问题。谐波 CLBF 利用谐函数满足的最大原理，对控制 Lyapunov 屏障函数的性质进行编码。因此，它们可以在实验开始时初始化，而无需基于样本轨迹进行训练。选择控制输入以最大化系统动力学与谐波 CLBF 最陡下降方向的内积。通过对不同到达 - 避免环境下四个不同系统的数值结果进行呈现，谐波 CLBF 表现出进入不安全区域的风险显著降低，并具有进入目标区域的高概率。

用于约束最优控制和达到避障规范的谐波控制李亚普诺夫屏障函数

Harmonic Control Lyapunov Barrier Functions for Constrained Optimal  Control with Reach-Avoid Specifications

Recently, it was posited that disparate optimization algorithms may be
coalesced in terms of a central source emanating from optimal control theory.
Here we further this proposition by showing how coordinate descent algorithms
may be derived from this emerging new principle. In particular, we show that
basic coordinate descent algorithms can be derived using a maximum principle
and a collection of max functions as "control" Lyapunov functions. The
convergence of the resulting coordinate descent algorithms is thus connected to
the controlled dissipation of their corresponding Lyapunov functions. The
operational metric for the search vector in all cases is given by the Hessian
of the convex objective function.

最近，有人提出可以从最优控制理论中统一不同的优化算法。我们通过展示如何从这一新兴原理中导出坐标下降算法来进一步证明此命题。特别地，我们展示了基本的坐标下降算法可以通过使用最大原理和一组最大函数作为 “控制” Lyapunov 函数来导出。因此，由导致的控制系统的 Lyapunov 函数的耗散与相应的坐标下降算法的收敛性相关。所有情况下，搜索向量的运算度量均由凸目标函数的 Hessian 给出。

从最优控制理论推导坐标下降算法

Derivation of Coordinate Descent Algorithms from Optimal Control Theory

Physics informed neural networks (PINNs) are deep learning based techniques
for solving partial differential equations (PDEs) encounted in computational
science and engineering. Guided by data and physical laws, PINNs find a neural
network that approximates the solution to a system of PDEs. Such a neural
network is obtained by minimizing a loss function in which any prior knowledge
of PDEs and data are encoded. Despite its remarkable empirical success in one,
two or three dimensional problems, there is little theoretical justification
for PINNs.
As the number of data grows, PINNs generate a sequence of minimizers which
correspond to a sequence of neural networks. We want to answer the question:
Does the sequence of minimizers converge to the solution to the PDE? We
consider two classes of PDEs: linear second-order elliptic and parabolic. By
adapting the Schauder approach and the maximum principle, we show that the
sequence of minimizers strongly converges to the PDE solution in $C^0$.
Furthermore, we show that if each minimizer satisfies the initial/boundary
conditions, the convergence mode becomes $H^1$. Computational examples are
provided to illustrate our theoretical findings. To the best of our knowledge,
this is the first theoretical work that shows the consistency of PINNs.

该论文介绍了新型 PINNs 的理论和实践研究，证明了其在解决偏微分方程方面的有效性和可靠性。