SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2693-2716, January 2021.
The linear quadratic regulator problem is central in optimal control and has been investigated since the very beginning of control theory. Nevertheless, when it includes affine state constraints, it remains very challenging from the classical “maximum principle” perspective. In this study we present how matrix-valued reproducing kernels allow for an alternative viewpoint. We show that the quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories. Drawing upon kernel formalism, we introduce a strengthened continuous-time convex optimization problem which can be tackled exactly with finite-dimensional solvers, and which solution is interior to the constraints. When refining a time-discretization grid, this solution can be made arbitrarily close to the solution of the state-constrained linear quadratic regulator. We illustrate the implementation of this method on a path-planning problem.

中文翻译：

---

从核方法的角度看线性约束线性二次调节器

SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2693-2716 页，2021 年 1 月。
线性二次调节器问题是最优控制的核心问题，自控制理论诞生之​​初就一直在研究。然而，当它包含仿射状态约束时，从经典的“最大值原理”的角度来看，它仍然非常具有挑战性。在这项研究中，我们展示了矩阵值再现内核如何允许另一种观点。我们展示了与线性动力学配对的二次目标对相关内核进行编码，定义受控轨迹的希尔伯特空间。利用核形式主义，我们引入了一个强化的连续时间凸优化问题，该问题可以用有限维求解器精确解决，并且该解决方案是约束的内部解决方案。在细化时间离散化网格时，该解可以任意接近状态约束线性二次调节器的解。我们说明了这种方法在路径规划问题上的实现。