关于线性二次最优控制问题中指数趋近现象的平坦性证明
A flatness proof of the exponential turnpike phenomenon for linear-quadratic optimal control problems
AI总结：
从微分平坦性角度重新审视有限维线性二次最优控制问题，通过参数化轨迹和简化欧拉-拉格朗日条件，揭示了指数趋近现象的形成机制，以及半正定权重对结果的影响。
摘要：
我们从微分平坦性的角度重新审视有限维线性二次最优控制问题。如果对 (A, B) 可控，则线性控制系统是平坦的，且每条轨迹都可以用平坦输出和其有限多个导数参数化。将此参数化代入二次函数时，欧拉-拉格朗日条件变为具有常系数的微分方程，或更一般地为多项式矩阵微分方程。经过降维到史密斯标准形后，该方程分解为标量常系数方程，其解为指数多项式。这为趋近现象提供了与经典哈密顿-里卡蒂分析截然不同的视角：趋近机制直接来自降维平坦方程的稳定-不稳定分裂。特别是当降维欧拉-拉格朗日算子无纯虚特征根且端点约束在稳定和不稳定模式上非退化时，最优轨迹由左边界层、右边界层和长期内部弧组成，且与静态最优值指数接近。同样的视角也澄清了当某些权重仅为半正定时的变化：降维方程的阶数可能降低，某些端点条件可能变得不相容，多项式或振荡模式可能破坏指数趋近。它还赋予了某些对控制和控制有限导数的端点约束以自然意义：这些痕迹不在 ambient L² 控制空间上定义，但在由降维欧拉-拉格朗日方程选出的光滑极值中是有意义的。我们将其原理作为一般定理提出，并在双积分器上详细说明。
英文摘要：
We revisit finite-dimensional linear-quadratic optimal control from the viewpoint of differential flatness. If the pair (A, B) is controllable, then the linear control system is flat, and every trajectory can be parametrized by a flat output and finitely many of its derivatives. Once this parametrization is inserted into the quadratic functional, the Euler-Lagrange condition becomes a linear differential equation with constant coefficients, or more generally a polynomial matrix differential equation. After reduction to Smith normal form, this equation decouples into scalar constant-coefficient equations, and its solutions are exponentialpolynomials. This yields a viewpoint on the turnpike phenomenon that is quite different from the classical Hamiltonian-Riccati analysis: the turnpike mechanism appears directly from the stable-unstable splitting of the reduced flat equation. In particular, when the reduced Euler-Lagrange operator has no purely imaginary characteristic roots and when the endpoint constraints act nondegenerately on the stable and unstable modes, the optimal trajectory consists of a left-boundary layer, a right-boundary layer, and a long interior arc exponentially close to the static optimum. The same viewpoint also clarifies what changes when some weights are only semidefinite: the order of the reduced equation may drop, some endpoint conditions may become incompatible, and polynomial or oscillatory modes may destroy the exponential turnpike. It also gives a natural meaning to certain endpoint constraints on the control and on finitely many derivatives of the control: such traces are not defined on the ambient L 2 control space, but they are meaningful on the smooth extremals selected by the reduced Euler-Lagrange equation. We formulate this principle as a general theorem and illustrate it in detail on the double integrator.
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