AI 中文总结
研究具有标量等周约束的有限时域最优控制问题，从函数空间对偶性角度出发，证明约束值函数相关性质并得出超微分公式，还处理线性 - 二次问题，分析表明即使算子奇异敏感性仍可能存在。
AI 中文摘要
我们从函数空间对偶性的角度研究具有标量等周约束的有限时域最优控制问题。将控制视为 \(L^\infty(0,T;\mathbb R^m)\) 弱紧子集的元素，状态方程诱导出一个从控制到 \(W^{1,\infty}(0,T;\mathbb R^n)\) 的控制 - 状态映射。对于线性动力学、凹收益和仿射等周泛函，我们证明约束值函数具有区间域、是凹的且允许Fenchel - Moreau对偶表示。这产生了一个超微分公式，将对偶乘数的负值与值函数关于约束水平的敏感性联系起来。然后使用约束限定来将对偶乘数与增广等周系统的正常庞特里亚金乘数进行识别。我们还通过将具有单个二次等式约束的线性 - 二次问题简化为希尔伯特空间上的二次形式来处理。所得分析将包络公式的有效性与黎卡提合成所需的正则性分开，表明即使修改后的控制权重算子变得奇异，敏感性仍可能存在。
英文摘要
We study finite-horizon optimal control problems with scalar isoperimetric constraints from a function-space duality perspective. Controls are treated as elements of weakly compact subsets of \(L^\infty(0,T;\mathbb R^m)\), while the state equation induces a control-to-state map into \(W^{1,\infty}(0,T;\mathbb R^n)\). For linear dynamics, concave payoff, and an affine isoperimetric functional, we prove that the constrained value function has an interval domain, is concave, and admits a Fenchel--Moreau dual representation. This yields a superdifferential formula identifying the negative of the dual multiplier with the sensitivity of the value function with respect to the constraint level. A constraint qualification is then used to identify the dual multiplier with the normal Pontryagin multiplier of the augmented isoperimetric system. We also treat linear-quadratic problems with a single quadratic equality constraint by reducing them to quadratic forms on a Hilbert space. The resulting analysis separates the validity of the envelope formula from the regularity needed for Riccati synthesis, showing that sensitivity may persist even when the modified control-weight operator becomes singular.
