最优控制中分段正则极值的二阶最优性条件
Second order optimality conditions for piecewise regular extremals in Optimal Control
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中文总结 AI 辅助
研究具有积分成本的最优控制问题二阶最优性条件，通过连接正则弧得极值，证明相关必要和充分条件，借助雅可比曲线框架，其在分段正则时具间断点，以一维自由控制系统为例展示二阶测试。
中文摘要 AI 辅助
我们研究具有积分成本的最优控制问题的二阶最优性条件。考虑通过连接有限多个正则弧得到的极值，证明了该类中弱局部最优性的必要条件和强局部最优性的充分条件。关键对象是雅可比曲线，它是拉格朗日子空间的曲线，编码沿参考极值的二阶变分。在正则情况下，此曲线是光滑的，可与合适的极值场的切空间等同。对于分段正则极值，极值场只是分段光滑的，相关的雅可比曲线在切换时刻有间断点。这使得垂直相交、共轭点和最优性之间的关系更加微妙。我们发展了这种间断雅可比曲线框架，并展示了它如何产生有效的二阶测试。详细研究了一维自由控制系统的情况，作为自然出现分段正则极值的一个说明性类别。
英文摘要
We study second-order optimality conditions for optimal control problems with integral cost. We consider extremals obtained by concatenating finitely many regular arcs and prove both a necessary condition for weak local optimality and a sufficient condition for strong local optimality within this class. The key object is the Jacobi curve, a curve of Lagrangian subspaces encoding the second variation along the reference extremal. In the regular case, this curve is smooth and can be identified with the tangent spaces to a suitable field of extremals. For piecewise regular extremals, the field of extremals is only piecewise smooth, and the associated Jacobi curve has discontinuities at the switching times. This makes the relation between vertical intersections, conjugate points, and optimality more delicate. We develop this discontinuous Jacobi-curve framework and show how it yields effective second-order tests. The case of one-dimensional free-control systems is studied in detail as an illustrative class in which piecewise regular extremals arise naturally.
