The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed‐endpoint problems for linear time‐varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free‐endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed‐endpoint and variable‐endpoint problems to equivalent free‐endpoint ones is possible under the assumption that the linearized system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free‐endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time‐varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the L^∞ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact L^p‐penalization of state constraints with finite p is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to L^p′, where p^′ is the conjugate exponent of p, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly.

中文翻译：

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精确惩罚函数，用于最优控制问题II：终端和点状状态约束的精确惩罚

我们的研究的第二部分致力于分析惩罚函数的精确度，以解决带有终端状态和点状态约束的最优控制问题。我们证明，使用精确惩罚函数方法，可以将线性时变系统和线性发展方程的固定端点问题简化为对控制输入具有凸约束的线性演化方程，如果终端状态为完全等效的自由端点最优控制问题属于可达集的相对内部。在非线性情况下，我们证明在线性化系统是完全可控的假设下，可以将固定端点和可变端点问题局部减少到等效的自由端点问题，并指出了非线性系统的一些一般性质，在这些性质下，可以实现等效自由点问题的整体归约。对于具有逐点状态不等式约束的问题，我们证明了具有时滞状态的线性时变系统和线性发展方程的此类问题可以简化为没有状态约束的等价问题，只要使用大号^∞惩罚项，并斯莱特的条件成立，而对于非线性系统局部减少是可能的，如果一个自然的约束条件满足。最后，我们表明，确切大号^p的有限状态约束-penalization p是可能的凸问题，如果对应于该状态约束拉格朗日乘数属于大号^{P' ，}其中p ^'是的共轭指数p，和一般非线性问题，如果成本函数不明确地取决于控制输入。