论文标题
半线性最佳控制问题的最小化器的非唯一性
Nonuniqueness of minimizers for semilinear optimal control problems
论文作者
论文摘要
给出了半线性最佳控制问题全球最小化器的唯一性的反例。在成本功能中，特殊选择国家目标是出现缺乏独特性的。我们的论点还表明，对于某些国家目标，存在并非全局的当地最小化器。当发生这种情况时，梯度型算法可能会被局部最小化器捕获，从而丢失了全局。此外，在抽象环境中分析了最佳控制中二次功能的凸度问题。 作为最小化器非唯一性的推论，推导了耦合椭圆系统的非唯一性结果。 已经进行了数值模拟，以说明理论结果。 我们还讨论了最小化器在长期范围内对收费公路属性的多样性的可能影响。
A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments show also that, for some state-targets, there exist local minimizers, which are not global. When this occurs, gradient-type algorithms may be trapped by the local minimizers, thus missing the global ones. Furthermore, the issue of convexity of quadratic functional in optimal control is analyzed in an abstract setting. As a Corollary of the nonuniqueness of the minimizers, a nonuniqueness result for a coupled elliptic system is deduced. Numerical simulations have been performed illustrating the theoretical results. We also discuss the possible impact of the multiplicity of minimizers on the turnpike property in long time horizons.
