Optimal design of nonlinear dynamical systems with uncertain delays and uncertain parameters

我们提出了一种适用于一大类技术相关问题的时滞动力系统优化方法。时滞系统在许多工程学科中都会发生。例如，延迟被用来对供应链中原材料交付的机器安装时间和提前期进行建模。在化工和生化行业无处不在的反应器-分离器系统中，每当出于经济或生态原因将未使用的原材料从产品中分离出来并回收利用时，就会出现延误。除了供应链和反应堆-分离器系统，该项目还考虑了人口和收获模型中的延迟，以及具有光反馈的激光的延迟，作为额外的例子。在所有情况下，延迟对系统的稳定性和最优运行都有很大的影响。例如，它们既可能是稳定的，也可能是不稳定的。从技术上讲，所提出的方法适用于一大类具有多个不确定参数和多个不确定状态相关和状态独立时滞的有限维光滑时滞微分方程。文中以上述四类应用为例说明了该方法的广泛适用性，该方法属于正规向量法，已成功地应用于连续时间系统的稳态和暂态优化，以及离散时间和周期系统的稳态优化(无时滞)。对于这里所讨论的这类时滞微分系统，发展一种法矢方法是具有挑战性的，因为在这种情况下，稳定性性质是由无穷多个特征值决定的。所有其他类型的法向量法都解决了特征值数目有限的问题。当延迟依赖于状态时，会出现进一步的复杂情况。事实上，状态无关时滞已经需要无穷多个特征值，但在其他方面类似于不确定模型参数。相比之下，依赖状态的延迟与不确定的模型参数有根本的不同。事实上，依赖国家的延误情况是该项目中处理的要求最高的情况。虽然这种情况在技术上是困难的，但它实际上是相关的，如示例应用程序类所展示的，特别是项目中涉及的供应链示例。项目中开发的方法将适用于其他延迟微分方程系统的示例。

We propose a method for the optimization of dynamical systems with delays that applies to a large class of technically relevant problems. Systems with delays occur in many engineering disciplines. Delays are used to model machine setup times and lead-times for raw material delivery in supply chains, for example. In reactor-separator systems, which are omnipresent in the chemical and biochemical industries, delays occur whenever unused raw material is separated from products and recycled for economic or ecologic reasons. Apart from supply chains and reactor-separator systems, the project considers delays in population and harvesting models, and in lasers with optical feedback, as additional examples. In all cases, delays have a nontrivial effect on the system stability and optimal operation. They can be both stabilizing and destabilizing, for example. Technically speaking, the proposed method applies to the large class of finite-dimensional smooth delay differential equations with multiple uncertain parameters and multiple uncertain state-dependent and state-independent delays. The four classes of applications mentioned above, which serve as examples throughout the project, are chosen to demonstrate the broad applicability of the proposed method.The proposed method belongs to the class of normal vector methods, which have successfully been applied to the steady state and transient optimization of continuous time systems, and the steady state optimization of discrete time and periodic systems (always without delays). Developing a normal vector method for the class of delay differential systems treated here is challenging, because the stability properties are determined by an infinite number of eigenvalues in this case. All other classes of normal vector methods have addressed problems with a finite number of eigenvalues. A further complication arises when delays are state-dependent. In fact, state-independent delays already require an infinite number of eigenvalues, but otherwise resemble uncertain model parameters. State-dependent delays, in contrast, are fundamentally different from uncertain model parameters. In fact, the case of state-dependent delays is the most demanding one treated in the project. While this case is technically difficult, it is practically relevant as demonstrated with the sample application classes, in particular the supply chain examples addressed in the project.The method developed in the project will be made available for application to other examples of delay differential equation systems.