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

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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.