On a State-Constrained PDE Optimal Control Problem arising from ODE-PDE Optimal Control

Abstract

Summary. The subject of this paper is an optimal control problem with ODE as well as PDE constraints. As it was inspired, on the one hand, by a recently investigated flight path optimization problem of a hypersonic aircraft and, on the other hand, by the so called "rocket car on a rail track"-problem from the pioneering days of ODE optimal control, we would like to call it "hypersonic rocket car problem". While it features essentially the same ODE-PDE coupling structure as the aircraft problem, the rocket car problem's level of complexity is significantly reduced. Due to this fact it is possible not just to obtain easily interpretable results but also a certain degree of insight into the structure of the adjoints. Therefore, the rocket car problem can be seen as a prototype of an ODE-PDE optimal control problem. The main objective of this paper is the derivation of first order necessary optimality conditions.

Optimal control of coupled multiphysics problems: Guidelines for real‐life applications demonstrated for a complex fuel cell model

“…Especially for simpler models one gets a reliable method for the direct treatment of such state constraints, cf. [29]. Unfortunately, not all codes can be treated by black-box AD, particularly adaptive codes can not be differentiated at present time.…”

“…Nevertheless, method 2 is the only method that allows a direct inclusion of state constraints. Therefore, especially smaller (diffusion dominated) PDAE models with state constraints can be solved comfortably and with small implementational effort, e. g. the rocket car problem in [29].…”

Time Dependent Scanning Path Optimization for the Powder Bed Fusion Additive Manufacturing Process

“…The broken line path can be defined through several variables. A first idea, developed in [9,10], is derived from optimal control: most of the resolution techniques for problems mixing partial and ordinary differential equations [16,32,50,51] choose the path curvature as control. In [9,10], a path control based on the direction of each broken line segment has been experimented.…”

Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes

“…The proposed model-based methodology for robust optimal control implies formulating and solving a large-scale dynamic optimization problem (DOP) constrained by PDEs [28]. Optimal design, optimal control, and parameter estimation of systems governed by PDE give rise to a class of problems known as PDE-constrained optimization [29,30]. The size and complexity of the discretized PDEs often pose significant challenges for contemporary optimization methods.…”

“…As discussed in Section 4.3, the validity of the robustification strategy is assessed by means of Monte Carlo simulations. A set of stochastic control signals,ũ, given by Equation (29) was generated by sampling a uniform distribution on the interval from ±σ w . For each case presented in Figure 6, 5.0 × 10 3 forward simulations were carried out for the uncertainty level of 5.0%.…”