Stochastic optimization of parabolic PDE systems under chance constraints with application to temperature control of a bar

The state variables of many engineering processes are both space and time dependent. Such processes are frequently described with parabolic partial differential equations (PDEs). In most practical applications there exist uncertainties which are characterized through model parameters and random disturbances. The optimization of parabolic PDE systems with state constraints poses a great challenge. In this study, such state constraints are modeled as chance constraints. The stochastic optimization problem formulated in this way is solved by the inner and outer approximation method. The effectiveness and efficiency of the proposed approach is demonstrated by the temperature control of a bar with an uncertain heat transfer coefficient.