The turnpike phenomenon for dynamic optimal control problems provide insights about the relation between the dynamic optimal control and the solution of the corresponding static optimal control problem. In this talk we give an overview about different turnpike structures for optimal control problems with ordinary differential equations (ODEs) and partial differential equations (PDEs).
For optimal control problems with ODEs an exponential turnpike inequality can be shown by basic control theory. These results can be extended to an integral turnpike inequality for optimal control problems with linear hyperbolic systems. For an optimal control problem with non differential tracking term in the objective function, that is exactly controllable, we can show under certain assumptions that the optimal system state is steered exactly to the desired state after finite time.
Further we consider an optimal control problem for a hyperbolic system with random boundary data and we show the existence of optimal controls.
A turnpike property for hyperbolic systems with random boundary data can be shown numerically.