Oberseminar Wissenschaftliches Rechnen (Prof. Martin Gugat)

Abstract:

The finite-time turnpike property is an extreme form of the turnpike property where the optimal state reaches a desired steady state after finite time. This steady state is in turn a solution of a static optimal control problem corresponding to the initial dynamic optimal control problem on a finite time interval [0, T].
This extreme situation can occur if the control cost is given by a non-smooth norm, for example with an L1-norm control cost. Moreover, to enforce the convergence to a steady state, a tracking term should be a part of the objective functional. The situation is clearly related to exact penalization, which is only possible with non-smooth penalty term.
We show that even if the tracking term is differentiable, the non-smooth control cost can lead to a finite-time turnpike phenomenon. A strictly convex tracking term has the advantage that it can enforce uniqueness of the optimal control.
We also discuss the approximation of the non-smooth control cost by smoothing kernels. The theory shows that the finite-time turnpike property is lost. But with increasing smoothing parameter, it is approximated, which is illustrated in numerical results.

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