- In optimal control problems, often initial data are required
that are not known exactly in practice.
In order to take into account this uncertainty,
we consider optimal control problems for a system with an uncertain initial
state. A finite terminal time is given. On account of the uncertainty of the
initial state, it is not possible to prescribe an exact terminal state.
Instead, we are looking for controls that steer the system into a given
neighborhood of the desired terminal state with sufficiently high
probability. This neighborhood is described in terms of an inequality for
the terminal energy. The probabilistic constraint in the considered optimal
control problem leads to optimal controls that are robust against the
inevitable uncertainties of the initial state.
We show the existence of such optimal controls.
Numerical examples with
optimal Neumann control of the wave equation are presented.