The topic of the master thesis is linear-quadratic optimal control of
time-varying and time-invariant differential-algebraic equations (DAEs).
The thesis is divided into two main parts: in the first part, we
investigate linear time-varying DAEs. We recall the solution theory of DAEs
and introduce the optimal control problem. We then proceed to prove that
the optimal value is a quadratic function and fulfils Bellman's principle
of optimality. Using these results, we can characterize the optimal value
as an extremal solution of the Kalman-Yakubovich-Popov inequality. In the
second part, we turn our attention towards time-invariant, regular DAEs. We
first derive a differentiability condition that the control input of the
system needs to fulfil. Using these results, we introduce an augmented
system that includes certain derivatives of the input as system states. For
this augmented system, an optimal control problem equivalent to the one of
the nominal system is defined that can be solved easily using results from
the theory of optimal control for ordinary differential equations. This
enables us to explicitly calculate the optimal control of the nominal DAE,
which can be implemented as a state feedback as well.