This thesis is an elaboration of the paper "The Kalman-Yakubovich-Popov
inequality for differential-algebraic systems" by Timo Reis, Olaf Rendel,
and Matthias Voigt from 2015. Based on their results, we aim to handle the
linear-quadratic optimal control problem with differential-algebraic
constraints. The considered approach uses the Kalman-Yakubovich-Popov
lemma, which relates the positive semi-definiteness of the Popov function
on the imaginary axis to the solvability of a linear matrix inequality.
Particular solutions of this inequality are provided by the Lur'e equation,
which in turn can be solved via deflating subspaces of certain matrix
pencils. These solutions enable both the calculation of the optimal costs
and the characterization of the solution of the optimal control problem.