49-02 Research exposition (monographs, survey articles)
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In this paper a general form of the infinite-horizon linear quadratic control problem is considered. We will discuss quadratic cost functionals which involve not only the state and input-variables but also derivatives of the state and input-variables of arbitrary order under constraints given by linear systems of higher order. We will examine two results that relate the linear quadratic control problem to an optimality system, which is given through a para-Hermitian matrix polynomial. The results can be applied to general rectangular descriptor systems (see Subsection 6.1) to obtain results which so far were only known for quadratic descriptor systems. Also we will see that the notion of dissipativity (when introduced in the proper way) is equivalent to the solvability of the linear quadratic control problem.