In this paper, we extend classical approach to linear quadratic (LQ) optimal control via Popov operators to abstract linear differential-algebraic equations in Hilbert spaces. To ensure existence of solutions, we assume that the underlying differential-algebraic equation has index one in the pseudo-resolvent sense. This leads to the existence of a degenerate semigroup that can be used to define a Popov operator for our system. It is shown that under a suitable coercivity assumption for the Popov operator the optimal costs can be described by a bounded Riccati operator and that the optimal control input is of feedback form. Furthermore, we characterize exponential stability of abstract differential-algebraic equations which is required to solve the infinite horizon LQ problem.

No abstract

No abstract

We study stable differential-algebraic equations. Besides characterizing the stability in terms of a generalized Lyapunov inequality, we show that these systems can always be rewritten as port-Hamiltonian systems on the subspace where the solutions evolve.

We consider linear port-Hamiltonian differential-algebraic equations (pH-DAEs). Inspired by the geometric approach of Maschke and van der Schaft [11] and the linear algebraic approach Mehl, Mehrmann and Wojtylak [12], we present another view by using the theory of linear relations. We show that this allows to elaborate the differences and mutualities of the geometric and linear algebraic views, and we introduce a class of DAEs which comprises these two approaches. We further study the properties of matrix pencils arising from our approach via linear relations.