A Linear Relation Approach to Port-Hamiltonian Differential-Algebraic Equations

Abstract

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.

On the stability of port-Hamiltonian descriptor systems

“…If (13) is stable then sE − DQ is regular. Hence, [GHR21] yields the regularity of sE − DQ. As a consequence of Lemma 4.3, sE − Q has row minimal indices at most zero.…”

“…Here, for simplicity, we include the port and the resistive variables already in the state. The DAE is then explicitly given by the range representation of the following linear relation A more general characterization of the eigenvalues of the index of port-Hamiltonian DAEs which are given by maximally dissipative subspaces that are not necessarily graphs of dissipative matrices can be found in [GHR21,Sec. 6].…”

“…This geometric formulation of pH-systems was studied extensively for classical systems, i.e., E = I n , in [vdSJ14]. Recently, the authors compared this geometric and the explicit formulation (1) in [GHR21] but for DAEs…”

Control of port-Hamiltonian differential-algebraic systems and applications

“…the columns of K L are associated with a Dirac structure. A further generalization is discussed in [86]. Clearly, if K = S = I, then P = P T and S = −S T and we are in the case of dHDAE systems with E = E T = P , S = −S T = J, where the extra condition E ≥ 0 has to be assumed.…”

“…For LTI DAE systems the characterization of stability via different generalized Lyapunov equations and the relation to pHDAE systems has recently been studied in different contexts e.g. in a behavior context in [85], via generalized Kalman-Yakubovich-Popov inequalities in [190,193], or via linear relations in [86].…”