On an Invariance Principle for the Solution Space of the Differential Riccati Equation

Abstract

The differential Riccati equation appears in different fields of applied mathematics like control theory and systems theory. For large-scale systems the numerical solution comes with a large amount of storage requirements. This motivates the use of Krylov subspace and projection based methods [1-3].In the present paper we apply an invariance theorem for ODEs to the differential Riccati Equation.We show that the solution is contained in a Krylov like subspace and extend our results to certain time-varying cases.

Order reduction methods for solving large-scale differential matrix Riccati equations

“…This provides enormous memory savings whenever the approximate solution is required at different time instances in [0, t f ]. Theoretical motivation for keeping the approximation space independent of the time-stepping is contained in [5], where it is shown that the solution of the DRE lives in an invariant Krylov-subspace 1 .…”

“…investigated in the literature; see, e.g., [36], [18]. The adaptive choice of shifts was tailored to the ARE in [32] by the inclusion of information of the term BB T during the shift selection; see also [43] for a more detailed discussion 5 . In our numerical experiments we used this last adaptive strategy, where the approximate solution at timestep t f is used.…”