  In this paper, we propose a method for estimating the algebraic Riccati
equation (ARE) with respect to an unknown discrete-time system from the system
state and input observation. The inverse optimal control (IOC) problem asks,
``What objective function is optimized by a given control system?'' The inverse
linear quadratic regulator (ILQR) problem is an IOC problem that assumes a
linear system and quadratic objective function. The ILQR problem can be solved
by solving a linear matrix inequality that contains the ARE. However, the
system model is required to obtain the ARE, and it is often unknown in fields
in which the IOC problem occurs, for example, biological system analysis. Our
method directly estimates the ARE from the observation data without identifying
the system. This feature enables us to economize the observation data using
prior information about the objective function. We provide a data condition
that is sufficient for our method to estimate the ARE. We conducted a numerical
experiment to demonstrate that our method can estimate the ARE with less data
than system identification if the prior information is sufficient.
