The linear-quadratic optimal regulator for descriptor systems

In this paper we investigate the linear-quadratic optimal regulator problem for the continuous-time descriptor system
E\dot{x} = Ax + Bu
where
E
is, in general, a singular matrix. We solve first a general finite-horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor, codescriptor, and control vectors. From this trajectory the optimal feedback gain relating the control and descriptor variable can be computed. By transforming to a coordinate system which can be computed by performing a singular value decomposition of
E
we derive several Riccati differential equations, all of which have the same solution; this solution gives the optimal cost. The steady-state optimal feedback gain can be computed by solving an eigenvalue-eigenvector problem formulated from the untransformed system parameters. In general, there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of
E
.