Descriptor variable systems and optimal state regulation

Linear systems of the form
E\dot{x} = Ax + Bu
with
E
singular are treated. It is desired to find a control which drives the system asymptotically to the origin, minimizing a quadratic cost functional. No restrictions are placed on initial conditions. The cost associated with the impulsive behavior of the system is examined as well as existence and uniqueness of the optimal control. Through a sequence of coordinate transformations it is proven that the optimal control can be found by solving a reduced order Riccati equation.