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A quadratic regulator problem related to identification problems and singular systems

We report some results concerning the quadratic regulator problem with point penalization of the control for a class of distributed systems. This new form of the quadratic regulator problem is suggested by certain identification problems related to delay systems and by the quadratic regulator problem for singular systems. The problem studied does not admit an optimal control. Hence we study the properties of the value function and we characterize the set of initial conditions which admit an optimal control. Finally, we show that it is always possible to construct minimizing sequences which are "robust" under perturbations on the initial conditions (and reference signals)