REGULARITY AND SINGULARITY IN LINEAR-QUADRATIC CONTROL SUBJECT TO IMPLICIT CONTINUOUS-TIME SYSTEMS

A linear-quadratic (LQ) control problem subject to a standard continuo us-time system is called regular if the input weighting matrix is inve rtible, and singular if this is not the case. Consequently, optimal in puts for regular LQ problems are ordinary functions (state feedbacks), whereas optical controls for singular problems are in general distrib utions, e.g., impulses. We will show that regularity and singularity i n LQ problems subject to a general (implicit) system depends not so mu ch on the input weighting matrix, as on the property that the integran d of the cost criterion is a function only if inputs and state traject ories are, as is the case for LQ problems, subject to a standard syste m. In particular, we will provide a simple criterion for distinguishin g between regularity and singularity in LQ problems subject to a gener al system. Our criterion is expressed in the system coefficients only and reduces to the classical one if the underlying system is standard.