Sequential decomposition of the matrix Riccati equation and its application to the linear quadratic regulator problem

The differential matrix Riccati equation for the multi-input-multi-output linear quadratic optimal regulator problem is considered. Two methods are presented, successive and parallel, that decompose this equation into a set of Riccati equations that correspond to optimal regulator problems of possibly reduced dimensions. The additivity of the solutions to the equations obtained by the successive sequential decomposition method (SDM) and the additivity of the inverses of the solutions to the equations obtained by the parallel SDM are established. Some duality relations between the successive and the parallel methods are presented via the use of the adjoint Riccati equation. The theory developed is extended to the algebraic matrix Riccati equation as a limiting case. The application of the SDMs in the infinite-time linear quadratic regulator problem is investigated. Special attention is paid to the partially ‘cheap’ problem where the cost of some of the regulator controls or of their combinations tends asymptotically to zero. Explicit expressions for the asymptotic optimal cost are derived and the behaviour of the asymptotic optimal root loci is investigated.