Nonlinear differential equations of Riccati type on ordered Banach spaces

In this paper we consider a general class of time-varying nonlinear differential equations on infinite dimensional ordered Banach spaces, which includes as special cases many known differential Riccati equations of optimal control. Using a linear matrix inequalities (LMIs) approach we provide necessary and sufficient conditions for the existence of some global solutions such as maximal, stabilizing and minimal solutions for this class of generalized Riccati equations. The obtained results extend to infinite dimensions and unify corresponding results in the literature. They provide useful tools for solving infinite-time linear quadratic (LQ) control problems for linear differential systems affected by countably-infinite-state Markovian jumps and/or multiplicative noise.