We consider a linear quadratic regulator problem in a general Banach space. The control is by an operator with range in the extrapolated Favard class. The observation operator is unbounded. We prove the existence of a Riccati operator which describes the value function for the optimal control and can be used to synthesize optimal feedback, similarly as in Hilbert spaces.
"Riccati operators in non-reflexive spaces." Differential Integral Equations 15 (12) 1493 - 1510, 2002. https://doi.org/10.57262/die/1356060709