A representation formula for the solutions to operator Riccati equation
Viorel Barbu, Giuseppe Da Prato · Differential and Integral Equations · 1992
We are here concerned with the following problem Minimize:over all controls u E L 2 (t, T; U), subject to the state equation:Here Hand U are Hilbert spaces with the norms I• I and I• lu,A: D(A) C H -t H is the infinitesimal generator of a strongly continuous semigroup etA in H and B E £(U; D(A*)').We shall denote by ( •, •) and ( •, •) the scalar products in H and U, respectively.Moreover, P0 , C E £(H) and Po is a symmetric nonnegative operator.We shall take B of the form B = (>.0-A)E where .\0 is a point in the resolvent set of A and E E £(U, H).We shall assume, following [5], that: { There exists J{ > 0 such that J 0 T IE* A*esA* xl 2 ds :"::: Klxl 2 , Vx E D(A*).