A representation formula for the solutions to operator Riccati equation (Q1198705)

We are concerned with the Riccati equation (1) \(P'=A^*P+PA- PBB^*P+C^*C\) \((t\geq 0)\), \(P(0)=P_ 0\), and with the dual equation (2) \(Q'=AQ+QA^*-QC^*CQ+BB^*\) \((t\geq 0)\), \(Q(0)=0\), where \(P_ 0\in\Sigma^ +(H)\), the set of all symmetric nonnegative operators on a Hilbert space \(H\). The linear operators \(A,B,C\) also verify usual assumptions in order that equations (1) and (2) have unique solutions \(P,Q\in C_ s([0,+\infty[;\Sigma^ +(H))\), the space of all strongly continuous mappings from \([0,+\infty[\) into \(H\). We present a simple formula which gives \(P\) in terms of \(Q\), then we give an application to boundary control problems for hyperbolic equations. The formula reeds as follows: \(P(T)=U_{G_ T}(T,0)[I+P_ 0Q(t)]^{- 1}P_ 0U^*_{G_ T}(T,0)+\int^ T_ 0U_{G_ T}(T,s)C^*CU^*_{G_ T}(T,s)ds\), where \(T>0\) is fixed and \(U_{G_ T}\) is the evolution operator associated with the family of linear operators \(G_ T(t)=A^*-C^*CQ(T-t)\), \(t\in[0,T]\).