Infinite horizon Riccati operators in nonreflexive spaces (Q2708568)

In this paper the following control problem is considered: Minimize \(J (u, x_0) = \int^{\infty}_0 [ \|C x (s) \|^2 + \|u (s) \|^2 ] ds \) over all \(u \in L^2 (R_+ , {\mathcal H})\) subject to NEWLINE\[NEWLINE\dot{x} = A x + B u , x (0) = x_0.NEWLINE\]NEWLINE Here \(A\) is the generator of an exponentially stable \(C_0\)-semigroup \(\exp(tA)\)in a non-reflexive Banach space \(X\), and \(B\) is a bounded linear operator from a Hilbert space \({\mathcal H}\) into the extrapolated Favard class \(F_{-1}\) corresponding to the semigroup, whereas \(C\) is a bounded linear operator from \(X\) to \({\mathcal H}\). NEWLINENEWLINENEWLINEUnder these hypotheses it is proved, that there exists a unique \(P \in {\mathcal L} (X, X^{\odot}) \) (where \(X^{\odot}\) is the sun dual of \((X, \exp ( t A))\) such that the following relations are fulfilled:NEWLINENEWLINENEWLINE(i) \(\langle P x , x \rangle \geq 0\) for all \(x \in X\).NEWLINENEWLINENEWLINE(ii) \(\langle P x , y \rangle = \langle P y , x \rangle\) for all \(x, y \in X\).NEWLINENEWLINENEWLINE(iii) If \(x, y \in F\) (the Favard class of \(\exp(tA)\)) and if \(u, v \in {\mathcal H} \) satisfy \(A_{-1} x + B u \in X , A_{-1} y + B v \in X \)NEWLINENEWLINENEWLINEthen NEWLINE\[NEWLINE\langle A_{-1} x , P y \rangle + \langle A_{-1} y ,P x \rangle + \langle C x , C y \rangle - \langle B^* P x , B^* P y \rangle = 0.NEWLINE\]NEWLINE Moreover, the optimal control \(\widehat{u}\) is given by \(\widehat{u} (t) = - B^* P \widehat{x} (t)\) where \(\widehat{x}\) is the mild solution of \(x = A x - B B^* P x , x (0) = x_0\), and \(\min_{u \in L^2 (R_+ , {\mathcal H})} J (u, x_0) = \langle P x_0 , x_0\rangle\) holds.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].