Past-time state feedback solution of infinite dimensional LQ optimal regulator problem (Q2732547)

In the paper the author considers a past-time linear quadratic optimal control problem on the infinite-time interval, i.e., considers the following infinite dimensional system: NEWLINE\[NEWLINEy_{s,x}(t,u(\cdot))=e^{(t-s)A}+\int_{s}^{t} e^{(t-\tau)A} Bu(\tau) d\tau, s\leq t<\infty,NEWLINE\]NEWLINE with the quadratic cost functional NEWLINE\[NEWLINE J_{s,x}(u(\cdot))=\frac 12\int_s^{\infty} [\langle Qy_{s,x}(t,u(\cdot)), y(t,u(\cdot))\rangle +\langle Ru(t),u(t)\rangle ]dt. NEWLINE\]NEWLINE He, first, establishes the link between the infinite-dimensional integral Riccati equation and the corresponding Fredholm integral equation. Using the above result and the Bellman optimality principle, he obtains the past-time state feedback solution for the above-mentioned problem.