AN OPTIMAL-CONTROL PROBLEM WITH UNBOUNDED CONTROL OPERATOR AND UNBOUNDED OBSERVATION OPERATOR WHERE THE ALGEBRAIC RICCATI EQUATION IS SATISFIED AS A LYAPUNOV EQUATION

We provide an optimal control problem for a one-dimensional hyperbolic equation over Omega = (O,infinity), with Dirichlet boundary control u (t) at x = 0, and point observation at x = 1, over an infinite time ho rizon. Thus, both control and observation operators B and R are unboun ded. Because of the finite speed of propagation of the problem, the in itial condition y(o)(x) and the control u(t) do not interfere. Thus, t he optimal control u(o)(t) drop 0. A double striking feature of this p roblem is that, despite the unboundedness of both B and R, (i) the (un bounded) gain operator BP vanishes over D(A), A being the basic (unbo unded) free dynamics operator, and (ii) the Algebraic Riccati Equation is satisfied by P on D(A), indeed as a Lyapunov equation (linear in P ).