An optimal control problem with unbounded control operator and unbounded observation operator where the Algebraic Riccati Equation is satisfied as a Lyapunov equation Original Research Article

We provide an optimal control problem for a one-dimensional hyperbolic equation over Ω = (0, ∞), with Dirichlet boundary control u(t) at x = 0, and point observation at x = 1, over an infinite time horizon. Thus, both control and observation operators B and R are unbounded. Because of the finite speed of propagation of the problem, the initial condition y0(x) and the control u(t) do not interfere. Thus, the optimal control u0(t) ≡ 0. A double striking feature of this problem is that, despite the unboundedness of both B and R, 1. (i) the (unbounded) gain operator B∗P vanishes over D(A), A being the basic (unbounded) free dynamics operator, and 2. (ii) the Algebraic Riccati Equation is satisfied by P on D(A), indeed as a Lyapunov equation (linear in P).