An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation (Q1372241)

The author considers the control system \[ \begin{aligned} y_t & =-y_x -\alpha y,\quad t>0,\;x>0, \\ y(0,x) & =y_0(x),\quad t\geq 0, \\ y(t,0) & =u(t),\quad t\geq 0,\end{aligned} \] the quadratic cost function \(J(u,y) = \int_0^\infty (| y(t,1)| ^2+| u(t)|^2)dt\) and the associated optimal control problem. Thus, the generator \(A\), the control operator \(B\) and the observation operator \(R\) are unbounded. It is shown that in despite of these bad properties, the problem reveals some striking features: (a) the unbounded gain operator \(B^\ast P\) vanishes on \(D(A)\); (b) the algebraic Riccati equation is satisfied on \(D(A)\) by \(P\), indeed as a Lyapunov equation.