After being discriminated into two classes according to respective abstract assumptions, the optional control problem, abstract equation \(y_ t=Ay+Bu\), \(y(0)=y_ 0\), together with the cost functional \[ J(u,y)=\int^ T_ 0[\| Ry(t)\|^ 2_ z+\| u(t)\|^ 2_ u]dt+\| Gy(t)\|^ 2_ w, \] is discussed in the notes, where \(R\) and \(G\) are observation operators. The main attention is put on the existence of Riccati operator \(p(t)\), which is the solution to the associated Riccati equation, differential when \(T<\infty\) or algebraic when \(T=\infty\). To justify the abstract theory, some illustrative examples of boundary/point control problems for partial differential equations and of numerical approximation are given.