<p>This paper studies both the finite-horizon and the infinite-horizon H-infinity optimal control for linear systems under imperfect state measurements using the game theoretic approach. In the finite-horizon case, the result to this problem is well-known that the solution exists under the existences of solutions to two generalized Riccati differential equations and that they satisfy the spectral radius condition. In the existing game theoretic approach to this problem, it calls for the separation principle, which is a deep principle that is not obvious for beginners. This paper offers a completely elementary solution to the problem that uses the completion of squares method and Riccati differential equation solutions. This solution method calls for estimation and control sequential design, and only after the controller has been obtained, do I relate the over all solution to the control Riccati differential equation. In the infinite-horizon case, the standard approaches call for the inner systems, which is something that is not familiar with the control community. I obtain the exact conditions under which the infinite-horizon problem will admit an internally stabilizing solution. This exact set of conditions basically solves the H-infinity optimal control problem completely in conjunction with Theorem&#xa0;9.8 of Başar and Bernhard (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\({H}^\infty\)</EquationSource> </InlineEquation>-optimal control and related minimax design problems: a dynamic game approach, Birkhäuser, Boston, 1995). I then obtain another set of exact conditions as a corollary, which is quite standard, under which the infinite-horizon problem will admit an internally stabilizing solution that is the appropriate limit of the finite-horizon solution.</p>