Lyapunov inequality for linear elliptic equation, an optimal control approach

This paper is devoted to establishing the Lyapunov inequality for PDE by means of an optimal control approach, via which we obtain the important results in paper [4], and some new results in this direction. In the supercritical case, through the necessary condition of the optimal control pair, we show that the solution space of the optimal control equation is 1-d, by which we obtain the optimal feedback control coinciding with the result of theorem 1; In the critical case, instead of the optimal feedback control, we show Pontraginpsilas Maximum Principle for the optimal control pair, which is the necessary condition for the optimal control pair.