ThispapergivesatutorialonhowtoproveLyapunovtypecriteriabyoptimalcontrolmethods.Firstly,weconsiderstabilitycriteriaonHill'sequationswithnonnegativepotential.Byoptimalcontrolmethodsdevelopedin1990s,weobtainseveralstabilitycriteriaincludingLyapunov'scriterion,Ne..gauzandLidski'scriterion.Secondly,wepresentstabilitycriteriaonHill'sequationswithsign-changingpotentialinwhichBrog'scriterionandKrein'scriterionareincluded.