Inthisnoteweshowthatforagivencontrollablepair(A,B)andanyλ＞0,againmatrixKcanbechosensothatthetransitionmatrixe(A+BK)tofthesystemx=(A+BK)xdecaysattheexponentialratee-λtandtheovershootofthetransitionmatrixcanbeboundedbyMλLforsomeconstantsMandLthatareindependentofλ.Asaconsequence,foranyh＞0,againmatrixKcanbechosensothatthemagnitudeofthetransitionmatrixe(A+BK)tcanbereducedby-1/2-(orbyanygivenportion)over[0,h].Aninterestingapplicafonoftheresultisinthestabilizationofswitchedlinearsystemswithanygivenswitchingrate(see[1]).
