Global Stability Proofs for Continuous-Time Indirect Adaptive Control Schemes

$2) E c b (a) x c b (2). By the second assumption of Theorem 2, the map S is a contraction and has a unique fixed point 6 = which is equivalent to (9). By (I), (91, and Ito's formula, we have e-afdl(x,)-@l(0)= l b e-" ' [ { b $ ; G (b Q ;) + F (o Q ;) }-f ~ l (x ~) 03 + 1' e-~'+;[au+bul(x,) d ~ + 1' e-aV;(x,) d ~. T h g the expectation and letting t go to m, we obtain, by the dominated convergence theorem, Jl(u, u)-dl(o)=E [ 1, e-" '{bQ;(G(bQ;)+u)}(x,) ds1 0 + E [ [m e-asHl(u)(xs) c l s ] (16) where H,(U)=(~/~)(U+~Q;)~+F(~Q;)-(~/~)(UQ;)~.