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With the help of Lemma 11, [mathematical expression not reproducible] are all uniformly ultimately bounded (UUB).
From Theorems 7 and 14, it is noted that the weight matrices adaptation laws (8)-(11) during the process of modeling and the designed controller (32) guaranteed that the closed-loop system (1) is UUB.
The trajectories of the modeling errors ([e.sub.1] = [x.sub.1]-[[??].sub.1], [e.sub.2] = [x.sub.2]-[[??].sub.2]) by the RNN are shown in Figure 2, where we can see that the state estimation errors are UUB around the equilibrium.
According to the Boundedness Theorem , we can get the error using dynamic neural network to approximate the nonlinear system which is uniformly ultimately bounded (UUB) and converges to a set containing origin with a rate at least as fast as [e-.sup.[eta]t/2].
Since the stability of system identification and adaptive control has already been proven in Theorems 5 and 7, respectively, we can make a conclusion that the errors between the real system states and the desired reference inputs are uniformly ultimately bounded (UUB).
It is obviously seen that V(t) is UUB for all V(0) [less than or equal to] [B.sub.0] with [B.sub.0] being any positive constant.
To prove that [e.sub.u] and [[epsilon].sub.[beta]] are UUB, the Lyapunov function is given by
It is obvious that [[integral].sup.t.sub.0] [e.sub.u] [d.sub.[tau]] and are UUB and will be arbitrarily small by choosing suitable parameters.
Based on Lyapunov stability theory, the UUB property of [??] is achieved.
Then, the estimate error [??] is uniformly ultimately bounded (UUB).
By adopting the observer in (15) and controller in (21), the control error of system states and estimation error of observer are locally uniformly ultimately bounded (UUB).
Consequently, the control error of system states and estimation error of observer are locally UUB.
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