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Additional resources for Differential Equations Volume 43, Number 6 June, 2007
One has the energy estimate ∞ 2 uh L2,∞ (Q) + Dt L η h 2 h L2 (Q) + E e L1,∞ (Q) Sr s0 , eh + 2 0 Proof. For m = 1, we set uhΓ = 1 − (τ ) uhΓ x=X = uX , 0 ≤ h ≤ 1, and Dt uhΓ = 1 − ) Dt u(τ 0 + h h h (τ ) h (τ ) uX , u0 + ≤ K(N ). 2) L1,∞ (Q) h (τ ) = Ih η h /V h . Note that uhΓ ) h h Dt u(τ X + dV u − uΓ , x=0 = u0 , Dˆ uhΓ = dV η h , where dV = uX − u0 /V h . We set uhΓ = uX for m = 2 and uhΓ = 0 for m = 3. Let us introduce the functions v h = uh − uhΓ and σΓ = (1 − x/X)σ0 + (x/X)σX . By taking the inner product of Eq.
6 2007 STABILIZATION OF PROGRAMMED MOTIONS BY THE VIRTUAL OUTPUT . . 769 The structure of system (3) is similar to that of system (2) with output y = z. Following , we say that the nonautonomous system (3) is represented in normal form, and the system of equations ψ˙ = q¯(0, ψ, t), (5) where q¯(0, 0, t) = 0, is referred to as the zero dynamics of the nonstationary system (3). If the equilibrium point ψ = 0 of the zero dynamics is (uniformly) asymptotically stable, then the aﬃne system (3) with output y = e is referred to as a (uniformly) minimal-phase system.
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