here P= P is an arbitrary symmetric 2-by-2 matrix defining storage function ViTI(2(), t)=2(t)'Pa(t) Among the non-trivial quadratic supply rates o valid for 2, the simplest are defined by 2)+g22(x1-x2), with the storage function L(2(),t)=0.25(q1x1()+g2x2(t)4) where dk>0. It turns out(and is easy to verify) that the only convex combinations of these supply rates which yield o<0 are the ones that make o= aLTI +oNL =0, for example 0.50 P=00,d1=d2=g=1,q1=0 The absence of strictly negative definite supply rates corresponds to the fact that the system is not exponentially stable. Nevertheless, a Lyapunov function candidate can be constructed from the given solution V(x)=x'Px+0.25(1x1+gx2)=0.5x2+0.25x2 This Lyapunov function can be used along the standard lines to prove global asymptotic stability of the equilibrium =0 in system(7. 16), (7.17) 7.4.1 The general case Now consider the case when T E[ 0, 0. 2 is an uncertain parameter. To show that the delayed system(7. 16), (7. 17) remains stable when T <0. 2,(7.16),(7. 17) can be represented by a more elaborate behavior set 2=2 with z={x1;x2;1;23;4;ws;∈R satisfying Lti relations and the nonlinear/infinite dimensional relations 1(t)=x1 =x2-(x2+m 7)-x2(1) )3 Some additional supply rates /storage functions are needed to bound the new variables These will be selected using the perspective of a small gain argument. Note that the11 where P = P∗ is an arbitrary symmetric 2-by-2 matrix defining storage function VLT I (z(·), t) = x(t) ∗ P x(t). Among the non-trivial quadratic supply rates ψ valid for Z, the simplest are defined by ψNL(z) = d1x1w1 + d2x2w2 + q1w1(−w1 − w2) + q2w2(x1 − x2), with the storage function VNL(z(·), t) = 0.25(q1x1(t) 4 + q2x2(t) 4 ), where dk ∗ 0. It turns out (and is easy to verify) that the only convex combinations of these supply rates which yield ψ → 0 are the ones that make ψ = ψLT I + ψNL = 0, for example � � 0.5 0 P = , d1 = d2 = q2 = 1, q1 = 0. 0 0 The absence of strictly negative definite supply rates corresponds to the fact that the system is not exponentially stable. Nevertheless, a Lyapunov function candidate can be constructed from the given solution: 4 4 2 4 V (x) = x∗ P x + 0.25(q1x1 + q2x2) = 0.5x1 + 0.25x2. This Lyapunov function can be used along the standard lines to prove global asymptotic stability of the equilibrium x = 0 in system (7.16),(7.17). 7.4.1 The general case Now consider the case when φ ≤ [0, 0.2] is an uncertain parameter. To show that the delayed system (7.16),(7.17) remains stable when φ → 0.2, (7.16),(7.17) can be represented by a more elaborate behavior set Z = {z(·)} with z = [x1; x2; w1; w2; w3; w4; w5; w6] ≤ R8 , satisfying LTI relations x˙ 1 = −w1 − w2 + w3, x˙ 2 = x1 − x2 and the nonlinear/infinite dimensional relations 3 3 3 w1(t) = x1, w2 = x2, w3 = x2 − (x2 + w4) 3 , 3 w4(t) = x2(t − φ ) − x2(t), w5 = w4, w6 = (x1 − x2) 3 . Some additional supply rates/storage functions are needed to bound the new variables. These will be selected using the perspective of a small gain argument. Note that the