Theorem 7.4 Assume that matrices Ap, Bp, Cp are such that Ap is a Hurwitz matriz, and there exists E >0 such that Re(1-G(ju)(1-H(ju)≥∈u∈R, where H is a Fourier transform of a function with L1 norm not exceeding 1, and G(s)=Cp(SI -Ap)B Then system i(t)=Ap.c(t)+Bpo(c(t)+u(t)) has finite L2 gain, in the sense that there exists ?>0 such that for all solutions 7. 4 Example with cubic nonlinearity and delay Consider the following system of differential equations with an uncertain constant delay parameter T ()=-x1(t)3-x2(t-7)3 i2(t)=x1(t)-x2(t) Analysis of this system is easy when T=0, and becomes more difficult when T is an arbitrary constant in the interval [0, To. The system is not exponentially stable for any value of T. Our objective is to show that, despite the absence of exponential stability, the method of storage functions with quadratic supply rates works The caseT=0 For T=0, we begin with describing(7. 16),(7.17) by the behavior set ere 1-t2,x2=1-2 Quadratic supply rates for which follow from the linear equations of 2 are given by oLTI (a)=2 P-wr sUggested by Petar Kokotovich� � � � 10 Theorem 7.4 Assume that matrices Ap, Bp, Cp are such that Ap is a Hurwitz matrix, and there exists γ > 0 such that Re(1 − G(jσ))(1 − H(jσ)) ∗ γ � σ ≤ R, where H is a Fourier transform of a function with L1 norm not exceeding 1, and G(s) = Cp(sI − Ap) −1 Bp. Then system x˙ (t) = Apx(t) + Bp�(Cx(t) + v(t)) has finite L2 gain, in the sense that there exists θ > 0 such that |x(t)| 2 dt → θ(|x(0)| 2 + |v(t)| 2 dt 0 0 for all solutions. 7.4 Example with cubic nonlinearity and delay Consider the following system of differential equations2 with an uncertain constant delay parameter φ : x˙ 1(t) = −x1(t) 3 − x2(t − φ ) 3 (7.16) x˙ 2(t) = x1(t) − x2(t) (7.17) Analysis of this system is easy when φ = 0, and becomes more difficult when φ is an arbitrary constant in the interval [0, φ0]. The system is not exponentially stable for any value of φ . Our objective is to show that, despite the absence of exponential stability, the method of storage functions with quadratic supply rates works. The case φ = 0 For φ = 0, we begin with describing (7.16),(7.17) by the behavior set Z = {z = [x1; x2; w1; w2]}, where 3 3 w1 = x1, w2 = x2, x˙ 1 = −w1 − w2, x˙ 2 = x1 − x2. Quadratic supply rates for which follow from the linear equations of Z are given by � �∗ � � x1 −w1 − w2 ψLTI (z) = 2 P x1 − x2 , x2 2Suggested by Petar Kokotovich