In the case when X=R", and f: R"bR is such that existence and uniqueness of solutions r:[0.∞)→ R" is guaranteed for all locally integrable inputs:0,∞)→ and V is continuously differentiable, the well-known dynamic programming conditio o W and all initial conditions ( to)= to E R, the infimum in(7. 12)(and hence, th corresponding storage function) do not depend on time. If, in addition, f is continue 1era1(,)-VY(),)0≤mr((n,a)-V()(n,a)y∈R (7.15) will be satisfied. However, using(7. 15)requires a lot of caution in most cases, since, even or very smooth f, o, the resulting storage function V does not have to be differentiable 7.3.3 Zames-Falb quadratic supply rate A non-trivial and powerful case of an implicitly defined storage function with a quadratic supply rate was introduced in late 60-s by G. Zames and P Falb Theorem 7. 3 Let A, B, c be matrices such that A is a Hurwitz matris, and Ceat Bdt 1 Let:R→→ R be a monotonic odd function such that 0≤00()≤|l2y∈R Then for all 8< 1 system (t)=Ar(t)+ Bw(t) has a non-megative storage function with supply rate 0+(,)=(-60(如)(-Cz and system i(t)=Ar(t)+B((t)-6o((t) has a non-negative storage function with supply rate a-(,)=(-60()-C) The proof of Theorem 7.3 begins with establishing that, for every function h: RHR with LI norm not exceeding 1, and for every square integrable function w: R HR the integr (w(t)-olw(t))y(t)dt where y= h*w, is non-negative. This verifies that the assumptions of Theorem 7.2 are satisfied, and proves existence of the corresponding storage function without actually finding it. Combining the Zames-Falb supply rate with the statement of the Kalman Yakubovich-Popov lemma yields the following stability criterion� � � � 9 In the case when X = Rn, and f : Rn ∈� Rn is such that existence and uniqueness of solutions x : [0,⊂) ∈� Rn is guaranteed for all locally integrable inputs w : [0,⊂) ∈� W and all initial conditions x(t0) = ¯x0 ≤ Rn, the infimum in (7.12) (and hence, the corresponding storage function) do not depend on time. If, in addition, f is continuous and V is continuously differentiable, the well-known dynamic programming condition x, ¯ x)f(¯ w)}0 → inf {ψ(¯ w)−⇒V (¯ x0, ¯ x0 ≤ Rn lim inf {ψ(¯ w)−⇒V (¯ x, ¯ x0, ¯ x0)f(¯ w)} � ¯ ��0,�>0 w¯→W,x¯→B�(¯x0) w¯→W (7.15) will be satisfied. However, using (7.15) requires a lot of caution in most cases, since, even for very smooth f, ψ, the resulting storage function V does not have to be differentiable. 7.3.3 Zames-Falb quadratic supply rate A non-trivial and powerful case of an implicitly defined storage function with a quadratic supply rate was introduced in late 60-s by G. Zames and P. Falb. Theorem 7.3 Let A, B, C be matrices such that A is a Hurwitz matrix, and |CeAtB|dt < 1. 0 Let � : R ∈� R be a monotonic odd function such that 0 → w�¯ (w¯) → |w¯ ¯ | 2 � w ≤ R. Then for all � < 1 system x˙ (t) = Ax(t) + Bw(t) has a non-negative storage function with supply rate ψ+(¯x, w¯) = (w¯ − ��(w¯))(w¯ − Cx¯), and system x˙ (t) = Ax(t) + B(w(t) − ��(w(t)) has a non-negative storage function with supply rate ψ−(¯x, w¯) = (w¯ − ��(w¯) − Cx¯)w. ¯ The proof of Theorem 7.3 begins with establishing that, for every function h : R ∈� R with L1 norm not exceeding 1, and for every square integrable function w : R ∈� R the integral (w(t) − �(w(t)))y(t)dt, −� where y = h � w, is non-negative. This verifies that the assumptions of Theorem 7.2 are satisfied, and proves existence of the corresponding storage function without actually finding it. Combining the Zames-Falb supply rate with the statement of the Kalman￾Yakubovich-Popov lemma yields the following stability criterion