fort1≥to,30∈B. Combining this with v≥0 yields I(a, to, t1)2-V(a, to)=-V(2o, to for all z, zo defining same state of B at time to Now let us assume that(a) is valid. Then a finite infimum in(7. 12)exists(as an nfimum over a non-empty set bounded from below and is not positive(since I(2o, to, to) 0). Hence V is correctly defined and not negative. To finish the proof, let us show that (7. 13)holds. Indeed, if a1 defines same state as zo at time ti then o1(1)= x0(1),t≤t1 t>t1 defines same state as zo at time to ti(explain why). Hence the infimum of I(a, to, t)in the definition of V is not larger than the infimum of integrals of all such z01, over intervals of length not smaller than ti -to. These integrals can in turn be decomposed into two integral T(c01,to,t)=工(20,to,t1)+I(21,t1,t) which yields the desired inequality 7. 3.2 Storage functions for ODE models As an important special case of Theorem 7.2, consider the ODE model i(t)=f(ar(t), w(t)) 14 defined by a function f: XXWHR, where X, w are subsets of R" and R" espectively. Cxonsider the behavior model B consisting of all functions z(t)=x(t); u(t) where r: 0, oo)+X is a solution of(7 14). In this case, two signals 21=[1; U1 and 22=[2; v2] define same state of B at time to if and only if i(to)=a2(to). Therefore according to Theorem 7.2, for a given function o: X HR, existence of a function V:X×R R such th vx(2),t)-V(x(t1,t)≤/o(x(,0(t)dt for all0≤t1≤t2,[z;可]∈ B is equivalent to finiteness of the infimum of the integrals o(a(),u(r))dr over all solutions of(7. 14)with a fixed a (to)=To which can be extended to the time interval[0.∞)� � 8 for t1 ∗ t0, z0 ≤ B. Combining this with V ∗ 0 yields I(z, t0, t1) ∗ −V (z, t0) = −V (z0, t0) for all z, z0 defining same state of B at time t0. Now let us assume that (a) is valid. Then a finite infimum in (7.12) exists (as an infimum over a non-empty set bounded from below) and is not positive (since I(z0, t0, t0) = 0). Hence V is correctly defined and not negative. To finish the proof, let us show that (7.13) holds. Indeed, if z1 defines same state as z0 at time t1 then z0(t), t → t1, z01(t) = z1(t), t > t1 defines same state as z0 at time t0 < t1 (explain why). Hence the infimum of I(z, t0, t) in the definition of V is not larger than the infimum of integrals of all such z01, over intervals of length not smaller than t1 − t0. These integrals can in turn be decomposed into two integrals I(z01, t0, t) = I(z0, t0, t1) + I(z1, t1, t), which yields the desired inequality. 7.3.2 Storage functions for ODE models As an important special case of Theorem 7.2, consider the ODE model x˙ (t) = f(x(t), w(t)), (7.14) defined by a function f : X × W ∈� Rn, where X, W are subsets of Rn and Rm respectively. Cxonsider the behavior model B consisting of all functions z(t) = [x(t); v(t)] where x : [0,⊂) ∈� X is a solution of (7.14). In this case, two signals z1 = [x1; v1] and z2 = [x2; v2] define same state of B at time t0 if and only if x1(t0) = x2(t0). Therefore, according to Theorem 7.2, for a given function ψ : X × W ∈� R, existence of a function V : X × R+ ∈� R+ such that t2 V (x(t2), t2) − V (x(t1), t1) → ψ(x(t), v(t))dt t1 for all 0 → t1 → t2, [x; v] ≤ B is equivalent to finiteness of the infimum of the integrals � t ψ(x(φ ), v(φ ))dφ t0 over all solutions of (7.14) with a fixed x(t0) = ¯x0 which can be extended to the time interval [0,⊂)