A time-varying ODE model i1(t)=a1(x1(t),t) (5.6) can be converted to (5.2) by introducing r(t)=[x1(t);t,a(barx;r)=[a1(Z,7);1], n which case the Lyapunov function V= v((t))=v(i(t), t) can naturally depend on time 5.1.3 Storage functions for ODE models Consider the ode model i(t)=f(ar(t), u(t) with state vector a()∈XcR", input u(t)∈UcR", where f:XxU→R"isa given function.Leta:X×U→→ R be a given functional. A function v:X→Ris called a storage function with supply rate o for system(5.7) v(a(ti))-v(a(to)) (x(t),(t)) for every pair of integrable functions x: [to, tibX, u: to, ti H U such that the composition tH f(a(t), u(t)satisfies the identity ar(t)=r(to)+/f(r(t),u(t)dt ∈[o,t When X is an open set, f and o are continuous, and v is continuously differentiable verifying that a given f is a valid storage function with supply rate g is straightforward it is sufficient to check that VV·f(,a)≤0(z,)VE∈X,i∈U When V is locally Lipschitz, the following generalization of Theorem 5. 1 is available Theorem5.2 If X is an open set in R,V:X→ R is locally lipschitz,f,a:X×U→ R lim V(I+tf(,i))-V(r) ≤(x,)Vz∈X,∈ (5.8) is satisfied then V(r(t)) is a storage function with supply rate o for system(5.7) The proof of the theorem follows the lines of Theorem 5. 1. Further generalizations to discontinuous functions f, etc, are possible� 5 A time-varying ODE model x˙ 1(t) = a1(x1(t),t) (5.6) can be converted to (5.2) by introducing x(t) = [x1(t);t], a([barx; � ]) = [a1(¯x, � ); 1], in which case the Lyapunov function V = V (x(t)) = V (x1(t),t) can naturally depend on time. 5.1.3 Storage functions for ODE models Consider the ODE model x˙ (t) = f(x(t), u(t)) (5.7) with state vector x(t) ≤ X � Rn, input u(t) ≤ U � Rm, where f : X × U ≡� Rn is a given function. Let � : X × U ≡� R be a given functional. A function V : X ≡� R is called a storage function with supply rate � for system (5.7) t1 V (x(t1)) − V (x(t0)) ∀ �(x(t), u(t))dt t0 for every pair of integrable functions x : [t0,t1] ≡� X, u : [t0,t1] ≡� U such that the composition t ≡� f(x(t), u(t)) satisfies the identity � t x(t) = x(t0) + f(x(t), u(t))dt t0 for all t ≤ [t0,t1]. When X is an open set, f and � are continuous, and V is continuously differentiable, verifying that a given f is a valid storage function with supply rate � is straightforward: it is sufficient to check that ∈V · f(¯x, u¯) ∀ �(¯x, u¯) � x¯ ≤ X, u¯ ≤ U. When V is locally Lipschitz, the following generalization of Theorem 5.1 is available. Theorem 5.2 If X is an open set in Rn, V : X ≡� R is locally Lipschitz, f, � : X×U ≡� Rn are continuous, and condition V (¯x + tf(¯x, u¯)) − V (¯x) lim sup ∀ �(¯x, u¯) � x¯ ≤ X, u¯ ≤ U (5.8) ��0,�>0 0<t<� t is satisfied then V (x(t)) is a storage function with supply rate � for system (5.7). The proof of the theorem follows the lines of Theorem 5.1. Further generalizations to discontinuous functions f, etc., are possible