156

■Appendices

with transversality conditions

λ^∗

i ^{(tf) = 0,}

i = 1,...,4.

Furthermore,

u^∗(t) = min



max



0, ^λh(b)S∗

h^(t)

C

(λ^∗

2^(t)−λ∗

1^(t))



,1



.

(7.A.6)

Proof 7.A.1 Existence of an optimal solution (S^∗

h^,I∗

h^,S∗

v^,I∗

v^{) associated to an optimal con-}

trol u^∗comes from the convexity of the integrand of the cost function J with respect to the

control u and the Lipschitz property of the state system with respect to state variables

(Sh,Ih,Sv,Iv) (see, e.g., [11, 13]). System (7.A.5) is derived from the Pontryagin maxi-

mum principle (see (7.A.2), [34]) and the optimal controls (7.A.6) come from the mini-

mization condition (7.A.3). The optimal control pair given by (7.A.6) is unique due to the

boundedness of the state and adjoint functions and the Lipschitz property of systems (7.1)

and (7.A.5) (see, e.g., [17] and references cited therein).