Appendices

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7.A

UNIQUE OPTIMAL SOLUTION

According to the Pontryagin Maximum Principle [34], if u^∗(·) ∈Ωis optimal for

the problem (7.1), (7.3) with the initial conditions given in Table 7.1 and fixed final

time tf, then there exists a nontrivial absolutely continuous mapping λ : [0,tf] →R⁴,

λ(t) = (λ1(t),λ2(t),λ3(t),λ4(t)), called adjoint vector, such that

˙Sh = ^∂H

∂λ1

,

˙Ih = ^∂H

∂λ2

,

˙Sv = ^∂H

∂λ3

,

˙Iv = ^∂H

∂λ4

(7.A.1)

and

˙λ1 = −^∂H

∂Sh

,

˙λ2 = −^∂H

∂Ih

,

˙λ3 = −^∂H

∂Sv

,

˙λ4 = −^∂H

∂Iv

,

(7.A.2)

where function H defined by

H = H(Sh,Ih,Sv,Iv,λ,u)

= A1Ih + ^C

2 ^u2

+λ1 (Λh −(1−u)λhSh +γhIh −µhSh)

+λ2 ((1−u)λhSh −(µh +γh +δh)Ih)

+λ3 (Λv −λvSv −µvbSv)

+λ4 (p2λvSv −µvbIv)

is called the Hamiltonian, and the minimization condition

H(S^∗

h^(t),I∗

h^(t),S∗

v^(t),I∗

v^{(t),λ∗(t),u∗(t))}

= min

0⩽u⩽1^H(S∗

h^(t),I∗

h^(t),S∗

v^(t),I∗

v^{(t),λ∗(t),u)}

(7.A.3)

holds almost everywhere on [0,tf]. Moreover, the transversality conditions

λi(tf) = 0,

i = 1,...,4,

(7.A.4)

hold.

Theorem 7.A.1 Problem (7.1), (7.3) with fixed initial conditions Sh(0), Ih(0), Sv(0) and

Iv(0) and fixed final time tf, admits an unique optimal solution (S^∗

h^(·),I∗

h^(·),S∗

v^(·),I∗

v^(·))

associated to an optimal control u^∗(·) on [0,tf]. Moreover, there exists adjoint functions

λ^∗

1^{(·), λ∗}

2^{(·), λ∗}

3^{(·) and λ∗}

4^{(·) such that}































˙λ^∗

1^{(t) = λ∗}

1^{(t)((1−u∗(t))λh +µh)−λ∗}

2^{(t)λh(1−u∗(t))}

˙λ^∗

2^{(t) = −A1 −λ∗}

1^{(t)γh +λ∗}

2^{(t)(µh +γh +δh)}

˙λ^∗

3^{(t) = λ∗}

3^{(t)(λv +µvb)−λ∗}

4^(t)(λv))

˙λ^∗

4^{(t) = λ∗}

4^{(t)µvb ,}

(7.A.5)