Proof: Because of (24), γ ≥ 0, ρ > r, κ < κ, and βH > 0 it holds that
ρ - R + nβH ≥ вн + ' ' > 0.
κ — κ
This proves the lemma. <ι
Proof of Theorem 3: Suppose that all players j = i use the equilibrium strategies φj .In
this situation, player i faces the following optimal control problem Q:
+∞
- κxi(t)] dt
maximize
subject to
e-ρt[U(ci(t),yi(t)+γz(t))
0
. , 4 ~ , 4
z( t ) = Rz ( t ) — Xi ( t )
Уi ( t ) = ryi ( t ) + Xi ( t ) — Ci ( t )
yj(t) = ryj(t) + βHz(t) - ft[yj(t) + Yz(t)] for j = i
0 ≤ xi(t) ≤ βHz(t) , ci(t) ≥ 0 , y(t) ≥ 0 , z(t) ≥ 0,
where R = R — (n — 1)βH. The Lagrangian function of this problem is
L = U ( ci,yi+γz ) — κxi + λ ( R z—Xi )+μi ( ryi+Xi — ci )+∑ μj [ ryj + вн z — ft( yj+γz )] + ν ( вн z — Xi ),
j=i
where λ and μk, k = 1, 2,... ,n, have the same interpretation as in the proof of theorem 1 and
where ν is the Lagrange multiplier for (23). Note that the Lagrangian is jointly concave in
(z, y, ci, Xi). The theorem is therefore proved if there exist costate trajectories λ :[0, +∞) →
[0, + ∞ ) and μk : [0, + ∞ ) → [0, + ∞ ), k = 1, 2, ...,n, as well as a multiplier ν : [0, + ∞ ) →
[0, +∞) such that the feasibility conditions of problem Q as well as the following optimality
conditions are satisfied by ci(t) = ft[yi(t) + γz(t)] and xi(t) = βHz(t):
u' ( Ci ( t ) / [ yi ( t ) + γz ( t )]) — μi ( t ) = 0, (25)
μi ( t ) — λ ( t ) — κ — ν ( t ) = 0, (26)
λ(t) = (ρ — R)λ(t) — γw(Ci(t)/[yi(t) + γz(t)]) — (вн — γq) ∑ μj(t) — внν(t), (27)
j=i
μ i( t ) = ( ρ — r ) μ( t )— w ( ci( t ) /[ yi( t )+ Yz ( t )]), (28)
μ j( t ) = ( ρ — r+ ft) μj( t ) (29)
ν ( t )[вн z ( t ) — Xi ( t )] = 0, (30)
lim e-ρt λ (t) z (t) + ∑ μk (t) yk (t) ≤ 0. (31)
t→+∞
k=1
We define the costate trajectories by
λ(t) =
γ ( ρ — r ) κ + вн ( ft — κ )
ρ — R + пвн
μi ( t ) = ft,
μj(t) = 0 for j = i.
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