Page 105 - Control Theory in Biomedical Engineering
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92 Control theory in biomedical engineering
ρT ∗
u ∗
∗
0
λ ðtÞ¼ 1+ λ 1 ðtÞ δ + a 1 ð1 e Þ + λ 2 ðtÞnT + λ 1 ðt + τÞχ μT ,
∗
1 ½0, t f τ
η +T ∗
u ∗
0
∗
∗
∗
λ ðtÞ¼ 1+ λ 2 r 2 +2r 2 β T + nE + c 1 N + a 2 ð1 e Þ
2 1
" #
ρE T ∗ ρE ∗
∗
+ λ 3 c 2 N + χ ½0, t f τ 1 ðt + τÞ + μE ∗ ,
λ
∗
∗ 2
ðη +T Þ η +T ∗
u ∗
0
∗
λ ðtÞ¼ λ 2 c 1 T λ 3 r 3 2r 3 β N c 2 T a 3 ð1 e Þ γ,
∗
∗
3 2
∗
∗
0
∗
λ ðtÞ¼ λ 1 ðtÞa 1 e u ∗ E + λ 2 ðtÞa 2 e u ∗ T + λ 3 ðtÞa 3 e u ∗ N + λ 4 ðtÞd 1 , ð9Þ
4
with transversality conditions
(
1 if t ½0, t f τ,
λ i ðt f Þ¼ 0, i ¼f1,2,3,4g and χ ¼ (10)
½0, t f τ
0 otherwise:
Furthermore, the following properties hold
λ 4 λ 1 s 1
∗
∗
v ¼ min v max , ,w ¼ min w max , : (11)
B v B w
Proof. The adjoint equations and transversality conditions can be
obtained by using Pontryagin’s minimum principle with delay in state
(Pontryagin et al., 1962) such that
∂H ∂H
0 ðtÞ χ
λ ðtÞ¼ ðtÞ ðt+ τÞ, λ 1 ðt f Þ¼ 0,
1 ½0, t f τ
∂E ∂E τ
∂H ∂H
0 ðtÞ χ
λ ðtÞ¼ ðtÞ ðt+ τÞ, λ 2 ðt f Þ¼ 0,
2 ½0, t f τ
∂T ∂T τ
∂H (12)
0
λ ðtÞ¼ , λ 3 ðt f Þ¼ 0,
3
∂N
∂H
0
λ ðtÞ¼ , λ 4 ðt f Þ¼ 0:
4
∂u
The optimality system consists of the state system (3) coupled with the
adjoint system (9) with initial and transversality conditions together with the
