Page 104 - Control Theory in Biomedical Engineering
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Modeling and optimal control of cancer-immune system 91
keep normal cells above the average of its capacity. Therefore, our objective
is to maximize the functional (see Rihan et al., 2014a)
Z
t f
B v 2 B w 2
max Jðv,wÞ¼ EðtÞ TðtÞ ½vðtÞ + ½wðtÞ dt, (4)
v,w 2 2
0
where B u , B w are, respectively, the weight factors that describe the patient’s
acceptance level of chemotherapy and immunotherapy with a constraint
kðE,T,N,u,E τ ,T τ ,vÞ¼ N 0:75 0, 0 t t f : (5)
We are seeking optimal control pair (v*, w*) such that
∗
∗
Jðv ,w Þ¼ maxfJðv,wÞ : ðv,wÞ Wg, (6)
where W is the control set defined by
W ¼fðv,wÞ : ðv,wÞ piecewise continuous, such that
(7)
0 vðtÞ v max < ∞,;0 wðtÞ w max < ∞, 8t ½0, t f g:
The existence of optimal controls v*(t) and w*(t) for this model is guaranteed
by standard results in optimal control theory (Fleming and Rishel, 1994).
Necessary conditions that the controls must satisfy are derived via Pontrya-
gin’s Maximum Principle (Pontryagin et al., 1962). The OCP given by
expressions (3)–(7) is equivalent to that of minimizing the Hamiltonian
H(t) (see the “Appendix” section):
dE dT dN du
B v 2 B w 2
H ¼ E T ½ vðtÞ ½ wðtÞ + λ 1 + λ 2 + λ 3 + λ 4 + γk
2 2 dt dt dt dt
(8)
and γ 0 with γ(t)k(t) ¼ 0, where
1 if NðtÞ 0:75,
γ ¼
0 otherwise:
A standard application of Pontryagin’s maximum principle leads to the fol-
lowing result:
Theorem 1. Given an optimal pair v*(t) and w*(t) and corresponding solu-
tions E*, T*, N*, u* and w* for system (3) that minimizes J(u(t), w(t)) over Ω.
The explicit optimal controls are connected to the existence of continuous specific func-
tions λ i for i ¼ 1, 2, 3, 4, satisfying the adjoint system
