Page 111 - Control Theory in Biomedical Engineering
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Modeling and optimal control of cancer-immune system 97
yðtÞ¼ ψðtÞ,t 0 τ max t < t 0 , and yðt 0 Þ¼ y 0 : (A.2)
It should be noted that ψ(t 0 ) need not be the same as y 0 . This immediately
introduces the possibility of a discontinuity in the state, y(t).
We mention here that there are many problems in biosciences (such as
epidemics, harvesting, chemostats, treatment of diseases, physiological con-
trol, vaccination) that can be addressed within an optimal control framework
for systems of DDEs (Banks, 1975; Kolmanovskii and Shaikhet, 1996;
Smith, 2005). However, the amount of real experience that exists with
OCPs is still small. The DDE (A.1) can be converted into an OCP by adding
an m-dimensional control term u(t)
0
y ðtÞ¼ fðyðtÞ,yðt τ 1 Þ,yðt τ 2 Þ,…,yðt τ d Þ,uðtÞ,tÞ (A.3)
and a suitable objective functional (measure): J 0 (u)
t f
Z
Minimize J 0 ðuÞ¼ LðyðtÞ,yðt τ 1 Þ,yðt τ 2 Þ,…,yðt τ d Þ,uðtÞ,tÞdt,
0
(A.4)
and subject to control constraint a u(t) b, and state constant y(t) c,
where a and b are the lower and upper bounds. The integrand, Lð:Þ is called
the Lagrangian of objective functional, which is continuous in [0, t f ]. Addi-
tional equality or inequality constraint(s) can be imposed in terms of J i (u).
Pontryagin’s maximum principle (Pontryagin et al., 1962)gives
necessary conditions that the control and the state need to satisfy, and
introduces an adjoint function to affix to the differential equation to the
objective functional. The necessary conditions needed to solve the
OCP are derived from the so-called “Hamiltonian” H,which is given
by the equation
T
HðtÞ¼Lð:Þ + λ ðtÞfð:Þ (A.5)
T
Here, λ (t) is a vector of costate variables of the state variables y(t), which is
the solution of the equation
∂H ∂H
0 ðtÞ χ
λ ðtÞ¼ ðt+ τÞðAdjoint equationÞ (A.6)
½0, t f τ
∂yðtÞ ∂yðt τÞ
where
1 if t ½0, t f τ
χ ¼
½0, t f τ
0 otherwise
