Question 3 (12 marks) We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB) treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB infected individuals (including latent and infectious individuals) as well as the costs required to con- trol TB by treating latent and infectious individuals, over a certain time horizon [0,t]. The cost of each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the optimization problem below. Minimize the objective function J(u1(.), u2(.) = " A₁E(t) + A21(t) + B¹u²(t) + ¹²u²(t) 1 subject to d.S = dt dV A(1 p) BSI+kV-ds, =Ap-kVdV, dt dE = dt BSI (u(t) +e+d)E+OR, dI == EE (d-8-u2(t))I, dt dR = u(t)Eu₂(t) (d+0)R dt S(0) So≥0, V(0) = Vo≥ 0, E(0) = Eo ≥ 0,1(0) = 10 ≥0, R(0) = Ro≥ 0. The control variables are assumed to be bounded: Given any t > 0, 0(t) b₁ and 0 < u2(t)
Question 3 (12 marks) We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB) treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB infected individuals (including latent and infectious individuals) as well as the costs required to con- trol TB by treating latent and infectious individuals, over a certain time horizon [0,t]. The cost of each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the optimization problem below. Minimize the objective function J(u1(.), u2(.) = " A₁E(t) + A21(t) + B¹u²(t) + ¹²u²(t) 1 subject to d.S = dt dV A(1 p) BSI+kV-ds, =Ap-kVdV, dt dE = dt BSI (u(t) +e+d)E+OR, dI == EE (d-8-u2(t))I, dt dR = u(t)Eu₂(t) (d+0)R dt S(0) So≥0, V(0) = Vo≥ 0, E(0) = Eo ≥ 0,1(0) = 10 ≥0, R(0) = Ro≥ 0. The control variables are assumed to be bounded: Given any t > 0, 0(t) b₁ and 0 < u2(t)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Question 3 (12 marks)
We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB)
treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB
infected individuals (including latent and infectious individuals) as well as the costs required to con-
trol TB by treating latent and infectious individuals, over a certain time horizon [0,t]. The cost of
each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the
optimization problem below.
Minimize the objective function
J(u1(.), u2(.) = " A₁E(t) + A21(t) + B¹u²(t) + ¹²u²(t)
1
subject to
d.S
=
dt
dV
A(1 p) BSI+kV-ds,
=Ap-kVdV,
dt
dE
=
dt
BSI (u(t) +e+d)E+OR,
dI
== EE (d-8-u2(t))I,
dt
dR
= u(t)Eu₂(t) (d+0)R
dt
S(0) So≥0, V(0) = Vo≥ 0, E(0) = Eo ≥ 0,1(0) = 10 ≥0, R(0) = Ro≥ 0.
The control variables are assumed to be bounded: Given any t > 0,
0(t) b₁ and 0 < u2(t) <b₂.
Use Pontryagin's Maximum Principle to derive the necessary conditions for an optimal control u(t) and u(t)
that minimizes the objective functional J(u1()), 2(.).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F044d874c-7b63-4457-9ef3-c7a82eb00b30%2F6185569a-a4f9-4a42-bc8f-acf81eeeb5e8%2F03z407n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 3 (12 marks)
We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB)
treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB
infected individuals (including latent and infectious individuals) as well as the costs required to con-
trol TB by treating latent and infectious individuals, over a certain time horizon [0,t]. The cost of
each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the
optimization problem below.
Minimize the objective function
J(u1(.), u2(.) = " A₁E(t) + A21(t) + B¹u²(t) + ¹²u²(t)
1
subject to
d.S
=
dt
dV
A(1 p) BSI+kV-ds,
=Ap-kVdV,
dt
dE
=
dt
BSI (u(t) +e+d)E+OR,
dI
== EE (d-8-u2(t))I,
dt
dR
= u(t)Eu₂(t) (d+0)R
dt
S(0) So≥0, V(0) = Vo≥ 0, E(0) = Eo ≥ 0,1(0) = 10 ≥0, R(0) = Ro≥ 0.
The control variables are assumed to be bounded: Given any t > 0,
0(t) b₁ and 0 < u2(t) <b₂.
Use Pontryagin's Maximum Principle to derive the necessary conditions for an optimal control u(t) and u(t)
that minimizes the objective functional J(u1()), 2(.).
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