Electronic Journal of Differential Equations, Vol. 2007(2007), No. 171, pp. 1–24.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
SEEKING BANG-BANG SOLUTIONS OF MIXED
IMMUNO-CHEMOTHERAPY OF TUMORS
LISETTE G. DE PILLIS, K. RENEE FISTER, WEIQING GU,
CRAIG COLLINS, MICHAEL DAUB, DAVID GROSS, JAMES MOORE, BEN PRESKILL*
Abstract. It is known that a beneficial cancer treatment approach for a single patient often involves the administration of more than one type of therapy.
The question of how best to combine multiple cancer therapies, however, is still
open. In this study, we investigate the theoretical interaction of three treatment types (two biological therapies and one chemotherapy) with a growing
cancer, and present an analysis of an optimal control strategy for administering
all three therapies in combination. In the situations with controls introduced
linearly, we find that there are conditions on which the controls exist singularly.
Although bang-bang controls (on-off) reflect the drug treatment approach that
is often implemented clinically, we have demonstrated, in the context of our
mathematical model, that there can exist regions on which this may not be the
best strategy for minimizing a tumor burden. We characterize the controls in
singular regions by taking time derivatives of the switching functions. We will
examine these representations and the conditions necessary for the controls to
be minimizing in the singular region. We begin by assuming only one of the
controls is singular on a given interval. Then we analyze the conditions on
which a pair and then all three controls are singular.
1. Introduction
The goal of applying optimal control theory to mathematical models representing
the interaction between tumor, immune system, and chemotherapy is to determine
the ideal mix of treatments that minimizes both tumor mass and negative effects
upon the health of the patient. Recent research into the mixture of chemo- and
immunotherapy regimens shows a great deal of potential for the success of such
treatment schedules, c.f. [13], [32], [34], [22], [25], [26], [35], [15].
The logic behind the development of a combination chemo-immunotherapy strategy is intuitive - use as little chemotherapy drug as possible to effectively kill tumor
cells while utilizing immunotherapy to bolster the patient’s immune system, thus
strengthening the body’s natural defenses against both the tumor cells and the
dangerous side effects of the chemotherapy.
2000 Mathematics Subject Classification. 37N25, 49J15, 49J30, 49N05, 62P10, 92C50.
Key words and phrases. Cancer modelling; mixed immuno-chemo-therapy; immunotherapy;
chemotherapy; linear optimal control.
c
2007
Texas State University - San Marcos.
Submitted October 3, 2007. Published December 6, 2007.
Supported by grant NSF-DMS-041-4011 from the National Science Foundation.
*The second line of authors are students who worked on the project.
1
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L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
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The application of optimal control theory to mathematical models incorporating
the interaction between tumors and treatments has provided valuable information
in the past. Works by Kim et al. [18], Swan and Vincent [37], and Murray [30] have
successfully utilized control theory to maximize the effectiveness of chemotherapy
against tumor cells and minimize the toxic effects of such treatments. Optimal
control has also been effectively applied to immunotherapy models; for example,
Swan [36], Kuznetsov and Knott [24], and Kirschner et al. [20] have contributed
research on applications to immunotherapy strategies for cancer and HIV. The
more recent approach of combining chemo- and immunotherapy is being explored
by several researchers; Kirschner and Panetta [21] present a model detailing tumorimmune interaction with chemotherapy, while Burden et al. [2] apply quadratic
control to that model. In addition, in the works by de Pillis and Radunskaya [5],
[6], and de Pillis et al. [7], [4] the authors explore various approaches to combining
chemo- and immunotherapies through numerical simulation and the implementation
of numerical linear controls.
The model presented in this paper is an expansion of the model presented by
de Pillis et al. [4]. The modifications are introduced in response to newer research
on the kinetics of IL-2 and immune cell populations. Another alteration to this
model is the inclusion of constraints to limit both the tumor mass after treatment
and the total concentration of lymphocytes during treatment. These additional
constraints introduce a slight variation on the typical optimal control problem and
must be dealt with by other means than the standard application of Pontryagin’s
Maximum/Minimum Principle [31]. We turn to works by Kirk [19], Hartl et al.
[14], and Kamien and Schwartz [17] for the methods necessary to incorporate these
constraints.
The interest in applying control in a linear fashion to this model is somewhat
pragmatic, given standard chemotherapy treatments. A widely used approach to
cancer treatment is to give a maximum dosage of chemotherapy drug for some
period of time, followed by a period of recuperation in which no drug is given.
This type of treatment correlates theoretically with bang-bang control. However,
when dealing with linear controls, we investigate the possibility of singular arcs and
which conditions are necessary for those arcs to be optimal. In this paper, we use
the Generalized Legendre-Clebsch conditions - as given by Krener [23] - to generate
the higher order necessary conditions for the optimality of the singular arcs. A
more detailed discussion of this is given in Section 3.
The outline of the paper is as follows. In Section 2, we present the model and
discuss the model’s components. Section 3 deals with the application of optimal
control to the model, beginning with the description of the objective functional.
The section continues with the proof of existence of an optimal control in this context and concludes with the conditions under which singular controls are optimal.
Section 4 summarizes the conclusions reached from analysis of the data. In the
appendices we provide full statements of certain theorems and detailed equations
used to develop the work in this paper.
2. Three-Cell Three-Chemical Model
2.1. Model Development and Cellular Dynamics. Medical advances have
given doctors more options in cancer treatment. Traditional chemotherapy schedules may now be supplemented with various forms of immunotherapy. The existence
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BANG-BANG AND SINGULAR CONTROLS
3
of more options, however, can make it more challenging to find the best treatment.
A mathematical model of all the processes involved can help to reveal improved
treatment strategies. In this paper we analyze a model exploring the possible dynamics that can occur in different regions of parameter space. The hope is that
the analysis will provide intuition into the interactions of the tumor, chemotherapy,
and immunotherapy.
The model tracks 6 quantities
•
•
•
•
Tumor Cells, T (t) (Units: Number of Cells)
Natural Killer Cells, N (t) (Units: Number of Cells per Liter)
Circulating Lymphocytes, C(t) (Units: Number of Cells per Liter)
Tumor Specific CD8+ T -Lymphocytes, L(t) (Units: Number of Cells per
Liter)
• Interleukin 2, I(t) (Units: International Units (IUs) per Liter)
• Medicine, M (t) (Units: Milligrams per Liter)
The circulating lymphocyte population represents all B and T lymphocytes in
the bloodstream. Natural killer cells, which are also lymphocytes, are tracked as
a separate population. The quantity L represents the concentration of cells that
have been activated by a tumor related antigen.
2.2. Tumor Equation (T ). Simple logistic growth of the tumor cell population,
in the absence of medicine and immune interactions, is used. Death of tumor cells
due to natural killer cells is given by a mass action term cN T , whereas death
due to CTLs is given by a ratio dependent term, D. Note that unlike other cell
populations, T is measured as an absolute number and not a concentration.
2.3. Natural Killer Cell Equation (N ). A constant source term of Natural
Killer cells differentiating from Circulating Lymphocytes and a linear death term
are both assumed. We also assume that natural killer cells die when they have
interacted with a tumor cell; so we include a mass action death term identical to
that in the T equation. For N , L, and C the quantity given is cells per liter of
blood.
2.4. Circulating Lymphocyte Equation. This population has the simplest dynamics: a constant source term and a linear death rate.
2.5. Tumor Specific T Cell (CD8+T CTL) Equation. We assume that these
cells have a linear death rate, −mL, as well as a quadratic death rate, −uL2 C.
The latter term represents the activity of regulatory T-cells, which are a subset
of circulating lymphocytes. The CTLs may also die through interaction with the
tumor and this is represented by a mass action term,−qLT . Interactions of the tumor with the larger lymphocyte populations, N and C, stimulate CTL production.
These stimulatory terms are represented by the two positive mass action terms.
Additionally, tumor-related antigens stimulate the proliferation of CTLs through
TL
.
j k+T
2.6. Dynamics and Effects of Medicine. Once injected, medicine (chemotherapy) is assumed to have a linear decay rate. The medicine interacts with each of the
four cell populations, T , N , L and C through a term of the form KX (1−e(−δX M ) )X,
[12]. For each cell population, this term represents cell death due to the medicine.
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L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
2.7. Dynamics and Effects of IL-2. Although naturally produced, the cytokine
IL-2 is often used to treat cancer. This model assumes a linear decay rate and
a constant source from circulating lymphocytes. Additionally, when a T-cell is
stimulated by IL-2, the T-cell will secrete more IL-2 as represented by ω gILI
+L .
IL-2 also stimulates the proliferation of Natural Killer cells and CTLs. This
XI
, in each equation.
stimulation is represented by the Michaelis-Menten terms, p g+X
Also, IL-2 inhibits the linear death term but stimulates the quadratic death term
in the CTL equation.
2.8. The Equations. The system of differential equations describing the growth,
death, and interactions of these populations with a chemotherapy treatment is given
by
dT
= aT (1 − bT ) − cN T − DT − KT (1 − e−δT M )T
(2.1)
dt
e
pN N I
dN
= f C − N − pN T +
− KN (1 − e−δN M )N
(2.2)
dt
f
gN + I
dL
θmL
pI LI
u0 L2 CI
=−
− qLT + r1 N T + r2 CT +
−
dt
θ+I
gI + I
κ+I
jT L
− KL (1 − e−δL M )L + η1 vL (t)
(2.3)
+
k+T
α
dC
=β
− C − KC (1 − e−δC M )C
(2.4)
dt
β
dM
= −γM + η2 vM (t)
(2.5)
dt
ωLI
dI
= −µI I + φC +
+ η3 vI (t)
(2.6)
dt
ζ +I
`
)
where D = d s/n(L/T
` +(L/T )` , T (0) = T0 , N (0) = N0 , L(0) = L0 , I(0) = I0 , C(0) = C0 ,
and M (0) = M0 .
In Table 1, we have provided a summary of equation term descriptions, and in
Table 2 we have a list of parameters with their units and biological interpretation.
Table 1: Equation Descriptions
Eq.
dT
dt
dN
dt
Term
aT (1 − bT )
−cN T
−DT
−KT (1 − e−δT M )T
eC
−f N
−pN T
pN N I
gN +I
δN M
−KN (1 − e
− mθL
θ+I
)N
Description
Logistic tumor growth
NK-induced tumor death
CD8+ T cell-induced tumor death
Chemotherapy-induced tumor death
Production of NK cells from circulating lymphocytes
Natural killer breakdown
Natural killer death by exhaustion of tumor-killing
resources
Stimulatory effect of IL-2 on NK cells
Death of NK cells due to medicine toxicity
CD8+T cell breakdown
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BANG-BANG AND SINGULAR CONTROLS
−qLT
dL
dt
CD8+T cell death by exhaustion of tumor-killing
resources
CD8+ T cell stimulation by NK-lysed tumor cell
debris
Activation of naive CD8+T cells in the general
lymphocyte population
Stimulatory effect of IL-2 on CD8+T cells
r1 N T
r2 CT
pI LI
gI +I
L2 CI
− u0κ+I
jT L
k+T
dC
dt
dM
dt
dI
dt
−KL (1 − e−δL M )L
η1 vL (t)
α
−βC
−KC (1 − e−δC M )C
−γM
η2 vM (t)
−µI I
φC
ωLI
ζ+I
η3 vI (t)
5
Breakdown of surplus CD8+T cells in the presence
of IL-2
CD8+ T cell stimulation by CD8+ T cell-lysed
tumor cell debris
Death of CD8+ T cells due to medicine toxicity
External TIL therapy, controllable
Lymphocyte synthesis in bone marrow
Lymphocyte breakdown
Death of lymphocytes due to medicine toxicity
Excretion and breakdown of medicine
External chemotherapy, controllable
IL-2 breakdown
Production of IL-2 due to naive CD8+T cells and
CD4+T cells
Production of IL-2 from activated CD8+T cells
External IL-2, controllable
Table 2: Parameter Descriptions
Eq.
dT
dt
Param.
a
b
c
KT
δT
e/f
f
p
dN
dt
pN
gN
KN
δN
m
θ
Description
Growth rate of tumor
Inverse of carrying capacity
Rate of NK-induced tumor death
Rate of chemotherapy-induced tumor death
Medicine efficacy coefficient
Ratio of rate of NK cell creation with rate of
cell death
Rate of NK cell death
Rate of NK cell death due to tumor interaction
Rate of IL-2 induced NK cell genesis
Concentration of IL-2 for half-maximal NK
cell genesis
Rate of NK depletion from medicine toxicity
Medicine toxicity coefficient
Natural decay rate of CD8+T cells
Concentration of IL-2 to halve effectiveness
of CD8+T self-regulation
Units
1/day
1/cells
liter/(cells·day)
1/day
liter/mg
unitless
1/day
1/(cells·day)
1/day
IU/liter
1/day
liter/mg
1/day
IU/liter
6
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
q
r1
r2
dL
dt
pI
gI
u0
κ
j
k
KL
δL
η1
vL (t)
α/β
dC
dt
dM
dt
β
KC
δC
γ
η2
vM (t)
µI
φ
dI
dt
ω
ζ
D
η3
vI
d
l
s
n
EJDE-2007/171
Rate of CD8+T cell death due to tumor in1/(cells·day)
teraction
Rate of NK-lysed tumor cell debris activation
1/(cells·day)
of CD8+T cells
Rate of CD8+T cell production from circu1/(cells·day)
lating lymphocytes
Rate of IL-2 induced CD8+T cell activation
1/day
Concentration of IL-2 for half-maximal
IU/liter
CD8+T cell activation
CD8 self-limitation feedback coefficient
liter2 /(cells2 ·day)
Concentration of IL-2 for half-maximal IL-2IU/liter
dependent CD8+T self-regulation
Rate of CD8+T-lysed tumor cell debris acti1/day
vation of CD8+T cells
Tumor size for half-maximal CD8+T-lysed
cells
debris CD8+T activation
Rate of CD8+T depletion from medicine tox1/day
icity
Medicine toxicity coefficient
liter/mg
TIL therapy administration rate
1/day
Externally administered TIL CD8+T thercells/liter
apy concentration
Ratio of rate of circulating lymphocyte procells/liter
duction to death rate
Rate of decay of circulating lymphocytes
1/day
Circulating lymphocyte-toxicity of medicine
1/day
Medicine toxicity coefficient
liter/mg
Rate of decay of medicine
1/day
Chemotherapy administration rate
1/day
Externally administered chemotherapy conmg/liter
centration
Rate of decay of IL-2
1/day
Rate of IL-2 production from circulating
IU/(cells·day)
lymphocytes
Rate of IL-2 production from CD8+T cells
IU/(cells·day)
Concentration of IL-2 for half-maximal
IU/liter
CD8+T IL-2 production
Exogenous IL-2 administration rate
1/day
Externally administered IL-2 conentration
IU/liter
Immune system strength coefficient
liter/day
Immune strength scaling coefficient
unitless
Concentration of CD8+T cells in tumor for
cells/liter
half-maximal tumor death
Number of CD8+T cells that infiltrate tumor
cells
In addition to the system of differential equations, there are two constraints
associated with this model. The first is a terminal condition in which the tumor
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BANG-BANG AND SINGULAR CONTROLS
7
population is required to be limited by an upper bound,
T (tf ) ≤ ΩT ,
(2.7)
where ΩT is constant.
The second constraint is a condition on the control vM in which the total drug
administered is limited by a constant. The constant is divided by 2 for mathematical
convenience.
Z tf
τ
−
vM (t) ≥ 0,
(2.8)
2
0
where τ is constant.
3. Linear Control
The goal of implementing linear control on the model described in Section 2 is
to determine theoretically the optimal treatment schedule for a cancer patient. We
prove the existence of such a control using the Filippov-Cesari Theorem, as stated
in Hartl, Sethi, and Vickson [14]. Constraints (2.7) and (2.8) prevent the direct
application of Pontryagin’s Minimum Principle to find the first order necessary
conditions for the control to be optimal. We therefore use conditions given by
Kamien and Schwartz [17] and Hartl et al. [14] to deal with the terminal inequality
conditions generated by constraint (2.7). Constraint (2.8) is incorporated into the
state system using a method detailed by Kirk [19].
When implementing linear control, we must investigate the possibility of singular
arcs. A singular arc is one for which one or more of the control variables vα satisfies
∂H
= 0,
∂vα
where H is the Hamiltonian. In this situation, first order necessary conditions
are inadequate; therefore, we use the generalized Legendre-Clebsch conditions (see
Appendix A, (4.3)) as given by Krener [23] to generate these higher order necessary
conditions for optimality.
3.1. Objective Functional. Now, seeking a bang-bang solution, we wish to minimize the objective functional
Z tf
J(vL , vM , vI ) =
(T (t) + L vL (t) + M vM (t) + I vI (t)) dt,
(3.1)
0
which is linear in the three controls and where L , M and I are weight factors.
3.2. Existence. We first establish the existence of an optimal control building on
the existence theorem of Filippov-Cesari, the full statement of which is provided in
Appendix A, Theorem (4.1).
Theorem 3.1 (Existence of a Linear Optimal Control). Given the objective functional (3.1), subject to system (2.1)-(2.6) with T (0) = T0 , N (0) = N0 , L(0) = L0 ,
Rt
C(0) = C0 , M (0) = M0 , I(0) = I0 , T (tf ) ≤ ΩT , and τ2 − 0 f vM dt ≥ 0, and the
control set
U = {vL (t), vM (t), vI (t) piecewise cont. : 0 ≤ vL (t), vM (t), vI (t) ≤ 1, ∀t ∈ [0, tf ]},
8
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
the conditions (1) − (4) of Theorem (4.1) are met, and therefore there exists an
~ ∗ (t) = (v ∗ (t), v ∗ (t), v ∗ (t)) such that
optimal control V
L
M
I
min J(V ) = J(V ∗ ).
~ ∈[0,1]
V
Applying the notation of Theorem 4.1 to the optimal control problem (2.1)-(2.6),
(3.1), we have
 
T
N 
 
L

x=
C 
 
M 
I
N (x, t)




 θmL
=
− θ+I



f
T (t) + L vL (t) + M vM (t) + I vI (t)
−δT M )T
“aT (1 − bT
” ) − cN T − DT − KT (1 − e
e
C
f
− N − pN T +
LI
− qLT + r1 N T + r2 CT + gpI +I
−
I ”
“
α
β β −C −

pN N I
− KN (1 − e−δN M )N
gN +I
jT L
u0 L2 CI
+ k+T
− KL (1 − e−δL M )L
κ+I
KC (1 − e−δC M )C
−γM + η2 vM (t)
−µI I + φC + ωLI
+ η3 vI (t)
ζ+I




+ η1 vL (t)




where γ̃ ≤ 0, and vL , vM , vI ∈ Ω(x, t).
Proof. We know there exists an admissible solution pair for the state and controls
as seen in previous work, [8]. For the second condition, we define w1 =





 θmL
− θ+I




T (t) + L vL1 (t) + M vM1 (t) + I vI1 (t)
−δT M )T
aT
(1
− bT
” ) − cN T − DT − KT (1 − e
“
pN N I
e
f f C − N − pN T + g +I − KN (1 − e−δN M )N
− qLT + r1 N T



N

pI LI
jT L
u0 L2 CI
+ r2 CT + g +I − κ+I + k+T − KL (1 − e−δL M )L + η1 vL1 (t) ,

I
”
“

−δC M )C
−
C
−
K
(1
−
e
β α
C

β

−γM + η2 vM (t)
−µI I + φC +
and w2 =




 θmL
− θ+I




f
ωLI
ζ+I
1
+ η3 vI1 (t)
T (t) + L vL2 (t) + M vM2 (t) + I vI2 (t)
−δT M )T
aT
(1
− bT
“
” ) − cN T − DT − KT (1 − e
pN N I
− KN (1 − e−δN M )N
gN +I
2
LI
jT L
L CI
r2 CT + gpI +I
− u0κ+I
+ k+T
− KL (1 − e−δL M )L
I
“
”
α
−δ
M
β β − C − KC (1 − e C )C
e
C
f
− qLT + r1 N T +
− N − pN T +
−γM + η2 vM2 (t)
−µI I + φC + ωLI
+ η3 vI2 (t)
ζ+I





+ η1 vL2 (t)




for some γ˜1 , γ˜2 ≤ 0, and vLj , vMj , vIj ∈ Ω(x, t), with j = 1, 2.
We then let w3 = λw1 + (1 − λ)w2 for λ ∈ [0, 1]. To prove that N (x, t)is convex,
we need to show that w3 ∈ N (x, t) However, since the controls appear linearly, we
note that this occurs naturally.
For the third condition we need to show that there exists a number δ such that
~ We need to determine an upper
kxk ≤ δ, ∀t ∈ [0, tf ] and all admissible pairs (~x, V).
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BANG-BANG AND SINGULAR CONTROLS
9
bound on the right hand sides of the differential equations (2.1)-(2.6). To do this,
we first consider the right hand side of (2.1), the tumor cell population. With Tmax
as an upper bound solution associated with T and T (t) ≥ 0, then Tmax = T0 eatf .
By using the bound Tmax , we can form a set of supersolutions for system (2.1)(2.6). This set of supersolutions T̄ , N̄ , L̄, C̄, M̄ , I¯ of
dT
= aT
dt
dN
= pN N + eC
dt
dL
= r1 N Tmax + (pI + j)L + r2 CTmax + η1 vL
dt
dC
=α
dt
dM
= η2 vM
dt
dI
= ωL + φC + η3 vI
dt
is bounded on a finite time interval. We see that system (2.1)-(2.6) can be written
as

  
 0 
0
T̄
T̄
a
0
0
0
0 0

  
 N̄ 
0
pN
0
e
0 0
  N̄   0 
 

 L̄ 
0 r1 Tmax pI + j r2 Tmax 0 0  L̄   η1 vL 

  + 
  =

  
 C̄ 
0
0
0
0
0 0
  C̄   α 
 

M̄ 
0
0
0
0
0 0 M̄  η2 vM 
I¯
η3 vI
I¯
0
0
ω
φ
0 0
Since this supersolution system involves only constants then it has a finite upper
bound. Letting this upper bound be δ satisfies the third condition.
The fourth condition is satisfied by definition, since 0 < vα < 1, for all controls
vα .
3.3. Characterization of the Optimal Control. We now develop the representations of the optimal control, using Pontraygin’s Maximum/Minimum Principle
(see Appendix A, (4.2)), conditions from Kamien and Schwarz[17], and the generalized Legendre-Clebsch conditions (see Appendix A, (4.3)).
Before proceeding with the characterization of the optimal control, we must
consider the constraints that accompany the model. The terminal constraint (2.7)
generates additional transversality conditions which are included in the following
theorem. However, the treatment of the integral control constraint (2.8) requires
some explanation, so we will show the method used to deal with it before stating
the representation of the optimal control.
By introducing a state variable A, we can express constraint (2.8) equivalently
as
dA
= vM (t),
(3.2)
dt
with boundary conditions A(0) = 0 and A(tf ) ≤ τ2 . Note that this modification
produces a second terminal inequality condition, which will be dealt with in the
same manner as (2.7).
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L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
We will now treat equation (3.2) as an additional state equation, adding it to
system (2.1)-(2.6) along with the conditions T (tf ) ≤ ΩT , A(0) = 0, and A(tf ) ≤ τ2 .
With these modifications, we state the representation of the optimal control.
Theorem 3.2 (Characterization of the Optimal Control). Given an optimal control
~ = (vL ∗ (t), vM ∗ (t), vI ∗ (t)), and solutions of the corresponding state system,
triple, V
there exist adjoint variables λi for i = 1, 2, . . . , 7 satisfying the following:
`
s
d` TL
dλ1
−δT M
n`
= −1 − λ1 a − 2abT − cN − D + −
K
(1
−
e
)
T
` 2
s
dt
+ TL
n`
jkL
(3.3)
+ λ2 pN − λ3 −qL + r1 N + r2 C +
(k + T )2
dλ2
pN I
= λ1 cT + λ2 f + pT −
+ KN (1 − e−δN M ) − λ3 r1 T
(3.4)
dt
gN + I
s d` L `−1 dλ3
ωI
`
T
= λ1 n
−
λ
(3.5)
6
` 2
s
dt
ζ +I
+ TL
n`
pI I
2u0 LCI
jT
θm
−δL M
+ qT −
+
−
+ KL (1 − e
)
+ λ3
θ+I
gI + I
κ+I
k+T
dλ4
u0 L2 I
= −λ2 e − λ3 r2 T −
+ λ4 (β + KC (1 − e−δC M )) − λ6 φ
(3.6)
dt
κ+I
dλ5
= λ1 δT KT T e−δT M + λ2 δN KN N e−δN M + λ3 δL KL Le−δL M
dt
+ λ4 δC KC Ce−δC M + γλ5
(3.7)
2
dλ6
pI gI L
u0 κL C
pN gN N
θmL
= −λ2
− λ3
+
−
dt
(gN + I)2
(θ + I)2
(gI + I)2
(κ + I)2
ωζL
+ λ6 µI −
(3.8)
(ζ + I)2
dλ7
=0
(3.9)
dt
where λi (tf ) = 0 for i = 2, 3, . . . , 6. In addition, there are transversality conditions
imposed from the constraints:
λ1 (tf ) = −ρ1
λ7 (tf ) = −ρ2 ,
where ρ1 , ρ2 ≤ 0, and
ρ1 K 1 = 0
ρ2 K2 = 0,
where K1 = ΩT − T (tf ), K2 = τ2 − A(tf ), and Ki ≥ 0 for i = 1, 2.
Furthermore, the representations of the controls are determined by the switching
functions
ΦL = L + λ3 η1 ,
ΦM = M + λ5 η2 + λ7 , and
ΦI = I + λ6 η3 .
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BANG-BANG AND SINGULAR CONTROLS
The representations of the optimal controls are then given by


if ΦL > 0
0
vL (t) = 1
if ΦL < 0


singular if ΦL = 0,


if ΦM > 0
0
vM (t) = 1
if ΦM < 0


singular if ΦM = 0,


if ΦI > 0
0
vI (t) = 1
if ΦI < 0


singular if ΦI = 0.
11
(3.10)
(3.11)
(3.12)
Proof. The Lagrangian associated with this problem is
L = H − W1 (t)vL (t) − W2 (t)(1 − vL (t)) − W3 (t)vM (t) − W4 (t)(1 − vM (t))
− W5 (t)vI (t) − W6 (t)(1 − vI (t)),
where H is the Hamiltonian given by
H = T (t) + L vL (t) + M vM (t) + I vI (t)
+ λ1 (aT (1 − bT ) − cN T − DT − KT (1 − e−δT M )T )
e
pN N I
+ λ2 (f C − N − pN T +
− KN (1 − e−δN M )N )
f
gN + I
θmL
pI LI
u0 L2 CI
jT
+ λ3 (−
− qLT + r1 N T + r2 CT +
−
+
L
θ+I
gI + I
κ+I
k+T
− KL (1 − e−δL M )L + η1 vL (t))
α
+ λ4 (β
− C − KC (1 − e−δC M )C) + λ5 (−γM + η2 vM (t))
β
ωLI
+ λ6 (−µI I + φC +
+ η3 vI (t)) + λ7 vM (t)
ζ +I
The adjoint equations are formed from differentiating the Hamiltonian with re∂H
1
spect to the corresponding state variables as dλ
dt = − ∂T .
Since system (3.3)-(3.9) has bounded coefficients and the solutions are bounded
on the finite time interval, we know that adjoint variables satisfying (3.3)-(3.9) exist
from Lukes [29], p.182. Also, the final conditions are free for all variables with the
exception of T (t) and A(t); thus we have that λi (tf ) = 0, for i = 2, . . . , 6. The
additional conditions imposed by the terminal inequality constraints are taken from
Kamien and Schwarz [17].
We find the representations of (3.10)-(3.12) of the optimal controls by looking
∂H
at the coefficients of the controls in H; i.e. we consider the sign of ΦL = ∂v
to
L
determine the values of vL .
The characterization of the controls in singular regions is determined by taking
time derivatives of the switching functions. We will examine these representations
and the conditions necessary for the controls to be minimizing in the singular region
provided that the representations of the singular controls satisfy the control bounds.
Note that each interval of singularity is a subinterval of [0, tf ]. We will begin by
12
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
assuming only one of the controls is singular on a given interval in each of Theorems
3.3-3.5.
Theorem 3.3. If vL is singular on the interval (sL , tL ), the singularity is of degree
two and the representation of the control is given by
PL
vL = −
, QL 6= 0,
QL
where QL is given by
L
I ∂2D
− 2u0 C
QL = η1 λ1 T
(3.13)
∂L2
η1 κ + I
and PL is given in Appendix B. Furthermore, if vL is minimizing on this interval,
it is necessary that
L
∂2D
I
≤ 2u0 C
.
λ1 T
∂L2
η1 κ + I
Proof. On the interval (sL , tL ), we know that time derivatives of ΦL are identically
L
d
d2
0; i.e. dt
ΦL = 0, dt
in this
2 ΦL = 0, etc. In addition, we have that λ3 = − η
1
region. We use this information to find vL . To begin, notice that
d
dλ3
d
ΦL = η1 λ3 = 0, or
= 0.
dt
dt
dt
Then, from the corresponding adjoint equation (3.5), we have that
`−1 !
s
ωI d` TL
dλ3
n`
= λ1 −
λ
6
s
L ` 2
dt
ζ +I
` +
T
n
θm
pI I
2u0 LCI
jT
−δL M
+ λ3
+ qT −
+
−
+ KL (1 − e
)
θ+I
gI + I
κ+I
k+T
=0
Note that
s
∂D
` d`
T = n
s
∂L
` +
n
L `−1
T
L ` 2
T
This will simplify the notation in the following calculation.
Taking a second time derivative yields
2
∂ D dL
∂ 2 D dT
dλ1 ∂D
d 2 λ3
T + λ1 T
+
=
dt2
dt
∂L
∂L2 dt
∂T ∂L dt
∂D dT
dλ6
ωI
ωζ
dI
+ λ1
−
− λ6
∂L dt
dt ζ + I
(ζ + I)2 dt
h
θm dI
dT
pI gI dI
κ
dI
+ λ3 −
+q
−
+ 2u0 LC
(θ + I)2 dt
dt
(gI + I)2 dt
(κ + I)2 dt
I
dC
dL
jk
dT
dM i
+ 2u0
L
+C
−
+ δL KL e−δL M
2
κ+I
dt
dt
(k + T ) dt
dt
= 0.
Grouping like terms (by state derivative) and substituting equation (2.3) results in
d 2 λ3
dλ1 ∂D
dλ6
ωI
=
T−
dt2
dt ∂L
dt ζ + I
EJDE-2007/171
BANG-BANG AND SINGULAR CONTROLS
13
∂2D
dT
∂D
jk
+ λ1 T
+ λ1
+ λ3 q −
∂T ∂L
∂L
(k + T )2
dt
pI gI
κ
ωζ
dI
θm
−
+
2u
LC
−
λ
+ λ3 −
0
6
(θ + I)2
(gI + I)2
(κ + I)2
(ζ + I)2 dt
I
dC
dM
+ 2u0 λ3 L
+ λ3 δL KL e−δL M
κ + I dt
dt
2
I
∂ D
+ 2u0 λ3 C
+ λ1 T
∂L2
κ+I
h θmL
pI LI
u0 L2 CI
× −
− qLT + r1 N T + r2 CT +
−
θ+I
gI + I
κ+I
i
jT L
− KL (1 − e−δL M )L + η1 vL (t)
+
k+T
= 0.
Isolating the vL term and substituting λ3 = − ηL1 gives
d 2 λ3
L
∂2D
I
= η1 λ1 T
− 2u0 C
vL
dt2
∂L2
η1 κ + I
dλ6
ωI
dλ1 ∂D
T−
+
dt ∂L
dt ζ + I
dT
∂2D
∂D L
jk
+ λ1
−
+ λ1 T
q−
2
∂T ∂L
∂L
η1
(k + T )
dt
L
θm
pI gI
κ
dI
ωζ
+
+
− 2u0 LC
− λ6
2
2
2
2
η1 (θ + I)
(gI + I)
(κ + I)
(ζ + I) dt
L
I
dC
dM
L
− 2u0 L
− δL KL e−δL M
η1 κ + I dt
η1 dt
∂2D
L
I
+ λ1 T
−
2u
C
0
∂L2
η1 κ + I
h θmL
pI LI
u0 L2 CI
× −
− qLT + r1 N T + r2 CT +
−
θ+I
gI + I
κ+I
i
jT L
+
− KL (1 − e−δL M )L
k+T
= 0.
Now, let PL denote the above sum excluding the vL term. Notice that PL depends
on λ1 , λ2 , λ6 , T, N, L, C, M, I, vM , and vI . Next, define QL as
∂2D
L
I QL = η1 λ1 T
− 2u0 C
.
∂L2
η1 κ + I
Using this condensed notation, we see that
vL = −
PL
,
QL
if QL 6= 0.
Here the degree of the singularity is 2.
Furthermore, if vI and vM are assumed to be bang-bang on the interval (sL , tL ),
14
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
then the Legendre-Clebsch condition for the control to be minimizing is
(−1)
i.e.
∂ d2 ∂H
≥ 0,
∂vL dt2 ∂vL
L
∂2D
≤ 2u0 C
λ1 T
∂L2
η1
or QL ≤ 0;
I
κ+I
.
Theorem 3.4. If vM is singular on the interval (sM , tM ), the singularity is of
degree two and the representation of the control is given by
PM
, QM 6= 0,
vM = −
QM
where QM is given by
h
QM = −η2 δT KT λ1 T (δT e−δT M ) + δN KN λ2 N (−δN eδN M )
i
+ δL KL λ3 L(δL e−δL M ) + δC KC λ4 C(−δC eδC M )
(3.14)
and PM is given in Appendix B.
Furthermore, if vM is minimizing on this interval, it is necessary that
h
δT KT λ1 T (δT e−δT M ) + δN KN λ2 N (δN e−δN M )
i
+ δL KL λ3 L(δL e−δL M ) + δC KC λ4 C(δC e−δC M ) ≥ 0.
Proof. If vM is singular on some interval (sM , tM ), we know that time derivatives
of ΦM are identically zero in this region; in addition, we have that λ5 = −M /η2 .
As before, note that
d
d
d
dλ5
ΦM = η2 λ5 + λ7 = 0, or
= 0,
dt
dt
dt
dt
7
since dλ
dt = 0. Coupling this information with equation (3.7) yields
dλ5
= λ1 δT KT T e−δT M + λ2 δN KN N e−δN M + λ3 δL KL Le−δL M
dt
+ λ4 δC KC Ce−δC M + γλ5
= 0.
Taking a time derivative gives the result
dT
dλ1
d2 λ5
−δT M
−δT M dM
= δT KT λ1 T (−δT e
)
+e
λ1
+T
dt2
dt
dt
dt
dM
dN
dλ2
−δN M
−δN M
+ δN KN λ2 N (−δN e
)
+e
λ2
+N
dt
dt
dt
dM
dL
dλ
3
+ δL KL λ3 L(−δL e−δL M )
+ e−δL M λ3
+L
dt
dt
dt
dλ4
dC
−δC M dM
−δC M
+ δC KC λ4 C(−δC e
)
+e
λ4
+C
dt
dt
dt
dλ5
+γ
dt
= 0.
EJDE-2007/171
BANG-BANG AND SINGULAR CONTROLS
15
dM
5
Notice that the last term is zero, since dλ
dt = 0. Grouping the dt terms, substituting
equation (2.5), and isolating the vM term in the above equation gives
h
d 2 λ5
=
η
δT KT λ1 T (−δT e−δT M ) + δN KN λ2 N (−δN e−δN M )
2
dt2
i
+ δL KL λ3 L(−δL e−δL M ) + δC KC λ4 C(−δC e−δC M ) vM
dλ1
dN
dλ2
dT
+ δT KT e−δT M λ1
+T
+ δN KN e−δN M λ2
+N
dt
dt
dt
dt
dL
dλ3
dC
dλ4
+ δL KL e−δL M λ3
+L
+ δC KC e−δC M λ4
+C
dt
dt
dt
dt
h
− γM δT KT λ1 T (−δT e−δT M ) + δN KN λ2 N (−δN e−δN M )
i
+ δL KL λ3 L(−δL e−δL M ) + δC KC λ4 C(−δC e−δC M )
= 0.
Now, as before, let PM denote the above sum, excluding the vM term. Note that
PM is dependent on all the state and adjoint variables, as well as the control vL .
Define QM as
h
QM = −η2 δT KT λ1 T (δT e−δT M ) + δN KN λ2 N (−δN eδN M )
i
+ δL KL λ3 L(δL e−δL M ) + δC KC λ4 C(−δC eδC M ) .
Then we have
PM
, if QM 6= 0.
QM
Here the degree of the singularity is two.
Furthermore, if vL and vI are assumed to be bang-bang on the interval (sM , tM ),
then the Legendre-Clebsch condition for the control to be minimizing is
vM = −
(−1)
∂ d2 ∂H
≥ 0,
∂vM dt2 ∂vM
or in this situation,
QM ≤ 0.
Thus, if vM is singular on some interval, we have the additional necessary condition
h
δT KT λ1 T (δT e−δT M ) + δN KN λ2 N (δN e−δN M )
i
+ δL KL λ3 L(δL e−δL M ) + δC KC λ4 C(δC e−δC M ) ≥ 0,
since η2 6= 0, in general.
Theorem 3.5. If vI is singular on the interval (sI , tI ), the singularity is of degree
two and the representation of the control is given by
PI
vI = − , QI 6= 0,
QI
where QI is given by
2λ3 θmL 2λ3 pI gI L 2λ3 u0 κL2 C
2I ωζL
2λ2 pN gN N
+
+
−
−
QI = η3
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
(3.15)
16
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
and PI is given in Appendix B. Furthermore, if vI is minimizing on this interval,
it is necessary that
λ2 pN gN N
λ3 θmL
λ3 pI gI L
λ3 u0 κL2 C
I ωζL
+
+
≤
+
.
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
Proof. Assume vI is singular on the interval (sI , tI ); then all derivatives with respect
to time of ΦI are equal to zero and λ6 = − ηI3 . Then
d
d
dλ6
ΦI = η3 λ6 = 0, and so
= 0.
dt
dt
dt
Together with equation (3.8), we have
θmL
pI gI L
u0 κL2 C
dλ6
pN gN N
−
λ
+
−
= −λ2
3
dt
(gN + I)2
(θ + I)2
(gI + I)2
(κ + I)2
ωζL
+ λ6 µI −
(ζ + I)2
= 0.
Differentiate with respect to t for
h (g + I)2 p g dN − p g N · 2(g + I) dI i
dλ2
pN gN N
d 2 λ6
N
N N dt
N N
N
dt
=
−
−
λ
2
dt2
dt (gN + I)2
(gN + I)4
dλ3
pI gI L
u0 κL2 C
θmL
−
+
−
dt (θ + I)2
(gI + I)2
(κ + I)2
"
dI
(θ + I)2 θm dL
dt − θmL · 2(θ + I) dt
− λ3
4
(θ + I)
dI
(gI + I)2 pI gI dL
dt − pI gI L · 2(gI + I) dt
(gI + I)4
#
dL
dI
2
u0 κ(κ + I)2 L2 dC
dt + 2LC dt − u0 κL C · 2(κ + I) dt
−
(κ + I)4
h (ζ + I)2 ωζ dL − ωζL · 2(ζ + I) dI i
dλ6
ωζL
dt
dt
+
µI −
− λ6
2
dt
(ζ + I)
(ζ + I)4
= 0.
+
(3.16)
dλ6
dt
We substitute λ6 = −I /η3 , and note that the
term equals zero. Then we group
terms, substitute equation (2.6), and isolate the vI term to get
d 2 λ6
dt2
2λ2 pN gN N
2λ3 θmL 2λ3 pI gI L 2λ3 u0 κL2 C
2I ωζL
= η3
+
+
−
−
vI
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
pN gN N
dλ2
−
dt (gN + I)2
dλ3
θmL
pI gI L
u0 κL2 C
−
+
−
dt (θ + I)2
(gI + I)2
(κ + I)2
dN
λ2 pN gN
dC λ3 u0 κL2
−
−
dt (gN + I)2
dt (κ + I)2
0=
EJDE-2007/171
BANG-BANG AND SINGULAR CONTROLS
17
λ3 pI gI
2λ3 u0 κLC
I ωζ
dL
λ3 θm
−
+
+
−
dt
(θ + I)2
(gI + I)2
(κ + I)2
η3 (ζ + I)2
2λ2 pN gN N
2λ3 θmL 2λ3 pI gI L 2λ3 u0 κL2 C
2I ωζL
+
+
+
−
−
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
ωLI
× − µI I + φC +
.
ζ +I
Let PI denote the above sum excluding the vI term. Note that PI depends on all
the state and adjoint variables, as well as the control vL . Let QI be defined by
2λ3 θmL 2λ3 pI gI L 2λ3 u0 κL2 C
2I ωζL
2λ2 pN gN N
+
+
−
−
.
QI = η3
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
Then we have
PI
vI = − , if QI 6= 0.
QI
Here the degree of the singularity is 2. Furthermore, if vL and vM are assumed to
be bang-bang on the interval (sI , tI ), then the Legendre-Clebsch condition for the
control to be minimizing is
+
∂ d2 ∂H
≥ 0,
∂vI dt2 ∂vI
or in this situation, QI ≤ 0. Thus, if vI is singular on some interval, we have the
additional necessary condition
(−1)
λ3 θmL
λ3 pI gI L
λ3 u0 κL2 C
I ωζL
λ2 pN gN N
+
+
≤
+
,
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
since η3 =
6 0.
Having found the representations for each control on a singular interval, we turn
our attention to the necessary conditions generated when two of the controls are
simultaneously singular on the same interval. Note that the degree of singularity
of each control is two; both this fact and the representations found previously will
be used to generate the necessary Legendre-Clebsch conditions.
Theorem 3.6. If vL and vM are both singular on some interval (s, t), then the
following conditions must hold for vL and vM to be minimizing:
QL ≤ 0,
QL QM ≥ η1 η2 (λ3 δL KL e−δL M )2 ,
where QL and QM are defined as in (3.13) and (3.14), respectively.
Proof. Assuming that vL and vM are both singular on the interval (s, t), we use
the generalized Legendre-Clebsch conditions to form the 2 × 2 matrix
hi +hj
Aij = (−1)
hj +1
2
∂ d( 2 +1) ∂H
∂ d2 ∂H
= (−1)
,
hi +hj
∂vi dt( 2 +1) ∂vj
∂vL dt2 ∂vM
which is given by
AvL ,vM =
−η1 QL
−η1 η2 λ3 δL KL e−δL M
−η1 η2 λ3 δL KL e−δL M
−η2 QM
,
Note that QL and QM are defined as in (3.13) and (3.14), respectively. This matrix
is clearly symmetric. In order for vL and vM to be minimizing on the interval (s, t),
18
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
the matrix AvL ,vM must be positive definite. In other words, all upper left minors
of AvL ,vM must be positive; then the conditions
QL ≤ 0,
QL QM ≥ η1 η2 (λ3 δL KL e−δL M )2 .
must hold.
In a similar fashion, conditions for the pairs vL , vI and vM , vI to be singular on
a given interval are found as noted in the following theorems without proof.
Theorem 3.7. If vL and vI are both singular on some interval (s, t), then the
following conditions must hold for vL and vI to be minimizing:
QL ≤ 0,
QL QI ≥ η1 η3 (~)2 .
where QL and QI are defined as in (3.13) and (3.15), respectively, and
λ3 θm
λ3 pI gI
2λ3 u0 κLC
λ6 ωζ
~=
+
−
+
.
(θ + I)2
(gI + I)2
(κ + I)2
(ζ + I)2
(3.17)
Theorem 3.8. If vM and vI are both singular on some interval (s, t), then the
following conditions must hold for vM and vI to be minimizing:
QM ≤ 0,
QM QI ≥ 0.
where QM and QI are defined as in (3.14) and (3.15), respectively.
Finally, we will use the Legendre-Clebsch conditions to find the conditions necessary for all three controls to be minimizing on the same interval.
Theorem 3.9. If vL , vM , and vI are simultaneously singular on some interval
(s, t), then the following conditions must hold for these controls to be minimizing:
QL ≤ 0,
QL QM ≥ η1 η2 ()2 ,
QL QM QI ≤ η1 η3 QM ~ +η1 η2 QI ,
where QL , QM , QI and ~ are defined by (3.13), (3.14), (3.15), and (3.17), respectively, and
 = −λ3 δL KL e−δL M .
(3.18)
Proof. Assuming vL , vM , and vI are all singular on the interval (s, t), we use the
generalized Legendre-Clebsch conditions to form the 3 × 3 matrix whose entries are
given by
hi +hj
Aij = (−1)
Here we have

AvL ,vM ,vI
hj +1
2
∂ d( 2 +1) ∂H
.
j
∂vi dt( hi +h
+1) ∂vj
2
−η1 QL
=  η1 η2 
η1 η3 ~
η1 η2 
−η2 QM
0

η1 η3 ~
0 ,
−η3 QI
where QL , QM , QI , ~, and  are defined previously in (3.17) and (3.18).
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BANG-BANG AND SINGULAR CONTROLS
19
This matrix is symmetric; in order for AvL ,vM ,vI to be positive semidefinite, we
must have that
QL ≤ 0,
QL QM ≥ η1 η2 ()2 ,
QL QM QI ≤ η1 η3 QM ~ + η1 η2 QI  .
Conclusion. If the controls are singular either as a unit, in pairs, or as a triple,
conditions are given for those controls to be minimizing. First, existence of the control triple is established. Within the characterization, the introduction of singular
and/or bang-bang controls is given. It is worth noting that clinicians commonly
give treatments in a bang-bang type scenario, i.e. give the drug combination for a
period and then do not given any drug, c.f. [15], [33]. However, in this work, conditions are given such that a singular control vector would minimize the proposed
objective functional. This could change the dynamic in which the chemotherapy
and immunotherapy treatments are administered. Due to the interdependence on
the conditions, we note that the characterizations of the singular control on the
selected intervals are not explicitly determined. In future work, numerical analyses will be investigated to aid in graphically characterizing these singular control
expressions.
4. Appendix A
In this section, we will give precise statements of the theorems used for proving
the existence and finding the characterizations of optimal quadratic controls.
Theorem 4.1 (Fillipov-Cesari Theorem [14]). Consider the following optimal control problem:
Z T
min J =
F (x(t), u(t), t)dt + S(x(T ), T )
0
ẋ(t) = f (x(t), u(t), t),
x(0) = x0
g(x(t), u(t), t) ≥ 0
h(x(t), t) ≥ 0
a(x(T ), T ) ≥ 0
b(x(T ), T ) = 0
where T is free on [0, tf ]. Assume that F,f,g,h,S,a, and b are continuous in all their
arguments at all points (x, u, t). Define the (state-dependent) control region
Ω(x, t) = {u ∈ Rm | g(x, u, t) ≥ 0} ⊂ Rm
and the set
N (x, t) = {(F (x, u, t) + γ, f (x, u, t))| γ ≤ 0, u ∈ Ω(x, t)} ⊂ Rn+1
where m and n are the number of control and state variables, respectively. Suppose
that the following conditions hold:
(1) There exits an admissible solution pair.
(2) N (x, t) is convex for all (x, t) ∈ Rn × [0, tf ].
20
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
EJDE-2007/171
(3) There exists δ > 0 such that kx(t)k < δ for all admissible {x(t), u(t)} and
t.
(4) There exists δ1 > 0 such that kuk < δ1 for all u ∈ Ω(x, t) with kxk < δ.
Then there exists an optimal triple {T ∗ , x∗ , u∗ } with u∗ (·) measurable.
Theorem 4.2 (Pontraygin’s Maximum/Minimum Principle [17]).
Let u(t) = [u1 (t), . . . , um (t)] be a piecewise continuous control vector and x(t) =
[x1 (t), . . . , xn (t)] be an associated continuous and piecewise differentiable state vector defined on the fixed time interval [t0 , t1 ] that minimizes
Z t1
f (t, x(t), u(t))dt
t0
subject to the differential equations
xi (t) = gi (t, x(t), u(t)),
i = 1, . . . , n,
initial conditions
xi (t0 ) = xi0 ,
i = 1, . . . , n
(xi0 fixed),
terminal conditions
xi (t1 ) = xi1 ,
xi (t1 ) ≥ xit ,
i = 1, . . . , p,
i = p + 1, . . . , q
xi (t1 )free,
(xi1 , i = 1, . . . , q fixed),
i = q + 1, . . . , n,
and control variable restriction
u(t) ∈ U,
U a given set in Rm .
We assume that f, g, ∂f /∂xj , and ∂gi /∂xj are continuous functions of all their
arguments, for all i = 1, . . . , n and j = 1, . . . , n. Then there exists a constant λ0
and continuous functions λ(t) = (λ1 (t), . . . , λn (t)), where for all t0 ≤ t ≤ t1 we
have (λ0 , λ(t)) 6= (0, 0) such that for every t0 ≤ t ≤ t1 ,
H(t, x∗ (t), u(t), λ(t)) ≤ H(t, x∗ (t), u∗ (t), λ(t)),
where the Hamiltonian function H is defined by
H(t, x, u, λ) = λ0 f (t, x, u) +
n
X
λi gi (t, x, u).
i=1
Except at points of discontinuity of u∗ (t),
λ0 (t) = −∂H(t, x∗ (t), u∗ (t), λ(t))/∂xi ,
i = 1, . . . , n.
Finally, the following transversality conditions are satisfied:
λi (t1 )
λi (t1 ) ≥ 0
no conditions,
(= 0 if
x∗i (t1 )
λi (t1 ) = 0,
i = 1, . . . , p,
> xi1 ))
i = p + 1, . . . , q,
i = q + 1, . . . , n.
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BANG-BANG AND SINGULAR CONTROLS
21
In addition, the modifications to (4.2) generated by the terminal inequality are
given in Kamien and Schwartz [17], p. 160: If K(xq (t1 ), . . . , xn (t1 )) ≥ 0 is required,
then the transversality conditions
λi (t1 ) = p ∂K/∂x1 ,
p ≤ 0,
pK = 0
i = q, . . . , n,
are necessary.
Theorem 4.3 (Generalized Legendre-Clebsch Conditions [23]). Assume that u(t)
and x(t) are defined for (2.1)-(2.6) on [0, tf ]. Suppose u(t) ∈ interior Ω and each
ui is singular of degree hi2+1 on the subinterval (t0 , t1 ). If u(t) is minimal, then
there exists a λ(t) satisfying the PMP on [0, tf ] such that on the subinterval (t0 , t1 ),
∂ dk ∂
H(λ(t), x(t), u(t)) = 0
∂ui dtk ∂uj
h +h
for k = 0, . . . , i 2 j , 1 ≤ i, j ≤ l (where l is the dimension of the control space).
Moreover, if hi < ∞ for i = 1, . . . , k ≤ l, then the k × k matrix whose i, j entry is
hi +hj
(−1)
hj +1
2
∂ d( 2 +1) ∂
H(λ(t), x(t), u(t))
j
∂ui dt( hi +h
+1) ∂uj
2
where 1 ≤ i, j ≤ k, must be symmetric and nonnegative definite.
Note that a given symmetric real matrix is nonnegative definite (i.e. positive
semidefinite) if and only if all of its upper-left minors are positive; that is, for a
given n × n matrix


A11 . . . A1n

..  ,
..
An×n =  ...
.
. 
An1 . . . Ann
we have that
A11 A12 A13 A11 A12 ≥ 0, A21 A22 A23 ≥ 0,
A11 ≥ 0, A21 A22 A31 A32 A33 etc.
5. Appendix B
In this section, we will state the equations PL , PM , and PI used in the characterizations of controls vL , vM , and vI , respectively.
dλ6
ωI
dλ1 ∂D
T−
PL =
dt ∂L
dt ζ + I
2
∂ D
∂D L
jk
dT
+ λ1 T
+ λ1
−
q−
∂T ∂L
∂L
η1
(k + T )2
dt
L
θm
pI gI
κ
ωζ
dI
+
+
− 2u0 LC
− λ6
η1 (θ + I)2
(gI + I)2
(κ + I)2
(ζ + I)2 dt
L
I
dC
L dM
− 2u0 L
− δL KL e−δL M
η1 κ + I dt
η1 dt
22
L. G. DE PILLIS, K. R. FISTER, W. GU, ET AL.
PM
EJDE-2007/171
∂2D
L
I
+ λ1 T
−
2u
C
0
∂L2
η1 κ + I
h θmL
pI LI
u0 L2 CI
× −
− qLT + r1 N T + r2 CT +
−
θ+I
gI + I
κ+I
i
jT L
+
− KL (1 − e−δL M )L ,
k+T
dλ1
dλ2
dT
dN
+T
+ δN KN e−δN M λ2
+N
= δT KT e−δT M λ1
dt
dt
dt
dt
dL
dλ3
dC
dλ4
+ δL KL e−δL M λ3
+L
+ δC KC e−δC M λ4
+C
dt
dt
dt
dt
h
−δT M
−δN M
− γM δT KT λ1 T (−δT e
) + δN KN λ2 N (−δN e
)
i
+ δL KL λ3 L(−δL e−δL M ) + δC KC λ4 C(−δC e−δC M )
,
pN gN N
θmL
pI gI L
u0 κL2 C
dλ3
dλ2
+
−
−
PI = −
dt (gN + I)2
dt (θ + I)2
(gI + I)2
(κ + I)2
2
λ2 pN gN
dC λ3 u0 κL
dN
−
−
2
dt (gN + I)
dt (κ + I)2
dL
λ3 pI gI
2λ3 u0 κLC
I ωζ
λ3 θm
+
−
+
+
−
dt
(θ + I)2
(gI + I)2
(κ + I)2
η3 (ζ + I)2
2λ3 θmL 2λ3 pI gI L 2λ3 u0 κL2 C
2I ωζL
2λ2 pN gN N
+
+
−
−
+
(gN + I)3
(θ + I)3
(gI + I)3
(κ + I)3
η3 (ζ + I)3
ωLI
× −µI I + φC +
.
ζ +I
Acknowledgements. This work was supported in part by generous funding from
the W.M. Keck Foundation through the Harvey Mudd College Center for Quantitative Life Sciences as well as by the National Science Foundation Division of
Mathematical Sciences, Grant Number DMS-0414011, “Mathematical Modeling of
the Chemotherapy, Immunotherapy, and Vaccine Therapy of Cancer.”
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Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
E-mail address, L. G. de Pillis: depillis@hmc.edu
E-mail address, D. Gross: david gross@hmc.edu
E-mail address, W. Gu: gu@math.hmc.edu
E-mail address, J. Moore: jmoore@hmc.edu
E-mail address, B. Preskill: bpreskill@hmc.edu
Department of Mathematics and Statistics, Murray State University, Murray, KY
42071, USA
E-mail address, C. Collins: craig.collins@murraystate.edu
E-mail address, K.R. Fister: renee.fister@murraystate.edu
Department of Mathematics and Statistics, Williams College, Williamstown, MA
01267, USA
E-mail address, M. Daub: Michael.W.Daub@williams.edu