Target control by using feedback spreading control with application to immunotherapy

Vol. 79, No. 7, July 2006, 813–821

Target control by using feedback spreading control with application to immunotherapy

A. EL JAIy and K. KASSARA*z

yUniversity of Perpignan, 52 Avenue de Villeneuve, 66860 – Perpignan Cedex, France zDepartment of Mathematics, University of Casablanca 1, P.O. Box 5366, Casablanca, Morocco

(Received 23 August 2005; in final form 16 March 2006)

In this paper, we show that a target control problem for semi-linear distributed parameter systems can be investigated in the framework of feedback spreading control (FSC) under speed constraints. The control laws which are proposed for a solution are designed in such a manner that they generate a spread which reaches the terminal conditions, provided that a lower bound condition on its speed holds. As a numerical example, we consider a semi-linear parabolic equation. The last part of the paper is devoted to study a partial differential equation (PDE) model from immunotherapy in the context of spreading control.

1. Introduction and statement of the problem

Spreading controls (El Jai and Kassara 1994, El Jai et al. 1997) constitute the most appropriate setting by which spreads in space can be designed in distributed parameter systems.

At the theoretical level, recent studies (Kassara 2000) investigate the problem of spreading control by a set- valued approach in the context of monotonicity with respect to a pre-order. They show that a suitable selection procedure of a tangential map involving both the map to be spread and the dynamics of the system may provide a feedback spreading control law termed in short by FSC law.

More recently, the papers (Kassara 2002, Aubin and Kassara 2004) give a setting in which the speed of a spread which is generated by an FSC law can be expressed by a formula depending upon a measure.

There it is shown how the FSC laws which generate spreads either slower or quicker than a desired given speed can be constructed.

This paper continues the search, started in (Kassara 2000), for the FSC laws and their applications. Its main contribution is to demonstrate that when a lower bound condition on the speed of the spread is satisfied, then an

FSC law can be used in order that a target be reached by the state of a semi-linear distributed parameter system.

On the subject of the application of the approach, the paper is devoted to deal with a target control problem from cancer (Webb 2002, Friedman 2004, Matzavinos et al. 2004, Swan 1990) modeling. Namely, how protocols can be formulated for a cancer therapy in such a manner that tumor cells are destroyed, taking into account the patient’s quality of life? We will investigate the problem in the special case of the immunotherapy partial differential equation (PDE) system established by (Matzavinos et al. 2004), con- sidering that a convenient way to eliminate cancer cells from a tissue can be decided by seeking protocols which spread the zones without tumor cells to the whole tissue.

Now, we turn to set the reachability problem we are going to deal with in this paper. Let be an open bounded domain of R

ⁿ

(n ¼ 1, 2, or 3) with smooth boundary @. Assume A to be an unbounded densely defined linear operator which generates a C ₀ analytic (Balakrishnan 1981, Pazy 1983) semi-group (S(t))

t0

on Z ¼ L

²

(, R

^k

) (for an integer k) and consider the following semi-linear control system,

@z

@t þ Az ¼ ’ðz, vÞ in ð0, 1Þ, ð1Þ

*Corresponding author. Email: k_kassara@yahoo.com

International Journal of Control

ISSN 0020–7179 print/ISSN 1366–5820 online2006 Taylor & Francis http://www.tandf.co.uk/journals

DOI: 10.1080/00207170600693503

with initial data,

zðx, 0Þ ¼ z 0 ðxÞ in , ð2Þ where z

0

2 dom (A) and ’ denotes a possibly non-linear operator that maps D V into Z, with V a metric space and D a closed subset of Z. Then, we have to deal with the following reachability problem,

Find ðv, zÞ: ½0, t

f

! V D with (1-2) satisfied and

!ðzðt

_f

ÞÞ ¼ ,

ð3Þ

where t

f

is a positive number and ! denotes a set-valued map by which the terminal target conditions are expressed. It is given as follows,

!: D ! M , ð4Þ

where M

denotes the set of measurable subsets in . This paper is structured as follows. In x 2 we provide brief back-grounds on FSC, including the notion of the spread speed. The purpose of x 3 is to solve the control problem (3). Section 4 is devoted to a numerical example and x 5 is concerned with studying the immunotherapy problem which is mentioned above.

2. Feedback spreading control

In this section, we give an overview consisting of the basic definitions and theorems which involve the concept of FSC. Let ! be given as in (4).

Definition 2.1: The mapping &: D ! V is said to be an FSC law with respect to ! on the interval [0, t

f

) if, for all z

0

in D, system (1–2) with v ¼ &ðzÞ has a solution z which satisfies the following expression,

zðtÞ 2 D for all t 2 ½0, t

f

Þ and ð!ð zðtÞÞÞ _0t<t

_f

is non-decreasing:

In (Kassara 2000), we show that the FSC laws can be built by making selections of a set-valued map which is expressed via tangential conditions as in the following way: for each couple ð y, zÞ 2 Z D let us consider the tangential condition,

8 > 0, 9 0 < h < and k pk such that, SðhÞz þ hð y þ pÞ 2 D and,

!ðSðhÞz þ hð y þ pÞÞ !ðzÞ:

ð5Þ

Then, define the set-valued maps T

_!

and F

_!

for each z 2 D as follows,

T

_!

ðzÞ ¼ :

y 2 Z j Equation ð5Þ holds with ð y, zÞ

,

and,

F

!

ðzÞ ¼ v 2 V j ’ðz, vÞ 2 T

!

ðzÞ

: ð6Þ

Even we set,

!

¼ ð y, zÞ 2 D ² j !ð yÞ !ðzÞ :

Hence, we may present the following basic result which characterizes the FSC laws.

Theorem 2.1: Let &: D ! V be a measurable function and assume that,

(i) The semi-group S() is compact.

(ii) The set

!

is closed.

(iii) The mapping ’ð, &ðÞÞ sends convergent sequences of D into weakly convergent sequences in Z.

Then & is an FSC law with respect to ! if and only if & is a selection of the map F

_!

given by (6). i.e., &ðzÞ 2 F

!

ðzÞ for each z 2 D.

Proof: See [Kassara 2000, Theorem 3.1]. œ Remark 1: We stress that an FSC law derives a spreading control

v ¼ ð zðÞÞ,

for all initial states z 0 2 D. Note the dependence of

v upon z

0

because z depends of z

0

, contrary to the law . Remark 2: A notable advantage in applying these feedback control laws resides in the fact that they ensure existence of a solution to the semi-linear system.

The speed of the spread ð!ð zðtÞÞÞ which is generated by a spreading control v may be set for each t 2 ½0, t

f

Þ as follows,

speedðt, vÞ ¼ : lim inf

h#0

ð!ð zðt þ hÞÞÞ ð!ð zðtÞÞÞ

h 0,

where satisfies,

: M ! R

^þ

and _{1 2} ) ð ₁ Þ ð ₂ Þ

ð7Þ

For an FSC law , the speed of the spread generated by the control v ¼ ð zÞ will be computed by using the formula shown in (Kassara 2002),

speedðt, vÞ ¼ ð’ð zðtÞ, vðtÞÞ, zðtÞÞ,

where z ¼ zð, vÞ and is given by, ð y, zÞ ¼ :

lim inf

h#0,jjpjj!0

ð!ðSðhÞz þ hð y þ pÞÞÞ ð!ðzÞÞ

h ,

ð8Þ for each z 2 D and y 2 T

_!

ðzÞ. Furthermore, whenever D and

!

¼ :

! are convex we have, ð y, zÞ ¼ d

!

ðzÞð y AzÞ,

for each y 2 T

_!

ðzÞ and z 2 D \ domðAÞ. For sake of notation, we define the spread speed functional as follows,

ðz, vÞ ¼ :

ð’ðz, vÞ, zÞ for each z 2 D, v 2 F

_!

ðzÞ: ð9Þ Thus, the speed of the spread generated by a spreading control law v can be measured, at each time t, by the number ð zðtÞ, vðtÞÞ. Note its dependence upon the functional .

Let be a non-negative measurable function on D, then define the following maps by,

T

_!

ðzÞ ¼ :

y 2 T

_!

ðzÞ j ð y, zÞ ðzÞ

, and,

F

_!

ðzÞ ¼ :

v 2 V j ’ðz, vÞ 2 T

_!

ðzÞ

,

for each z 2 D. It follows that any selection of the map F

_!

which satisfies assumptions of Theorem 2.1 provides an FSC law which produces a spread having a speed greater than . Conditions of existence of such a law can be listed in the result below. Before that, we need to use the following notations: a set-valued map M on D is said to be lower semi-continuous whenever for each z 0 2 D and any sequence of elements z

n

2 D converging to z

0

, for every y 0 2 Mðz 0 Þ, there exists a sequence of elements y

n

2 Mðz

n

Þ which converges to y

0

. See for instance (Aubin 1990). The inverse of M is the map M

¹

which is defined by,

M

¹

ð yÞ ¼ fz 2 D j y 2 MðzÞg:

The directional derivative (Jahn 1983) of a mapping F: D ! R is defined by,

dFðzÞy ¼ : lim inf

h#0

Fðz þ hyÞ FðzÞ

h for each y 2 Z:

Theorem 2.2: Further assumptions (i) and (ii) of Theorem 2.1, assume that,

(iv) !

¹

has convex values.

(v) T

!

is a lower semi-continuous map.

(vi) For each sequences ðz

_n

Þ

_n

D and ð y

_n

Þ

_n

Z such that y

_n

2 T

_!

ðz

_n

Þ for every n, we have,

z

n^strong

! z y

_n^weakly

! y

) y 2 T

_!

ðzÞ and ð y

n

, z

n

Þ ! ð y, zÞ:

(vii) For each z 2 D and y 2 T

_!

ðzÞ, there exists v 2 V such that ’ðz, vÞ ¼ y.

(viii) is continuous.

(ix)

!

has a directional derivative.

(x) For each z 2 D, there exists y 2 T

!

ðzÞ such that ð y, zÞ > ðzÞ.

(xi) is upper semi-continuous.

Then there exists an FSC law with respect to !, which satisfies the following inequality,

ðz, ðzÞÞ ðzÞ for each z 2 D:

Proof: See [Kassara 2002, x 5].

Inspired by the works (Aubin 1990, Quincampoix and Saint-Pierre 1995] which investigate the viability kernel algorithms for differential inclusions on finite- dimensional spaces, a convenient use of the expressions (5) and (6) has led the authors (Aubin and Kassara 2004) to state the following algorithm.

Algorithm 1: Let h > 0 and N 2 N such that h ¼ t

f

/N.

I. Initialize r ¼ z

0

and ¼ !(z

₀

).

II. Iterate for k ¼ 1 to N.

II.a Find v such that:

!ðSðhÞr þ h’ðr, vÞÞ ð10Þ

ðr, vÞ ðrÞ: ð11Þ II.b Let v

k

¼ v and z

k

¼ SðhÞr þ h’ðr, vÞ.

II.c Put r ¼ z

k

, ¼ !(z

_k

) and go to II.a.

III. At each time t

k

¼ kh,

The approximated state is z

k

, the spreading control is v

k

, the generated spread is ð!ðz

k

ÞÞ

_k

:

It is of interest to note the key sequence II.a by which an approximate value of the spreading control is derived by selection.

Even we consider that convergence of Algorithm 1 means that:

ðaÞ ðbÞ ðcÞ

v

^h

! v

s

in L ² ð0, t

f

, V Þ when h ! 0, v

s

is a spreading control,

z

^h

! zð, v

s

Þ when h ! 0,

where v

^h

and z

^h

are given as follows,

ðv

^h

ðtÞ, z

^h

ðtÞÞ ¼ : ðv 1 , z 1 Þ on ½0, hÞ, .. .

ðv

_N

, z

_N

Þ on ½ðN 1Þh, t

_f

Þ:

Although Algorithm 1 has been tested with success (Aubin and Kassara 2004) for semi-linear parabolic systems, its convergence according to the above sense, is not yet proved rigorously. However, we stress that as it is easily implementable one may test its convergence usefully on a computer.

3. Control with terminal target

We turn next to study the reachability problem (3).

The key idea is to seek a solution among the FSC laws with respect to !. We have to find an FSC law v ¼ (z) that satisfies the terminal condition,

!ðzðt

f

ÞÞ ¼ : This gives rise to the following result.

Theorem 3.1: Let ! 0 ¼ :

!ðz 0 Þ be non-empty. Assume that the conditions of Theorem 2.2 are satisfied except for (x) and (xi) which are replaced by the following conditions,

(xii) For all z 2 D, there exists y 2 T

_!

ðzÞ such that, ð y, zÞ > ðÞ ð! 0 Þ

t

_f

:

where and are given respectively as in (7) and (8).

(xiii) is such that, and, ðÞ ¼ ðÞ

) lðnÞ ¼ 0,

where l denotes the Lebesgue measure on . Then, there exists on FSC law which solves Problem (3).

Proof: Let be defined as follows,

ðzÞ ¼ ðÞ ð! 0 Þ

t

_f

for each z 2 D,

then it follows that conditions (x) and (xi) in Theorem 2.2 are well verified. Thus, all of the assumptions in that theorem are satisfied and thereby there is an FSC law, say , which satisfies,

ðz,

ðzÞÞ ðzÞ for each z 2 D:

Now, denote by z a solution of system (1–2) with v ¼

ðzÞ and v ¼

ð zÞ. Then we get,

!

ð zðt

f

ÞÞ

!

ð zð0ÞÞ þ Z

_tf

0

d

!

ð zðtÞÞ_ zðtÞ dt, and by considering (1), (2) and (9) we get,

ð!ð zðt

_f

ÞÞÞ ð! ₀ Þ þ Z

tf

0

ð zðtÞ, vðtÞÞ dt:

It follows that,

ð!ð zðt

f

ÞÞÞ ð! 0 Þ þ t

f

ðÞ ð! ₀ Þ t

f

,

and hence ð!ð zðt

_f

ÞÞÞ ðÞ. Now, by using (xiii) we get,

lðn!ð zðt

_f

ÞÞÞ ¼ 0,

ending the proof. œ

Remark 3: For instance the function can be provided by any measure on which satisfies,

ðAÞ ¼ 0 ) lðAÞ ¼ 0,

so as condition (xiii) holds. Nevertheless, as required by (ix), it is needed that the function

!

¼ ! has a directional derivative.

Remark 4: It is of interest to notice that the FSC law , the existence of which is proved in Theorem 3.1 depends upon the number (() (!

0

))/t

f

. Moreover the functional has only a technical role, note the fact that it does not appear in the formulation of the problem.

4. A numerical example

Let ¼ (1; 1) (1; 1) and consider the following semi-linear parabolic equation,

@z

@t divðDrzÞ ¼ z ² þ vðtÞgðxÞ on ½0, t

_f

Þ, ð12Þ with Dirichlet boundary conditions,

zðtÞj

_@

¼ 0 on ½0, t

_f

Þ, ð13Þ and initial data,

zð0Þ ¼ z 0 on : ð14Þ

Here D, g, z

0

map into R and 2 R . D and g stand,

respectively, for the diffusion coefficient and the

actuator. We know (Balakrishnan 1981) that the linear

operator A: z ! divðDrzÞ with domain domðAÞ ¼

H ¹ ₀ ðÞ \ H ² ðÞ, generates a compact analytic C 0

semi-group on Z ¼ L

²

() as required by condition (i) of Theorem 2.1. Let the map ! be given by

!ðzÞ ¼ :

fx 2 j zðxÞ mg, ð15Þ for each z 2 D, where m stands for a positive number and,

D ¼ :

fz 2 Z j z 0 on g:

Table 1 contains all the data which concern the numerical example we are going to treat by means of Algorithm 1. The simulation results are presented in figures 1 and 2 which involve respectively the spread ð!ðzðtÞÞÞ _0tt

_f

at four instants and the spreading control over the interval [0, t

f

]. We emphasize that the value of h above are the smaller among a set of values which are taken under consideration in order to get the convergence of Algorithm 1.

Next, we show how the sequence IIa of Algorithm 1 can be implemented.

According to expression (15), the inclusion (10) can be rewritten for each z 2 Z and v 2 R , as follows,

SðhÞz þ h’ðz, vÞ m on !ðzÞ:

Then, by (12) we get,

SðhÞz þ hð z ² þ vgÞ m on !ðzÞ,

Table 1. The data.

1

D(x

1

, x

2

) 0.7

g(x

₁

, x

₂

) exp(x

₁

þ x

₂

) z

0

(x

1

, x

2

) 1

_fðx

10:6Þ²þð0:4þx2Þ⁴0:3g

(t

f

, h,m) (0.1, 0.1/30, 0.5)

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

− 1 0 1

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

− 1 0 1

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

− 1 0 1

− 1 − 0.8 − 0.6 − 0.4 − 0.2 0 0.2 0.4 0.6 0.8 1

− 1 0 1

Figure 1. The spread generated in up to down order at instants, 0, 0.033, 0.05 and terminal time 0.1.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 10

20 30 40 50 60 70 80

Figure 2. The spreading control.

or equivalently,

vg m SðhÞz h z ²

h on !ðzÞ:

But, as z m on !(z), the last inequality is satisfied if, vg z SðhÞz h z ²

h on !ðzÞ:

Now, by assuming that the actuator g satisfies the following condition, as in table 1,

gðx 1 , x 2 Þ > 0 on , it follows that,

v sup

!ðzÞ

z SðhÞz h z ²

hg :

In other words, an approximate spreading control will be provided by a suitable selection of the set-valued map,

z 2 D ! F

_h

ðzÞ, where,

F

_h

ðzÞ ¼ : sup

!ðzÞ

z SðhÞz h z ²

hg , þ1

" ! , for each z 2 D.

5. Application to immunotherapy

In this section, we investigate eliminating of cancer cells from an avascular tissue which is treated by immu- notherapy. Let ¼ :

ð0, LÞ, with L standing for the size of the tissue. According to (Matzavinos et al. 2004), the densities of immune cells ‘, tumor cells T, chemokine , and complexes , are governed by the following semi- linear PDEs,

@‘

@t ¼ D 1 r ² ‘ k 1 ‘ k 2 ‘T k 3 r ð‘rÞ þ k 4 k ₅ þ T þ k ₆ þ u

,

@T

@t ¼ D 2 r ² T þ k 7 ð1 k 8 T ÞT k 2 ‘T þ k 9 ,

@

@t ¼ D 3 r ² k 10 þ k 11 ,

@

@t ¼ D 4 þ k 2 ‘T,

ð16Þ

where r denotes the gradient operator and stands for the characteristic function which is given as follows,

ðxÞ ¼ 1 if x 2 0 if x 2 =

ð17Þ

where denotes a measurable subset of .

The D

i

and k

i

are positive parameters. k

3

is the chemotaxis, k

7

and k

8

express the logistic growth of the tumor and k

2

, k

6

, k

9

, and

4

define the terms of local kinetics. Now k

4

and k

5

compose the proliferation term and k

1

is a decay coefficient. The terms D

1

, and D

2

are random motility coefficients for the corresponding populations and D

3

measures the diffusion of chemokine. For precise information on the previous coefficients and their values we refer to (Matzavinos et al. 2004).

System (16) is augmented with boundary conditions of Neumann type as follows,

@‘

@n ðx, tÞ ¼ @T

@n ðx, tÞ ¼ @

@n ðx, tÞ ¼ 0 8x 2 @ and t 0, ð18Þ

where @/@n stands for the outward normal derivative on the boundary. The term u(t) represents the dosage of a source or supply of effector cells which is infused into the tissue over the zone (t). The latter refers to be as the therapy zone at time t. Thus the control in system (16–18) is represented by

v: t 2 ½0, t

f

; ðuðtÞ, ðtÞÞ 2 R M

,

where u 2 L ² ð0, t

f

Þ and M

stands for the set of measurable non-empty subsets of .

In order to keep the toxicity (Burden et al. 2004;

Matzavinos et al. 2004) to the normal tissue acceptable, the rate of external effector cells u() must satisfy the constraint,

uðtÞ u max for each t 2 ½0, t

f

Þ, ð19Þ where u

max

stands for a tolerance coefficient.

The interval t

f

> 0 and the map v respectively stand for the time horizon and the protocol of the therapy.

Obviously, system (16–18) may be written in the setting of system (1) by taking its linear operator as,

Az ¼ :

ð D 1 r ² ‘ D 2 r ² T D 3 r ³ D 4 Þ

⁰

, for each z ¼ ð‘, T, , Þ

⁰

2 domðAÞ, where,

domðAÞ ¼ W ¹ ₀ ðÞ W ¹ ₀ ðÞ W ¹ ₀ ðÞ W ¹ ₀ ðÞ,

and, W ¹ ₀ ðÞ ¼ :

f 2 L ² ðÞ j f

⁰

2 L ² ðÞ and @f

@n ¼ 0 on @

:

Moreover, the semi-linear term is given by,

’ðz, vÞ ¼ :

gðzÞ þ ð u 0 0 0 Þ

⁰

, where,

gðzÞ ¼ :

k ₁ ‘ k ₂ ‘T k ₃ r ð‘rÞ þ k ₄ =ðk ₅ þ T Þ þ k ₆ k ₇ ð1 k ₈ TÞT k ₂ ‘T þ k ₉

k ₁₀ þ k ₁₁ k ₂ ‘T 0 B B

@

1 C C A

for each z belonging to the subset, D ¼ :

z ¼ ð‘, T, , Þ

⁰

2 Z j ‘ 0, T 0, 0 and 0

: Now, assume that there is a positive small number so as,

Tðx, tÞ " () there are no tumor cells near x at time t

ð20Þ It follows that eliminating of cancer cells within the therapy horizon t

f

may hold whenever

!ðzð, t

_f

ÞÞ ¼ , where the map ! is given by,

!ðzÞ ¼ :

fx 2 j TðxÞ "g,

for each z 2 D. We know by (Balakrishnan 1981) that, for every c > 0 the operator cr ² with domain W ¹ ₀ ðÞ generates a compact analytic semi-group (S

c

(t))

t>0

on L

²

(). This is true with regards to the operator A which generates the following semi-group,

SðtÞz ¼ ð S

D1

ðtÞ‘ S

D2

ðtÞT S

D3

ðtÞ expðD 4 tÞÞ

⁰

, for each z ¼ ð‘, T, , Þ

⁰

2 Z and t > 0, where the state space Z coincides to the Hilbert space L

²

()

⁴

.

Thus a therapy protocol can be followed by implementing Algorithm 1.

In the sequel, we are going to show how the key sequence to execute at sub-step IIa can be implemented. Let h be a small positive number. For each k 1, let,

v

_k

¼ ðu

_k

,

_k

Þ and z

_k

¼ ð‘

_k

, T

_k

,

_k

,

_k

Þ

⁰

,

and suppose that z

k

and v

k

are determined for k ¼ 1, . . . , M 1, then we have,

T

_M

¼ S

_D2

ðhÞT

_M1

þ hðk ₇ ð1 k ₈ T

_M1

ÞT

_M1

k 2 ‘

M1

T

M1

þ k 9

M1

Þ,

and

_M

¼ expðD ₄ hÞ

_M1

þ k ₂ h‘

_M1

T

_M1

: Define for each v ¼ ðu, Þ 2 R M,

‘

^v_h

¼ :

S

D1

ðhÞ‘

M1

þ h

k 1 ‘

M1

k 2 ‘

M1

T

M1

k 3 r ð‘

M1

r

M1

Þ þ k ₄

_M1

k 5 þ T

M1

þ k 6

M1

þ u

,

ð21Þ

that we rewrite in the form,

‘

^v_h

¼ a

_h

ðz

M1

Þ þ hu , ð22Þ where,

a

h

ðzÞ ¼ :

S

D1

ðhÞ‘ þ h

k 1 ‘ k 2 ‘T

k ₃ r ð‘rÞ þ k 4 k ₅ þ T þ k ₆

ð23Þ

for each z ¼ ð‘, T, , Þ 2 Z. Then, according to (10) and (11), the protocol v

M

and the approximated state z

M

may be updated by seeking a protocol, v

M

¼ ðu, Þ, in such a manner that,

S

_D2

ðhÞT

_M

þ h k ₇ ð1 k ₈ T

_M

ÞT

_M

k ₂ ‘

^v_h

T

_M

þ k ₉

_M

", on the zone !(z

_M

). That is equivalent to the following inequality,

k ₇ ð1 k ₈ T

_M

ÞT

_M

k ₂ ‘

^v_M

T

_M

þ k ₉

_M

" S

_D2

ðhÞT

_M

h ,

on !(z

M

), which holds whenever,

k 7 ð1 k 8 T

_M

ÞT

_M

k 2 ‘

^v_M

T

_M

þ k 9

_M

T

M

S

D2

ðhÞT

M

h ,

on !(z

M

), as T

M

on !(z

M

). This yields,

‘

^v_h

T

M

b

h

ðT

M

,

M

Þ T

M

hk 2

on !ðz

M

Þ, where the operator b

h

is given by,

b

h

ðT, Þ ¼ :

S

D2

ðhÞT þ hðk 7 ð1 k 8 T ÞT þ k 9 Þ, ð24Þ for each ðT, Þ 2 L ² ðÞ ² . Then by considering (21) and (22) we get,

uT

_M

b

_h

ðT

M

,

M

Þ T

M

hk 2 T

M

a

_h

ðz

M1

Þ

h ² k ₂ on !ðz

_M

Þ

ð25Þ

But this implies that u may exist whenever !ðz

_M

Þ.

Now, since stands for a therapy zone, it should be reduced as well as possible leading us to take it as follows,

¼

_M

¼ :

!ðz

M

Þ: ð26Þ Subsequently equation (25) may be expressed as follows, u c

h,M

on

_M

, ð27Þ where,

_M

¼ :

M

nfx 2 j T

M

ðxÞ ¼ 0g, also given by,

_M

¼ fx 2 j 0 < T

_M

ðxÞ "g, ð28Þ and,

c

_h,M

¼ : b

_h

ðT

_M

,

_M

Þ=T

_M

1 hk ₂ a

_h

ðz

_M1

Þ h ² k 2

, ð29Þ where a

h

and b

h

are respectively given by (23) and (24). Let,

u

^h

_min ¼ : sup

_M

c

h,M

ð30Þ

Now, taking into account the constraint on the patient’s quality of life (19) and considering equation (27), it follows that u should satisfy,

u

^h

_min u u _max :

Next, we present a summarization of the result above.

Theorem 5.1: Both the state z

M

and the protocol v

M

can be provided by the following procedure,

(i) Compute, c

M1

¼ :

k 7 ð1 k 8 T

M1

ÞT

M1

k 2 ‘

M1

T

M1

þ k 9

M1

, T

M

¼ S

D2

ðhÞT

M1

þ hc

M1

,

M

¼ S

D3

ðhÞ

M1

þ hðk 11

M1

k 10

M1

Þ,

M

¼ expðD ₄ hÞ

M1

þ hk 2 ‘

M1

T

M1

:

(ii) Let

M

,

_M

, and u

^h

_min as given respectively by (26), (28), and (30). Assume that u

^h

_min < u max . (iii) Make a suitable selection u

M

2 ½u

^h

_min , u max . Then

let,

‘

_M

¼ a

_h

ðz

_M1

Þ þ hu

_M

_M

:

As an immediate consequence of applying Algorithm 1 to the immunotherapy model (16–18),

we point out the following facts:

(a) Due to Equation (26), one has to infuse the effector cells over the zone without tumor cells.

(b) The rate u

M

of these effector cells at time M depends upon the state of the tumor at time M 1 and has as a best value the minimal rate u

^h

_min given by equation (30).

6. Conclusion

This paper has investigated a target control problem in the context of FSC. It demonstrates that the FSC laws can be used in order to steer a semi-linear parabolic system to a desired target set at terminal fixed time.

The advantages of implementing these control laws can be listed as follows:

. Their feedback dependence upon the zones where the target condition holds. These zones will spread to the whole domain under the action of the designed spreading controls.

. By construction, an FSC law leads to a solution to the semi-linear system of PDEs. We know the relative difficulty of proving existence of a solution to such systems.

. They give rise to easy implementable and recursive algorithms.

The second part of the study concentrates on an immunotherapy PDE model in the framework of FSC.

It is shown how the feedback protocols can be designed in order to destroy the cancer cells from the treated tissue.

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