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IMPULSIVE CONTROL PROBLEM GOVERNED BY FRACTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS.

1. Introduction

There has been a continued effort in the mathematical modeling of the dynamics and control of human immunodeficiency virus (HIV) by various authors ([7], [9], [11], [14], [15], [24], [25], [33], [37], [39], [44], [48]). One of the earliest models dealing with HIV is due to Perelson, Kirschner and De Boer [33]. They consider the interaction of HIV with CD4+ T-cells where the CD4+ T-cells consist of four population groups: uninfected T-cells, latently infected T cells, actively infected T cells, and free virus. Much effort has been put toward the study of the global dynamics of the HIV differential equation models. There have also been a number of studies where optimal control techniques are employed [24], [6], [45], [15], [21]. Memory is an important feature in immune response ([14], [38]). To include memory in the model fractional differential equations have been used ([14], [15], [21]). Hou and Wong consider ([23]) an impulsive control problem with application to HIV treatment. The rational for impulsive formulations is that while treatment by medication can suppress the virus to a very low level, the cost of purchasing the drugs as well as the amount of damage done to the body due to the intake of drugs can greatly offset the benefit of suppressing the HIV virus. Thus, a treatment regime of taking medication and the amount of medication at optimal instants may be more beneficial. Thus, in the current paper an impulsive fractional models is considered. We have decision variables at the impulse times and between impulse times. We start with a general formulation useful for a wider application besides the HIV modeling.

Besides applications in HIV modeling fractional differential equations have proved to be valuable tools in the modeling of many phenomena in engineering, physics, and economics ([18], [19], [20], [28], [29], [34], [43]). Fractional differential equations have also been useful in biology, fluid mechanics, modeling of viscoelasticity. The most fundamental characteristics in these models is their nonlocal characteristics. That is, the future aspect of the model relates not only the present state, but also its historical states.

Impulsive control problems have also been useful in engineering and in finance, production control and inventory management. In production planning ([8], [16],[30], [32]), a decision maker may have to decide the proper quantity of products being produced at different times with the objective of maximizing profit over a planning horizon. The goals that a decision maker has to accomplish are generally complex and involve conflicting objectives. The decision maker must meet demands while adhering to industry requirement needs, capabilities, limitations, and restrictions. Depending on the particular application an appropriate model may be discrete or continuous time optimization problem.

In [30] a production-planning model conducive to optimization is developed and used with the preference-based optimization method: linear physical programming, multiobjective programming. In [16] a continuous-time aggregate production- planning is considered where the objective is to determine the total production-planning cost, which involves various sets of costs like production cost, subcontracting cost, overtime cost, hiring cost, firing cost, and inventory cost.

Mathematical aspects of impulsive hybrid control systems have been considered by engineers and mathematicians. In addition to the references in production planning above relevant references include [4], [5], [17], [35], [36], [42].

The organization of the paper is as follows. We first present preliminaries, then the problem statement. Then, we establish necessary conditions for the control problem, and finally present computational results.

2. Preliminaries

For information on fractional differential equation we recommend the reference [34]. Let f : [0, [infinity]) [right arrow] R. For -[infinity] < a < b < [infinity] the fractional integral of order [alpha] > 0 of f with lower limit zero is defined as

[sub.a][I.sup.[alpha].sub.t]f(t) = 1/[GAMMA]([alpha]) [[integral].sup.t.sub.0] f(s)/[(t - s).sup.1-[alpha]].

The left Riemann-Liouville fractional derivative of order [alpha] of f is given as

[mathematical expression not reproducible].

The right Riemann-Liouville fractional derivative of order a of f is given as

[mathematical expression not reproducible].

The right Caputo derivative of f of order [alpha] with lower limit zero is given as

[mathematical expression not reproducible].

The right and left Caputo derivatives, in integral form, are given as

[mathematical expression not reproducible].

The initial value problem

(2.1) [mathematical expression not reproducible]

is equivalent to the nonlinear Volterra integral equation ([34]):

x(t) = [x.sub.0] + 1/[GAMMA]([alpha]) [[integral].sup.t.sub.0] [(t - s).sup.[alpha]-1] f(s, x(s))ds.

In this paper we take [alpha] = 0.9.

3. Problem Statement

Let 0 = [t.sub.0] < [t.sub.1] < [t.sub.2] < ... < [t.sub.n-1] < [t.sub.n] = [t.sub.f] and, for i = 1, 2, ..., n the functions [f.sub.i] : [[t.sub.i-1], [t.sub.i]] x [R.sup.n] x [right arrow] R be such that [f.sub.i] (x, x, u) is measurable for fixed (x, u). For fixed t and u, the function [f.sub.i] is continuously differentiable in x. For fixed t, [f.sub.i] (t, x, x) is continuous. We also assume that

[mathematical expression not reproducible],

where K is a fixed constant.

Next let [h.sub.i], i = 1, 2, ..., n be an n x n matrix with continuously differentiable entries. That is, if the (k, j) entry of [h.sub.i] (n) is [a.sup.i.sub.kj]([eta]), then [a.sup.i.sub.kj] is a continuously differentiable of [eta].

Now, we consider the following fractional differential equation

(3.1) [sup.C.sub.0][D.sup.q.sub.t] [x.sub.1](t) = [f.sub.1](t, [x.sub.1](t), [u.sub.1](t)), 0 < q < 1, 0 = [t.sub.0] < t < [t.sub.1] x([t.sub.0]) = [h.sub.i] x [c.sub.1]

and for i = 2, ..., n

(3.2) [mathematical expression not reproducible]

We consider the objective function

[mathematical expression not reproducible].

The impulsive control problem we consider is

[mathematical expression not reproducible]

subject to

(3.3) [mathematical expression not reproducible].

Assume that problem (P) has a solution ([[bar.c].sub.1], ... [[bar.c].sub.n]), ([[bar.u].sub.1], ... [[bar.u].sub.n]). We denote the corresponding trajectories, [[bar.x].sub.i], i = 1, ..., n. Let U = [U.sub.1] x ... x [U.sub.n] be the control set containing the controls [[bar.u].sub.1], ... [[bar.u].sub.n]. Assume that U is a convex set. We can put constraints on the decision variables [c.sub.1], ..., [c.sub.n].

In the interval ([t.sub.n-1], [t.sub.n]) we have the fractional differential equation

(3.4) [mathematical expression not reproducible]

Let v [member of] [U.sub.n]. Consider

(3.5) [mathematical expression not reproducible]

Then,

(3.6) [mathematical expression not reproducible]

Set

(3.7) [mathematical expression not reproducible]

Then

[[parallel][x.sub.[theta]n](t) - [[bar.x].sub.n](t)/[theta] - [delta][x.sub.n](t)[parallel].sub.[infinity]] [right arrow] 0 as [theta] [right arrow] [0.sup.+].

Given u [member of] [U.sub.n] let

(3.8) [mathematical expression not reproducible]

Now, let [p.sub.n] [member of] [L.sub.2] ([[t.sub.n-1], [t.sub.n]]) such that

(3.9) [mathematical expression not reproducible]

Then,

(3.10) [mathematical expression not reproducible]

For ease of notation let us write

[mathematical expression not reproducible].

Then, using (3.7) and (3.10)

(3.11) [mathematical expression not reproducible]

We also have

(3.12) [mathematical expression not reproducible]

Thus,

(3.13) [mathematical expression not reproducible].

Let

(3.14) [mathematical expression not reproducible].

Next, we move to the interval [[t.sub.n-2], [t.sub.n-1]] and consider

(3.15) [mathematical expression not reproducible]

We have

(3.16) [mathematical expression not reproducible].

For v [member of] [U.sub.n-1], 0 < [theta] < 1, let

[mathematical expression not reproducible]

Proceeding as in (3.5-3.7) and taking limit as was done following (3.7) we arrive at the following two equations which are the changes in the states [x.sub.n-1], [x.sub.n] due to the change in [u.sub.n-1] from [u.sub.n-1] to [[bar.u].sub.n-1] - [theta]v while [[bar.u].sub.n] is unchanged.

(3.17) [mathematical expression not reproducible],

(3.18) [mathematical expression not reproducible].

For k = 2, 3, ..., n let [L.sub.k] be the solution of the fractional differential equation

(3.19) [mathematical expression not reproducible].

Next, for k = 2, 3, ..., n set

(3.20) [Q.sub.k] ([[bar.x].sub.k-1] ([t.sub.k-1]), [[bar.c].sub.k]) [[[h.sub.k] ([[bar.x].sub.k-1] ([t.sub.k-1])) x [[bar.c].sub.k]],.sub.x]) + I.

Using (3.19) and (3.20) the solution of (3.18) is given by

(3.21) [delta][x.sub.n] (t) [L.sub.n](t)[Q.sub.n] ([[bar.x].sub.n-1] ([t.sub.n- 1]), [[bar.c].sub.n])[delta][x.sub.n-1] ([t.sub.n-1])

Next, we note that

(3.22) [mathematical expression not reproducible]

The variation in the total cost due to the variation in [[bar.u].sub.n-1] [member of] [U.sub.n-1] involves the costs in the intervals [[t.sub.n-2], [t.sub.n-1]] and [[t.sub.n-1],[t.sub.n]] and is given by

(3.23) [mathematical expression not reproducible]

Similarly, the variation in the total cost due to the variation in [[bar.u].sub.n-2] [member of] [U.sub.n-2] involves the costs in the intervals [[t.sub.n-3], [t.sub.n-2]], [[t.sub.n-2], [t.sub.n- 1]] and [[t.sub.n-1], [t.sub.n]]. As in (3.8) given u [member of] [U.sub.n-1] let

(3.24) [mathematical expression not reproducible]

As in (3.9) define [p.sub.n-1] [member of] [L.sub.2] ([[t.sub.n-2], [t.sub.n- 1]]) by the equation

(3.25) [mathematical expression not reproducible]

Following the steps that we used to get (3.13) we obtain

(3.26) [mathematical expression not reproducible]

Set

(3.27) [mathematical expression not reproducible]

where [[gamma].sub.n] is defined in (3.14). Then, we may rewrite (3.26) as

(3.28) [mathematical expression not reproducible].

Setting

(3.29) [mathematical expression not reproducible],

the adjoint function in the interval [[t.sub.n-3], [t.sub.n-2]] is given by

(3.30) [mathematical expression not reproducible].

Next we proceed to give a formula for the adjoint function in any interval [[t.sub.n-(i+2)], [t.sub.n-(i+1)]]. Let

(3.31) [mathematical expression not reproducible],

Then,

(3.32) [mathematical expression not reproducible].

We now define the Hamiltonian in the interval [[t.sub.(i-1)], [t.sub.i]], i = 1, 2, ..., n by

(3.33) [H.sub.i] (t, [x.sub.i](t), [q.sub.i](t), [u.sub.i](t)) = [q.sub.i] (t) x [f.sub.i](t, [x.sub.i](t), [q.sub.i](t), [u.sub.i](t)) + [PHI](t, [x.sub.i](t), [q.sub.i](t), [u.sub.i](t))

Then, for any v [member of] [U.sub.i],

(3.34) [mathematical expression not reproducible]

To show the validity of (3.34) we will verify it in the last interval [[t.sub.(n-1)], [t.sub.n]]. When we perturb the control Un only the last term of the cost

(3.35) [mathematical expression not reproducible],

which is,

(3.36) [mathematical expression not reproducible]

is affected. Thus, if we perturb [[bar.u].sub.n] by adding [theta]v, v [member of] [U.sub.n] to it, then

(3.37) [mathematical expression not reproducible]

Thus,

(3.38) [mathematical expression not reproducible]

Next, using (3.10), and writing

[mathematical expression not reproducible]

we have

(3.39) [mathematical expression not reproducible]

Thus,

(3.40) [mathematical expression not reproducible].

Now, from (3.40), making needle-like variation, we obtain

(3.41) [mathematical expression not reproducible]

So far we have perturbed only the controls between impulse times. That is, we have assumed that problem (P) has a solution ([[bar.c].sub.1], ... [[bar.c].sub.n]), ([[bar.u].sub.1], ... [[bar.u].sub.n]), and perturbed only the controls ([[bar.u].sub.1], ... [[bar.u].sub.n]) between the impulse times and obtained the minimum principle (3.41) where the adjoint variables are as presented in (3.13), (3.26), and in general, in the interval [[t.sub.n-(i+2)], [t.sub.n-(i+1)]], by (3.32). Next we perturb the decision variables ([[bar.c].sub.1], ... [[bar.c].sub.n]). First we perturb only the decision variable [[bar.c].sub.n], while holding the other decision variables ([[bar.c].sub.1], ... [[bar.c].sub.n-2], [[bar.c].sub.n-1]), ([[bar.u].sub.1], ... [[bar.u].sub.n]) fixed. Only the last component

[mathematical expression not reproducible]

of the total cost

[mathematical expression not reproducible]

is affected. Next, we perturb only [[bar.c].sub.n-1] while holding the remaining decision variables ([[bar.c].sub.1], ... [[bar.c].sub.n-2], [[bar.c].sub.n]), ([[bar.u].sub.1], ... [[bar.u].sub.n]) fixed. Only,

[mathematical expression not reproducible]

of the total cost J([[bar.x].sub.1], [[bar.u].sub.1], ..., [[bar.x].sub.n], [[bar.u].sub.n]) is affected. Next we perturb [[bar.c].sub.n-2] and continue in this manner backwards. We obtain the following necessary conditions.

(3.42) [mathematical expression not reproducible]

4. Application

The following model of HIV-immune system with memory was considered in [21]. Here we extend this model to one where we consider impulsive model with added constraints at the impulse times. This extension is appropriate as stated in the introduction [23]. The model considered in [21] is given by the system

(4.1) [mathematical expression not reproducible],

where [x.sub.1] represents free virus, [x.sub.2] uninfected CD4+ T cells, [x.sub.3] lately infected CD4+ T cells, [x.sub.4] actively infected CD4+ T cells. The control [u.sub.1] is the concentration of protease inhibitor, [u.sub.2] fusion inhibitor, [u.sub.3] CD4+ T cell enhancer, [u.sub.4] reverse transcription inhibitor. The parameters [s.sub.i], [q.sub.i], i = 1, 2, 3, 4 and r are weight constants in the objective functional below.

Further,

[a.sub.1] = death rate of free virus, [a.sub.2] = rate CD4+ T cells become infected with virus. [a.sub.3] = number of free virus produced by actively infected CD4+ T cells. [a.sub.4] = death rate of actively infected CD4+ T cell population. [a.sub.5] = source term of uninfected CD4+ T cells. [a.sub.6] = death rate of infected (latently infected) CD4+ T cell population. [a.sub.7] = growth rate of CD4+ T cell population. [a.sub.8] = maximum population level of CD4+ T cells. [a.sub.9] = rate of latently infected cells becoming active.

Rational for this model has been presented [14], [15], [21]. In [21] necessary conditions for optimality were presented for a control problem with the dynamics given by the above model where the cost

(4.2) [mathematical expression not reproducible]

is to be minimized.

The objective in this paper is to deal with the impulsive control version of this control problem. We consider [t.sub.0] < [t.sub.1], ..., [t.sub.n] = [t.sub.f] where [t.sub.1], [t.sub.2], ..., [t.sub.n-1] are the impulse times, and constraints are imposed on the trajectories at these impulse times. We can add constraints at the initial and final times [t.sub.0] and [t.sub.f]. The material presented in the previous sections applies to more general models than we are considering in this section.

We now proceed to formulate the impulsion version of the above problem. First we divide the interval [[t.sub.0], [t.sub.f]] into n intervals: [[t.sub.i-1], [t.sub.i]], i = 1, 2, ..., n. In the interval [[t.sub.i-1], [t.sub.i]], we consider

(4.3) [mathematical expression not reproducible],

At the impulse times [t.sub.1], [t.sub.2], ..., [t.sub.n-1] we have

(4.4) [x.sub.i] ([t.sub.i-1]) = [h.sub.i]([x.sub.i-1]([t.sub.i-1]))[c.sub.i] + [x.sub.i-1] ([t.sub.i-1]).

At t = [t.sub.0]

(4.5) [x.sub.1] ([t.sub.0]) = [h.sub.1]([c.sub.1]).

We remark that the [h.sub.i], i = 1, 2, ..., n are 4 x 4 matrices and c = [([c.sub.i1], [c.sub.i2], [c.sub.i3, [c.sub.i4]).sup.T]. The cost is given by

(4.6) [mathematical expression not reproducible]

We now proceed to write the adjoint system. In the time interval [[t.sub.i-1], [t.sub.i]]

(4.7) [mathematical expression not reproducible],

Writing [f.sub.i] = [([f.sub.i1], [f.sub.i2], [f.sub.i3], [f.sub.i4]).sup.T] for the right hand side of (4.3) and [p.sub.i] = [([P.sub.i1], [P.sub.i2], [P.sub.i3], [P.sub.i4]).sup.T] the in [[t.sub.i-1], [t.sub.i]] is given by

(4.8) [mathematical expression not reproducible]

If [u.sub.i] = [([u.sub.i1], [u.sub.i2], [u.sub.i3], [u.sub.i4]).sup.T] were an interior point of the control constraint [U.sub.i] then, using (3.34) we have

(4.9) [mathematical expression not reproducible].

From (4.9) we get

(4.10) [mathematical expression not reproducible]

We remark that the optimal control may not be an interior point of [U.sub.i]. In the next section we take three intervals and carry out a numerical computation. Our numerical procedure is going to based on the method of steepest descent.

5. Numerical Computation and Simulation

In this section we take three intervals [[t.sub.0], [t.sub.1]], [[t.sub.1], [t.sub.2]], [[t.sub.2], [t.sub.3]], [t.sub.3] = [t.sub.f] and carry out a numerical simulation of the following impulsive control problem. For simplicity of notation we use different symbols for the states and controls in different intervals. All parameters will be given specific values later.

In the interval [[t.sub.0], [t.sub.1]], we consider

(5.1) [mathematical expression not reproducible].

In the interval [[t.sub.1], [t.sub.2]], we consider

(5.2) [mathematical expression not reproducible].

In the interval [[t.sub.2], [t.sub.3]], we consider

(5.3) [mathematical expression not reproducible].

The cost is given by

(5.4) [mathematical expression not reproducible]

Let [f.sup.(3)] (z, w) = ([f.sup.(3).sub.1] (z, w), [f.sup.(3).sub.2] (z, w), [f.sup.(3).sub.3] (z, w), [f.sup.(3).sub.4] (z,w)) where

(5.5) [mathematical expression not reproducible],

Let [f.sup.(2)] (y, v) = ([f.sup.(2).sub.1] (y, v), [f.sup.(2).sub.2] (y, v), [f.sup.(2).sub.3] (y, v), [f.sup.(2).sub.4] (y, v)) where

(5.6) [mathematical expression not reproducible],

Let [f.sup.(1)] (x, u) = ([f.sup.(1).sub.1] (x, u), [f.sup.(1).sub.2] (x, u), [f.sup.(1).sub.3](x,u), [f.sup.(1).sub.4] where

(5.7) [mathematical expression not reproducible],

Let [L.sup.(3)] be defined by the equation

(5.8) [mathematical expression not reproducible].

Let [L.sup.(2)] be defined by the equation

(5.9) [mathematical expression not reproducible].

Let L(1) be defined by the equation

(5.10) [mathematical expression not reproducible][down arrow].

Let [Q.sup.(3)] be be the matrix defined by

(5.11) [Q.sup.(3)] = diag(CC 1 + 1, CC2 + 1, CC3 + 1, CC4 + 1)

In (5.11) the notation "diag" means that matrix has all entries zero except the diagonal elements. Let [Q.sup.(2)] be be the matrix defined by

[Q.sup.(2)] = diag(cc1 + 1, cc2 + 1, cc3 + 1, CC4 + 1)

From the objective function in problem (P), (4.6), and (3.3)

(5.12) [mathematical expression not reproducible]

We now proceed to write the adjoint equations. We denote the adjoint variable by [p.sup.(3)] in the third interval, by [p.sup.(2)] in the second interval, and by [p.sup.(3)] in the first interval. In the third interval [p.sup.(3)] is the solution of the fractional differential equation

(5.13) [mathematical expression not reproducible][down arrow]

In the interval [[t.sub.1], [t.sub.2]] the adjoint is the solution of the fractional differential equation

(5.14) [mathematical expression not reproducible][down arrow].

In the interval [[t.sub.0], [t.sub.1]] the adjoint is the solution of the fractional differential equation

(5.15) [mathematical expression not reproducible]

Next we write the Hamiltonians in each of the intervals. In the interval [[t.sub.2], [t.sub.3]] we have

(5.16) [H.sub.3] (t, [z.sub.3](t), [p.sup.(3)] (t), v) [greater than or equal to] [H.sub.3](t, [z.sub.3](t), [p.sup.(3)] (t), w(t)), a.e. t [for all] v [member of] [U.sub.3]

The Hamiltonian in the interval [[t.sub.1], [t.sub.2]] we have

(5.17) [H.sub.2](t, [y.sub.2] (t), [p.sup.(2)](t), v) [greater than or equal to] [H.sub.2](t, [y.sub.2](t), [p.sup.(2)](t), v(t)), a.e. t [for all] v [member of] [U.sub.2]

The Hamiltonian in the interval [[t.sub.0], [t.sub.1]] we have

(5.18) [H.sub.1](t, [y.sub.1](t), [p.sup.(1)](t), v) [greater than or equal to] [H.sub.1](t, [y.sub.1](t), [p.sup.(1)](t),u(t)), a.e. t [for all] v [member of] [U.sub.1]

To carry out the numerical simulation we use the state equations (5.1), (5.2), (5.3), the adjoint equations (5.13), (5.14), (5.15) and the Hamiltonians (5.16), (5.17), (5.18). Specific values for the parameters in the state equations, and the cost are given below. The numerical procedure goes as follows. We start with the third interval, use the Hamiltonian to improve on the control. Using the improved control we update the state and the adjoint variables in the third interval. Then move to the second interval and use the Hamiltonian in the second interval to improve the control in the second interval. Then we use the improved control and update the states in the second interval and the third interval. The states in the third interval get updated because of the change in the state variables at [t.sub.2]. Finally move to the first interval and use the Hamiltonian there to improve on the control. Using the improved control update the states in the first interval. Due to the change in the states of the first interval at [t.sub.1] the states in the second interval, hence the states in the third interval are updated.
BEGIN pseudocode

Third interval:

Use Hamiltonian to improve control.
Using improved control update state, adjoint variables in
third interval.

Second interval:
Improve control in the second interval.
Update states in second and third interval.
First interval: Improve control.
Update states in all intervals.
END pseudocode


This procedure is essentially dynamic programming procedure.
The values of the parameters in (4.1) are given in the
following table.

Parameters                              Values

[a.sub.1] = Death rate of free virus    2.5d-1

[a.sub.2] = Rate CD4+ T cells become    2.4 x [10.sup.-5]
infected with virus                     [mm.sup.3]
                                        [d.sup.-1]

[a.sub.3] = Number of free virus        1200
produced by actively infected CD4+ T
cells

[a.sub.4] = Death rate of actively      0.24[d.sup.-1]
infected CD4+ T cells population

[a.sub.5] = Source term for             10[d.sup.-1]
uninfected CD4+ T cells                 [mm.sup.-3]

[a.sub.6] = Death rate of uninfected
(latently infected) CD4+ T cells        0.02[d.sup.-1]
population

[a.sub.7] = Growth rate of CD4+ T       0.02[d.sup.-1]
cells population

[a.sub.8] = Maximal population level    1500[mm.sup.-3]
of CD4+ T cells

[a.sub.9] = Rate latently infected      3 x [10.sup.-3]
cells become active                     [d.sup.-1]


In (5.4) the parameters in the cost are give the values

(5.19) [s.sub.1] = [s.sub.3] = [s.sub.4] = [q.sub.1] = [q.sub.3] = [q.sub.4] = [10.sup.3]

In (5.1) the parameters in the cost are give the values

(5.20) [mathematical expression not reproducible],

Next, we give particular values to the parameters in (5.2) and (5.3)

(5.21) [mathematical expression not reproducible].

The control variables take the values in the following table (Table 1).

Next, we give particular values to the parameters in (5.2) and (5.3) when there is no memory, that is the model is no more fractional differential equation. Thus, in (4.1), the left hand sides are ordinary derivatives of the states. The control variables in (5.2) and (5.3) are given in the following table (Table 1).

(5.22) [mathematical expression not reproducible].

Then, we get the following graphs (Figure 4-Figure 6).

6. Discussion of the results of the numerical computation

The numerical computation shows that in the fractional differential equation model that the virus at the intervention/impulse times should be killed 99%. The same is true in latently and actively infected CD4+ cells. Although there is damage to uninfected CD4+ cells the number rises to what is regarded as normal. The virus level and the infected and latently infected CD4+ cells also increase. However their number does not reach the number for uninfected CD4+ cells. In the differential equation model if 95% of the virus and the CD4+ cells are killed at the time of the intervention the number of uninfected CD4+ cells rises quickly to the normal number while the virus level and the infected CD4+ cells remain low. What one observes in these models is the importance of planned strong interventions.

7. Conclusion

We have considered an optimal control problem governed by fractional order differential equations modeling an HIV-immune system. The rational for using fractional differential equations is to account for the fact that the immune response involves memory. The impulse system formulation is to account for the fact a treatment regime of taking medication and the amount at optimal instants may be less damaging to the body and also less expensive. Some medications may still have to be taken regularly. Thus we have decision variables at impulse time and between impulse times. We have constructed necessary conditions for optimality and carried out numerical computation. Our results demonstrate that regardless of what we do between impulse times strong interventions are needed at impulse times, and the controls in between impulse times help increase the length of time between impulse times.

Received October 10, 2016

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N. G. MEDHIN (1) AND M. SAMBANDHAM (2)

(1) Department of Mathematics, North Carolina State University Raleigh, NC 27695-8205

(2) Department of Mathematics, Morehouse College, Atlanta, GA 30314

Caption: Figure 1. States in Time Interval 1.

Caption: Figure 2. States in Time Interval 2.

Caption: Figure 3. States in Time Interval 3.

Caption: Figure 4. States in Time Interval 1.

Caption: Figure 5. States in Time Interval 2.

Caption: Figure 6. States in Time Interval 3.
TABLE 1

Interval 1   Interval 2   Interval 3

u1 = 1.0       v1 = 1       w1 = 1
u2 = .95       v2 = 0       w2 = 1
u3 = 0.0       v3 = 1      w3 = 0.9
u4 = .95       v4 = 1       w4 = 1

TABLE 2

Interval 1         Interval 2       Interval 3

u1 = 0               v1 = 0       w1 = 0.0418538
u2 = 1               v2 = 1           w2 = 1
u3 = 0.0009521   v3 = 0.0009401   w3 = 0.0008798
u4 = 1.026987    v4 = 1.010854        w4 = 1
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