Rev. Acad. Canar. Cienc., XX (Núms. 1-2), 33-47 (2008) (publicado en septiembre de 2009)
OPTIMIZATION OF A FUNCTIONAL ON AN OPEN MANIFOLD
WITH DIFFERENTIABLE VARIETY AS ITS BOUNDARY
D. K. Bhattacharya, A. Gupta & T. E. Aman
AFISTRACT. The paper discusses control theoretic optimization of a functional
with restrictions on the domain of definition of the functional , in the sense
that, the domain is an open manifold with boundary, where the boundary is a
differcntiablc variety. Tbe paper also shows a realistic application of such an
optima! control problem.
Keywords and Phrases: Differentiable manifold with boundary, Differentiable
variety, Constrained optimal control problems with restricted domain.
Subject Classification Code [2005]: Primary 54C40; Secondary 46E25
l. INTRODUCTTON
Optimal control problems are of two types - ( i) when the restrictions are only
in the parameter domain and ( ii) when the restrictions are in the stable domain
and also in the parameter domain. The necessary condition of optimality type (i)
is known as Pontryagin's maximum principle (14], similar conditions of optimality
in type ( ii) is given in [l, 2, 13, 17]. So far as type ( i) optimal control problem are
concerned, their applications in real world problems and the corresponding analysis
are found in [6, 7, 8, 9]. But it is very difficult to have realistic applications of
type (ii) optima! control problems owing to arbitrary restrictions on the domain
of the fimctional expressed by inequalities of differentiable functions. In realistic
applications, it is found that the domain of restriction of the functional is a manifold
with boundary [10, 15], where the manifold is an open set of Rn (n > 1) and the
boundary is a differentiability variety in Rn [10]. Naturally it remains open to
formulate type (ii) optima! control problem with the aforesaid type of restrictions
on the state domain and apply it in realistic optima! control problem. The present
chapter dea ls with such control problems and their applications.
The whole matter of the paper is divided into four main sections, where section
1 gives the introduction. Section 2 gives the idea of differentiable manifold
and differentiable variety and contains sorne results in this connection. In section
3, sorne constrained optimal control problem in parameter domain are given and
discussing sorne real world problems on the domain of definition of the functional,
in the sense that, the domain is a manifold with boundary, where the boundary is ·
a d ifferentiable variety. Section 4 d iscusses the stability analysis and the controltheoretic
optimization of a functional of replicator dynamics on the same domain
of definitions.
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2. SOME KNOWN D EFINITIONS AND RESULTS [10]
Definition 2.1. [10] A differentiable variety in R"+l is defined as u-1 (O)},
where f : Rn+l ---+ R is a differentiable function such that at each z E M, the
matrix [f ,J(z)] has rank one, j = 1, 2, .... ., n
It can be shown that it is a differentiable manifold of dimension n.
Example 2.2. [10] A 2-sphere S2 = {(z1, z2, z3) E R3 : ZI + z~ + z§ - 1 =O} is a
differentiable variety in R3 and it is a manifold of dimension 2.
3. CONSTRAINED OPTIMAL CONTROL PROBLEM WITH RESTRICTIONS IN THE
STATE SPACE
Statement of the problem and its solution
(3.1) Maximize Jx(u) = cj>(t,xtJ + 1ti F(x,t,u)dt \f(x,u) E X
where X = {(x,u): corresponding to each u, x(t) is an integral curve of x'
f(t, x, u) and G(x, u) ~O}; G(x, u) ~O denotes a manifold with boundary in R 11+m
whose interior is open sub-manifold of Rn+m and whose boundary is a differentiable
variety given by G(x,u) =O; u E Rm, t E [O,t1], x = x(t) E R" is C1, x = Xo when
t = O; further f : R X Rn X Rm ---+ R11 ' e/> : R X R11 ---+ R, F : R X R11 X Rm ---+ R are
all C1-maps.
Proceeding as in [2, 17], we get the following
Theorem 3.1. A necessary condition that (x*, u*) minimizes the control problem
(3.1) is that there are costate vectors >. (t) and μ(t) such that the following holds:
(i) >.(ti)= (~)t=tu (ii) Fx + >.fx + μGx + ~ =O and
(iii) (Fu+ >.fu+ μGu)u=u* =O.
Remark 3.2. It is noted that the necessary conditions of optimality as given in
Theorem 3.1 reduce to the solution of a system of ordinary differential equations
in co-state variables. Naturally it becomes almost impossible to find the solution
analytically. This is why, steady state optimal solution is needed. Hence prior to
finding out the optimal solution, at least local asymptotic stability of the system is
to be assured.
Example 3.3. Suppose x, y are two non-interacting fishes and z is their predator
moving in a part of an ocean, that is given by x2 + y2 + z2 - a ~ O, x > O, y >
O, z > d > O, for some real a and d. Let the natural dynamics of motion be given
by (with standard meaningsj
(3.2)
x = rx(l - ..::_) - axz
K
y = sy(l - '}!_) - f3yz
L
Z = z( - f + /X + Óy)
Let the dynamics of exploited motion under control parameter u per unit biomass
be given by the following differential equations (with standard meaningsj
x = rx(l - ..::_) - axz - q1ux
K
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(3.3) y= sy(l - '}!_) - (Jyz - Q2UY
L
Z = z( - f + "(X + Jy) - Q3UZ
u = u(t) is the effort, q;, (i = 1, 2, 3) are the catchability coefficients of x, y, z
species, In this case, the state space of the natural dynamical system (3,2) is a
manifold with boundary; the interior is the three dimensional open sub-manifold of
R 3: x2 + y2 + z2 - a < O and the boundary is a differentiable variety of R 3, which
is the surface of a two dimensional sphere given by x 2 + y2 + z2 = a, The state
space of the exploited system (3,3) is x2 + y 2 + z2 + bu - a ::::; O which is also a
manifold with boundary; the interior is the four dimensional open sub-manifold of
R4 : x 2 + y2 + z2 + bu - a < O and the boundary is a differentiable variety of R4 ,
which is a paraboloid (of dimension 3) given by x2 + y2 + z2 +bu = a, Moreover,
the state space of the exploited system is bounded since (x, y , z, u) E R~.
Let c be the cost per unit efj'ort u. Let p1 , p2 , p3 be the prices of the species x, y , z
respectively. Then the profit function is taken as
(3.4)
where (x, y, z, u) belonging to the state space.
The control problem is as follows:
(3.5) M aximize 1T (P1Q1X + P2Q2Y + p3q3z - c)udt, u E U [the control set].
Remark 3.4. As mentioned in Remark 3.2, that first of all local stability analysis
of the system (3.3) is to be done and then corresponding optimal analysis is to be
performed. Far this problem local stability analysis is already known in connection
with fishery management problems, but the optimal analysis is definitely much
harder and completely new.
Proposition 3.5. The dynamical system (3.2) possesses equilibrium (x1 , y1 , z1 ) >
(O, O, O) , if
(3.6) a,("(K + JL - f) (J("f K + JL - f ) K JL f
r > p ,s > p ,"f + >
where P = -yoK + /3íiL.
1' s
Similarly (3.3) possesses equilibrium (x2 , y2 , z2 ) > (O, O, O) if (3.6), (3.7) and (3.8)
hold,
(3.7)
(3.8)
"(K +óL-f
u< Q
where Q = J!l.J!i. + íiq2 L r s + q3
Proof. It follows that
X1 = K[l - ,,':¡, ("f K + JL - f)], y1 = L[l - j3p ("f K + JL - !)], z1 = 1K +;L-f where
p = 10.K + {36L.
r s
So, (x1 , y1 , zi) > (O, O, O) , if (3.6) holds. Similarly, it follows that
Ku aQ
X2 = X1 - - (q1 - - ) r p
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where Q = 121~K + óq~L + q3 .
Clearly (x2, y2, z2) > (O, O, O), if (3.6), (3.7) and (3.8) hold. D
Proposition 3.6. The vanishing equilibrium (O, O, O) is always an unstable saddle
point and the interior equilibrium ( x1 , y1 , z1 ) is asymptotically stable for the system
(3.2).
Prooj. The variational matrix for the equation in model (3.2) at origin is
lo= ( ~ ~ ~ )
o o -f
which clearly shows that (O, O, O) is an unstable saddle point.
To show the interior equilibrium (x1 , y1 , z1 ) is asymptotically stable, we first rewrite
the model (3.2) as
(3.9)
. r
x = x[- K(x - x1 ) - a(z - zi)]
iJ = y[-~(y - Y1) - /3(z - z1)]
L
i = z[!'(x - x1) + 8(y - Y1)]
Consider the Liapunov function as
V(x, y, z) =X - X1 - X1 log(:,) + C1 [y - Yl - Y1 log(* )] + c2[z - Z1 - Z1 log( *)]
where c1 , c2 >O are constants to be determined suitably. It is obvious that V(x, y, z)
is positive definite.
. X - X1 y - YI z - Z1
V(x,y ,z) = (--)x + c1(--)iJ + c2(--)i
X y Z
= (x - x1)[- i(x - xi) - a(z - z1)] + c1(Y - Y1)[-f(y - Y1) - /3(z - z1 )] + c2(z -
z1)b(x - x1) + 8(y - Y1)]
= -i(x -x1)2 - f (y-y1)2 + (c2')'-a)(x -x1)(z - z1) + (c28 -c1/3)(y-y1)(z - z1)
Let us choose c2 = ~, then c1 = ~~ which implies that V(x, y , z) is negative and
consequently the interior equilibrium ( x 1 , y1 , z1 ) is asymptotically stable. D
Bionomic Equilibrium and its feasibility[3]
Let L denote the locus of dynamic equilibrium of the three species system (3.3)
and !et 7r = O denote the zero profit function. A feasible equilibrium is the point
of intersection of L = O and 7r = O, provided ali the coordinates of this point are
positive and also the value of the control parameter u(t) is positive at this point.
It is usually denoted by (x=, Y=, z=)·
The optimal steady state analysis is taken around the bionomic equilibrium of
the model, so its existence is to be assured. In this connection we prove the following
theorem.
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Theorem 3. 7. Let the dynarnic rnodel be given by (3.3) under the restrictions (3.6),
(3.7) and (3.8) . Let the objective function be given by (3.4). Then there exists a
feasible bionornic equilibriurn if (3.10) and (3.11) holds separately where
(3.10)
(3.11)
Proof. The locus of dynamic equilibrium (x2 , y2 , z2 ) is given by
(3.12) X - X1 y - Yl z - Z1 L. ·p- -----Q- -- ---R- -u
The zero profit function is given by
(3.13) 7r = P1q1x + p2q2y + p3q3z - e= O
If (3.12) intersects (3.13) at (x* , y* , z*) where u = u 1 , then it follows that
x = x* + Pu1, y* = Y1 - Qui , z* = z1 - Ru1
Again from 7r = O, it follows that
P1Q1X1 + p2q2Y1 + p3q3 z1 - e= (p2q2Q + p3q3R - P1Q1P)u1.
Hence
(3.l4) ui = P1q1x1 + P2q2Y1 + p3q3z1 - e
P2q2Q + p3q3R - P1Q1P
Obviously, we get u 1 >O if (3.10) and (3.11) hold.
Statement of the optimal control problem and its solution
Let the state space of the exploited system (3.3) be given by
o
X = { (x, y, z, u): corresponding to each u, (x (t) , y(t) , z(t)) is an integral curve of the
exploited system (3.3) and G(x, y , z, u) :::; O} where G(x, y, z, u) = x 2+y2+z2+bu-a
then G(x, y , z, u) :::; O denotes a manifold with boundary in R4 whose interior is
an open sub-manifold of R4 and whose boundary is a differentiable variety given
by G(x, y, z, u) = O; u E R, t E [O, T], (x(t) , y(t) , z(t)) E R3 is C 1 , (x, y, z) =
(xo, Yo, zo ) when t = O; further Jet f : R3 x R --> R3 , 7r : R3 x R --> R be all
C 1-maps where f = (f1, h ,f3)
X
fi(x , y ,z, u) = x[r(l - K) - az - Q1u]
y
h(x, y, z, u) = y[(l - -¡) - /3z - Q2u]
h(x, y, z, u)= z[-f + "(X+ 8y - q3u]
We assume that the total time taken to control the biomass is T . Then the
control problem is to maximize
(3.15) J = 1T 7r(x, y, z , u)dt V(x, y, z, u) E X
over the control parameter u , where u E (O, Umax ) and to find a suitable u= u* in
(O, Umax) for which J is maximum where
7r(x, y, z, u) = (p1 q1x + P2Q2Y + p3q3z - c)u(t)
Before going to the main theorem we want to find out the particular solution of a
3- system of ordinary differential equation with constant coefficients. Such solution
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of differential equation in co-state variable will be necessary in our subsequent
realistic example.
Particular Solution of a 3- system of ordinary differential equation
Let us consider the 3-system of ordinary differential equations
(1) % = a1.A1 + b1>.2 + C1A3 + d1
(! 1) ~ = a2.A1 + b2,\2 + C2A3 + d2
(IJI) 1if = a3>.1 + b3>.2 + c3.\3 + d3
Differentiating (!) with respect to t and using (JI) and (!JI), we get ,
(IV)
where X= d;t~1 - a1 % - (b1a2 + c1a3).A1 - (b1d2 + c1d3) ,
A= b1b2 + b3c1, B = b1c2 + c1c3.
Again differentiating (IV) with respect to t and using (JI), (I JI) , we get,
(V) Y= C>.2 + D>.3
where Y= 7J' >. - a1 7d, >. - ( b1a2 + c1a3 ) TJ>. i - ( a2 A + a3B ) >.1 - ( d2 A + d3B ) ,
C = b2A + b3B, D = c2A + c3B.
Solving (IV) and (V) we get,
DX-BY
>.2 f:t) = AD - BC
AY-CX
>.3 (t) = AD - BC
provided AD - BC f= O. Putting the values of >.2(t) and >.3(t) in (I) , we get,
s
>.1(t)=-R
h S = d1(c1A- b1B) _ (b1d2 + c1d3)(b1D - c1C) _ (d A d· B)
w ere AD - BC c1A - b1B 2 + 3
R _ a1(AD - BC) - (b1a2 + c1a3)(b1D - c1C) _ ( A B)
- C1 A - b1 B a2 + a3 .
Thus the values of .A2(t) and .\3(t) are given as follows
A2(t) = AD~BC [b1D(~ - d2) + c1D(~ - d3) + AB(d2 - ~) + B 2(d3 - ~)]
,\3(t) = AD~ 8c[A2 (~ - d2) + AB(~ - d3) + b1C(d2 - ~) + c1C(d3 - ~)].
Theorem 3.8. Let the dynamic model be given by (3.3) with restrictions (3.7) and
(3.8) and the profit function be given by (3.4) under restrictions (3.10) and (3.11).
The problem is to maximize
J = lT 7r(x, y, z, u)dt
where T is the total time. Then there exists u = u* satisfying (3.7) and (3.8) for
which J is maximum.
Proof. Hamiltonian for our model (3.3) is given by
H = (p1q1x + p2q2y + p3q3z - c)u(t) + .Aif1 + >-2h + ,\3f3 + μG
where ,\i(t) for i = 1, 2, 3 and μ(t) are co-state variables to be determined suitably.
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For steady state solution, we have
X
r ( 1 - K) - nz - q1 u = O
s(l - '}!_) - (Jz - q2u = O
L .
- f +/X+ Óy - Q3U = Ü
By applying Pontryagin's maximum principle we have (for a steady state solution)
d>.1 8H >-1r dt = - ax = -p1Q1U + Kx - A3/Z - 2μx
d>.2 8H >-2s
- = -- = -p2Q2U +-y- A3ÓZ - 2μy
dt 8y L
(3.16)
d>.3 8H dt = ---¡¡; = -p3q3u + nx>.1 + (Jy>.2 - 2μz
and
-8n + 8fi 8h 8f3 ac >-1- + >-2- + ;\3- +μ-=O
au au au au au
i.e.
(3.17)
Equation (3.16) can be rewritten as
d>.1 dt = a1>-1 + b1>-2 + c1>.3 + d1
(3.18)
d>.2
dt = a2>.1 + b2>.2 + c2>.3 + d2
d;\3
dt = a3A1 + b3>.2 + c3;\3 + d3
where a1 = (f- 29¿x)x, bi = - 291xy, c1 = -(r+ 29tx)z, d1 = 2~x -p1Q1U.
a2 = - 2q1xu,.b2 = (f - 2qbV)y, C2 = -(ó + 29tY)z, d2 = ~ - P2Q2U.
a3 = (n - 2qbz)x, b3 = ((3- 29bz)y, C3 = - 2qiz2
, d3 = 2~z -p3Q3U and
r = P1Q1X + P2Q2Y + p3q3z - c.
Using the particular solution of system of ordinary differential equations as given
in the previous section, we get the particular solution of (3.18) given by
1
>-1(t) = R(xu - V)
where p1qi(c1A-b1B) _ (b1P292+c1p3q3)(b1D-c1C) _ (Ap q + Bp q)
AD-BC c¡A- b1B 2 2 3 3
V= 2r [(c1A- b1B)x _ (b1y+c1z)(b1D-c1C) _(A + Bz)]
b AD-BC c 1 A - b1B y
>-2(t) =TU+ K
where T = BC~AD [( 'W- + P2E2)(b1D - AB) + ( 21'.ff- + p3E3)(c1D - B2)]
K = AD~ 8c[(~ - r)(b1D -AB) + (~ - r)(c1D - B2 )]
>.3(t) = r¡u+J
where r¡ = AD~ 8c[('W- + P2E2)(b1C ~ A2) + (21'.ff- +p3E3)(c1C - AB)]
J = AD~8c[(~ - r)(A2 - biC) + (~ - r)(AB - c1C)].
A, B, C, D, S, R have their usual meanings as discussed while giving the particular
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solutions of a 3- system of ordinary differential equations mentioned above. Finally
we take u---> U max as t---> T , in >..1, >..2 and A3, and have the limiting values given as
(3.19)
(3.20)
(3.21)
Using (3.17) the co-state vector μ(t) is given by
(3.22)
Now, if H is maximum at u = u* (say), O < u < Umax, then ~~ = O at u = u*.
Hence we have
(3.23) r - >..1(t)q1x - >..2(t)q2y - >..3(t)q3z + μ(t)b = O
where >..i (t) (i = 1, 2, 3) and μ(t) corresponds to u = u* . Hence from (3.19) to
(3.22) , we get
>..1(t) = ~(xu* -V)
>..2(t) = rn* + K
>..3(t) = r¡u* + J
1
μ(t) = ¡;[>..1(t)q1x + >..2(t)q2y + A3(t)q3z - r]
Again, as steady state optima! solution (x*, y*, z*) is desired, u* is given by
(3.24)
Thus finally we have
(3.25)
(3.26)
(3.27)
(3.28)
* X - X¡ y - Y1 z - Z ¡
u=--=--=--
p -Q - R
1 X - X ¡ >..1(t) = -R (x-p- - V)
where P , Q and R are given in proposition 3.5. Putting the values of x 1 , y1 and z1 ,
we obtain the equation of the optimal path. Solving (3.12) with the above optimal
path, we obtain the optimal values x*, y*, z* of x, y , z respectively and thus we find
the optimal value of u*. D
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4. STABILITY ANALYSIS AND THE CONTROL-THEORETIC OPTIMIZATION OF A
FUNCTIONAL OF REPLICATOR DYNAMICS
In this section we prefer to restrict our detailed analysis only to the following
replicator system of dynamics, which is completely new in all respects.
Definition 4.1. [16] Let
3
s3 = {x = (x1, x2, x3) E Rn: ¿ x., = c,x;;:::: O for 1 :::; i:::; 3}. It is called the con-
·i= l
centration simplex.
The dynamics on Sj is given by the differential equations
. 3 1>
(4.1 ) x; = x;[qi + L k;jXJ - -]
j=I C
3 3
where e> O, Qi and kij E R and </> = 'L: xi(q; + L kijXJ)· x; represents the
i= l j=l
concentration of the chemical or biological species i and q; E R corresponds to
the selfreproduction or decay of the species i and k;jXj represents the effect of the
species j on the reproduction of species i which is of mass action type, catalytic
if kiJ > O and inhibiting if k;J < O. ( 4.1) is called a replicator system on Sj; if it
keeps the boundaries and faces of s3 invariant.
Example 4.2. Let us consider an inhomogeneous hypercycle defined on 3- concentration
simplex s3 = { (x, y, z) E R3 : X+ y + z = e, x, y , z ::::: O} given by the system
of differential equation
(4.2)
x = x(q1 + k1Y - 1_)
e
Y = y( Q2 + k2Z - p_)
e
i = z( Q2 + k3x - p_)
e
where </> = k 1xy + k2yz + k3zx is called the dillution fiu.r. x, y , z repres ents the
concentrations of the chemical ar biological species. q1 , q2 , q3 denotes the self reproduction
ar decay of the species x, y and z respectively. k1 , k2 , k3 represents the
effect of the species x on y , y on z and z on x respectively.
Let the dynamics of the exploited system of (4.2) under control parameter u be
given by
(4.3)
x = x(q1 + k1y - p_) - E1UX
e
Y= y(q2 + k2Z - 1_) - E2UY
e
i = z (q2 + k3x - p_) - -E3UZ
e
where u is the effort of control per unit waste molecule.
E1 , E2, E3 : coefficients of degradation product (waste) from evolution reaction vessels
far the molecules x, y and z respectively. In this case, the state space of the
dynamical system is a 3-simplex, which is actually a manifold with boundary, the
manifold being the open submanifold in R3 given by x +y+ z < e and the boundary
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being the differentiable variety given by x + y + z = c. The exploited system ·under
the control parameter u is the 4 simplex x +y+ z +u - c ::::; O, u > O, which is also a
manifold with boundary¡ the interior is a 4- simplex which is an open submanifold
of R 4 given by x +y+ z +u - c < O and boundary is a 3-simplex x +y+ z +u = c.
Let P1,P2,p3 be the projected profit for degradation product (waste) of molecules
x, y , z respectively from evolution reactor coming out of the vessel to avoid risks of
breaking the walls of the reactor.
The total number of waste molecules x, y, z at time t taken by the control process
are given by E1 ux, E2uy, c3uz respectively.
Therefore, the net projected profit for degradation (waste) of molecules x, y and
z are respectively p1E1ux,p2E2uy and p3E3uz.
Let e be the cost per unit effort u at time t.
So total effort in the process is eu(t). Then the profit function is taken as
( 4.4)
where (x, y, z, u) belonging to the state space.
The control problem is as follows:
( 4.5) M aximize J = 1T 7r(x, y, z, u)dt V(x, y, z, u) E X [the control set]
Proposition 4.3. The replicator system (4.2) has equilibrium (x1 , y1 , z1) > (O, O, O)
if c is large enough. Similarly ( 4.3) possesses equilibrium if c is large enough as
well as ( 4.6) hold.
(4.6)
-€3 --€2- -E2 -- t+l l < Q
k3 ki
Proof. Equilibrium point of ( 4.2) can be obtained by solving the system of linear
equations given by
Q3 + k3X - <!!_ = Ü
c
where ef; = k1xy + k2yz + k3zx. It follows that
1 1 l
x = EL [c - !l.L!l1. + !1..c..'.t! ] y = .Ei. [c - !1..c..'.t! + <n..=!L!. J an d z = 5- [c - <n..=!L!. + !1.L!l2] 1 N k2 ki ' 1 N k3 k2 1 N k1 k3
where N = ~1 + ~2 + ¿. Thus for inhomogeneous hypercycle (where qi's are
unequal) the inner equilibrium (x1 , y1 , z1 ) > (O, O, O) if c is large enough.
For the exploited system (4.3), it follows that
X2 =Xi - ~u, Y2 = Yl - c;J·, Z2 = Z1 - 1{.,¡u
where
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Clearly (x2, y2, z2) >(O, O, O) if e is large enough as well as (4.6) hold. o
Proposition 4.4. The system (4.2) is globally asymptotically stable if (q1 +k1y)(xx1)
+ (q2 +k2z)(y-y1) +(q3 + k3x)(z-zi) <O V(x,y,z) > (0,0,0) andx i- x1,Y iY1,
z i- Z1.
Proof. To test the global stability analysis let us consider the following Lyapunov
function
V(x,y, z ) = x - X1 - x1log(~) + c1[Y - y¡ - y1log(.1L)] + c2 [z - Z1 - z1 log( zz )],
X1 Y1 1
where c1, c2 are positive constants to be determined suitably.
We have V(x, y, z) = ( x~xi ).i + c1 (Y~vi )y+ c2( z~zi )i. Using (4.2), we get
V = ( Q1 + k1y - ~ )(x - x1) + c1 ( Q2 + k2z - ~ )(y - yi) + c2 ( q3 + k3x - ~) (z - zi) =
(q1 + k1y)(x - x1) + c1 (q2 + k2z)(y-y1) + c2(q3 + k3x)(z - zi) - ~ [(x -xi)+ c1 (yYi)
+ c2(z - zi)].
Choosing ci = c2 = 1, we get
. ~ V= (q1 + k1y)(x -xi)+ (q2 + k2 z)(y - y1) + (q3 + k3x)(z - zi), since -c[(x - xi)+
c1(Y - Y1) + c2(z - z1)] = o for X+ y+ z = Zi + Yl + Z1 = c.
Thus by LaSalle's theorem it follows that (x1,y1,zi) for the system (4.2) is
globally asymptotically stable if (q1 + k 1y)(x - xi) + (q2 + k2z)(y - Yi) + (q3 +
k3x)(z - zi.) < O V(x,y,z) > (0,0, 0) and xi- xi, Y i- Yi,z i- zi. O
Theorem 4.5. Let the dynamic model be given by ( 4.3) under the restrictions ( 4.6)
ande be large enough. Let the objective function be given by (4.4), then there exists
a f easible bionomic equilibrium if ( 4. 7) and ( 4.8) holds where
(4.7)
(4.8)
Proof. The locus of dynamic equilibrium (x2, y2, z2) is given by
(4.9)
X - Xi y - Y1 z - Z¡
L:--=--=--=U
F -G - H
The zero profit function is given by
(4.10)
If (4.9) intersects (4.10) at (x*,y*, z* ) where u= u 1, then it follows that
x = x* + Fu1, y* = y1 - Gui, z* = zi - H u1
Again from 1T = O, it follows that
P1E1X1 + P2E2Y1 + p3E3Z1 - e= (P2E2Q + p3E3R - PiE1P)u1 .
Hence
(4.11)
Obviously, we get u1 > O if ( 4. 7) and ( 4.8) hold.
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Statement of the optimal control problem and its solution
Let the state space of the exploited system ( 4.3) be given by
X= {(x, y ,z,u): corresponding to each u, (x(t) ,y(t ),z(t)) is an integral curve of
the exploited replicator system (4.3) and G(x, y , z, u) :::; O} where G(x, y , z, u) =
x +y+ z +u - e then G(x, y , z, u) :::; O denotes a manifold with boundary in R 4
whose interior is open sub-manifold of R4 and whose boundary is a differentiable.
variety given by G(x, y , z, u) = O; u E R, t E [O, T ], (x(t) , y(t), z(t)) E R3 is C 1 ,
(x,y ,z) = (xo , y0 ,z0 ) when t =O; further let f: R3 x R _, R3, 7r: R3 x R _, R be
all C1-maps where f =(Ji, h , h)
ki XY + k2YZ + k3ZX
fi( x, y ,z,u)= x [q1+k1y- -E1u]
e
k1XY + k2YZ + k3ZX
h(x, y , z, u) = y[q2 + k2z - - E2u]
e
k1XY + k2yz + k3ZX
h(x, y, z, u) = z [q3 + k1x - - E3u]
e
We assume that the total time taken to control waste molecules is T. Then the
control problem is to maximize
( 4.12) J = 1T 7í(X, y , z, u)dt V(x, y , z, u) E X
over the control parameter u , where u E (O, Umax) and to find a suitable u = u* in
(O, Umax ) for which J is maximum where
7í(X, y , z, u) = (P1 E1X + P2E2Y + p3E3Z - c)u(t)
Theorem 4.6. Let the dynamic model be given by ( 4.3) with restrictions ( 4.6) and
the profit function be given by (4.4) under restrictions (4.7) and (4.8). The problem
is to maximize
J = 1T 7r(x, y, z, u)dt
where T is the total time. Then there exists u = u* satisfying ( 4.6) for which J is
maximum. Further the optimal path is given by
(E1 - Q1)x-k(x xp.x 1 - V)+ (E2 - Q2)y[r'1_::-J1 + K] + (E3 - q3)z[17 z_=-~1 + J] = O
where F , G and H are given in proposition (4.3) and (x 1 , y1 ,z1 ) is the nontrivial
equilibrium of the model ( 4.2) . Lastly, the optimal values of x, y and z are obtained
as the point of intersection of (4.9) with the above optimal path.
Proof. Hamiltonian for our model ( 4.3) is given by
H = (P1E1X + P2E2Y + p3E3Z - c)u(t) + )qfi + A2Í2 + A3f3 + μG
where Ai(t) for i = 1, 2, 3 and μ(t) are co-state variables to be determined suitably.
For steady state solution, we have
k k1XY + k2YZ + k3ZX Ü Q1+ '1Y- -E1U=
e
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By applying Pontryagin's maximum principle we have (for a steady state solution)
d)q aH .\1 dt = - ax = -(p1E1U - C(k1XY + k3zx) + μ]
( 4.13)
d.\2 aH .\2
- = - - = - (p2E2u - -(k1xy + k2yz) + μ]
dt ay e
d.\3 aH .\3 dt = --¡:¡; = -(p;3E3U - --;;(k2yz + k3zx) + μ]
and
a-rr aJi a12 ªh ac
-+.A1-+.A2-+.\3-+μ- =0
au au au au au
i.e.
(4. 14) P1E1x + P2E2Y + p3E3z - e - .A1E1x - .A2E2Y - A3E3z +μ=o
Equation ( 4.13) can be rewritten as
k1XY + k3ZX
(D- ).A1+P1E1u-μ=O
e
k1XY + k2yz
(D - ).\2 + P2E2U - μ = Ü
c
( 4.15)
k2YZ + k3ZX
(D - ).\3 + p3E3U - μ = Ü
e
where D = f,, . Again using (4.14) , the system (4 .15) becomes
d.\1 dt = a1.A1 + b1.A2 + c1 .\3 + d1
(4.16)
d.\2 dt = a2.A1 + b2.A2 + c2 .A3 + d2
d.\3 dt = a3A1 + b3A2 + C3 A3 + d3
where a1 = l = kixy~k3 zx + E1X, b1 = b3 = E2Y, C1 = C2 = E3Z, a2 = a3 = E1X,
b2 = rn = k¡x71~k2y z + E2Y, C3 = n = k21/ Z~ k3ZX + E:~z
L =-di= P1E1U + r, M = -d2 = P2 E2u + r , N = -d3 = p3E3U + r
r = P1E1X + P2E2Y + p3E3Z - c.
Solving the system of differential equation (5.4.16) we get the particular solutions
by using the particular solution of the system of ordinary differential equations as
given earlier given by
1
.A1(t) = R(xu - V)
where X = (Ap E + Bp E ) + b1p2<2+c1pa<3 _ P1<i(c1A-b1B)
2 2 3 3 c1 A-b1B AD-BC '
V= r[(c A - b B) - (b1+c1)(biD-c1C) - (A+ B)]
1 1 c1A-b1B
.A2 (t)=rn+K
where T = BC~AD [(:W- + P2E2)(b1D - AB) + (~ + p3E3)(c1D - B 2)]
K = AD~ 8c[(~ - r)(b1D -AB) + (~ - r)(c1D - B2 )]
A3(t) = 7]U + J
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where r¡ = AD~BC [(~ + P2E2)(b1C - A2) + ('7f + p3E3)(c1C - AB)]
.J = AD~ 80 [(~ - r)(A2 - b1C) + (~ - r)(AB - c1C)].
A, B, C, D, S, R have their usual meanings as deduced in previous sections.
Finally we take u---+ Umax as t---+ T, in )q, A2 and A3, we have the limiting values
as
(4.17)
( 4.18)
(4.19)
1
Ai(t) = R(XUmax - V)
A3(t) = r¡'Um~x + .J
Using (4.14) the co-state vector μ(t) is given by
(4.20) μ(t) = >-1(t)E1X + >-2(t)E2Y + A3(t)E3z - r
Now, if H is maximum at u = u* (say), O < u < Umax , then ~!/, = O at u = u*.
Hence we have
( 4.21)
where .>-;(t) (i = 1,2,3) and μ(t) corresponds to u= u*. Hence from (4.17) to
(4.20), we get
.A1(t) = ~(xu* - V)
.A2(t) = rn* + K
>.3(t) = r¡u* + .J
μ(t) = >-1(t)E1X + >-2(t)E2Y + A3(t)E3Z - r
Again, as steady state optimal solution (x*,y*,z*) is desired , u* is given by
( 4.22)
Thus finally we have
( 4.23)
( 4.24)
( 4.25)
( 4.26)
u* = x - X1 y - y1 z - z1·
F -G -H
1 X - Xi
>-1(t) = R(x-¡¡;- - T)
>-2(t) = Ty ~;1 + K
Z - Z ¡
A3(t) = r¡ -H + .J
μ(t) = .A1(t)E1X + .A2(t)E2Y + A3(t)E3Z - r
Under (4.23)- (4.26), (4.21) reduces to
(E1 - q1)x>-1(t) + (E2 - q2)Y>-2(t) + (E3 - q3)z>.3(t) =O.
This implies that
(Ei -qi)xj¡_(x 3'px1 -T) + (E2 - q2)Y[T 11_:::-61 + K] + (E3 - q3) z[r¡z_:::-~1 + .J] = 0
where F , G and H are given in proposition 4.3. Putting the values of x 1 , y1 and z1 ,
we obtain the equation of the optimal path. Solving ( 4.9) with the above optimal
path, we obtain the optimal values x*, y*, z* of x, y, z respectively and thus obtain
the optimal value of u*. D
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Discussion
Theory of constrained optimization of a functional on a subset of Rn was known
earlier. But realistic application of this theory and performance of corresponding
analysis to determine the optimal effort and optimal bio-masses was not attempted
earlier. This is the first instance where such problem of reality has been tackled
nicely.
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