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Chapter 12: Differential Games, Distributed System, and Impulse Control More than one decision maker, each having separate objective functions which each is trying to maximize, subject to a set of differential equations. The theory of differential games, Distributed parameter systems,
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More than one decision maker, each having separate
objective functions which each is trying to maximize,
subject to a set of differential equations.
The theory of differential games,
Distributed parameter systems,
Allow to make discrete changes in the state variables
at selected instants of time in an optimal fashion.
Different types of solutions such as minimax, Nash,
Pareto-optimal, along with possibilities of cooperation
12.1.1 Two Person Zero-Sum Differential Games
which player 1 wants to maximize and player 2 wants
The necessary conditions for u* and v* ,
is a saddle point of the Hamiltonian function H.
N players, represent the control
variable for the ith player,
denote the objective function which the ith player
wants to maximize.
A Nash solution:
Closed-Loop Nash Solution
we must recognize the dependence of the other
player’s actions on the state variable x. Therefore,
perturbation x in the state vector causes them to
revise their controls by the amount
12.1.3An Application to the Common-Property
Let denote the turnpike (or optimal biomass) level
given by (10.12).
We also assume that
which means that producer 1 is more efficient than
producer 2, i.e., producer 1 can make a positive profit
at any level in the interval , while producer 2
loses money in the same interval, except at where
he breaks even. For both producers make
his turnpike level , if If and if
then from (12.26) producer 2 will fish at his maximum
rate until the fish stock is driven to At this level it is
optimal for producer 1 to fish at a rate which maintains
the fish stock at level in order to keep producer 2
The direct verification involves defining a modified
And using the Green’s theorem results of Section
10.1.2. Since by assumption, we have
be shown that the new turnpike level for producer 1 is
which defines the optimal policy (12.27)-
(12.28) for producer 1. The optimality of (12.26) for
producer 2 follows easily.
Suppose that producer 1 originally has sole possession of the fishery, but anticipates a rival entry. Producer 1 will switch from his own optimal sustained yield to a more intensive exploitation policy prior to the anticipated entry.
A Nash competitive solution involving N 2 producers
results in the long-run dissipation of economic rents.
vi denote the capital stock of the ith producer and let
the concave function f(vi), with f(0)=0, denote the
fishing mortality function, for i=1,2,…,N. This requires
the replacement of in the previous model by f(vi).
Application of differential games to fishery
management, , Haurie, and Kaitala
(1984,1985) and , Ruusunen,and Kaitala
(1986,1990). Applications to problems in
environmental management, Carraro and Filar (1995).
Applications of marketing in general and optimal
advertising, Bensoussan, Bultez, and Naert(1978),
Jrgensen(1982a), Rao(1984,1990), Dockner and
Jrgensen(1986,1992), Chintagunta and
Vilcassim(1992), Chingtagunta and Jain(1994,1995),
and Fruchter(1999). A survey of the literature is done
by Jrgensen(1982a) and a monograph is written by
For applications of differential games to economics
and management science in general, see the book by
Dockner, Jrgensen , Long, Sorger(2000).
Systems in which the state and control variables are
defined in terms of space as well as time dimensions
are called distributed parameter systems and are
described by a set of partial differential or difference
In the analogous distributed parameter advertising
model we must obtain the optimal advertising
expenditure at every geographic location of interest at
each instant of time, see Seidman, Sethi, and
Derzko(1987). In section 12.2.2 we will discuss a
cattle-ranching model of Derzko, Sethi and
Thompson(1980), in which the spatial dimension
measures the age of a cow.
denote time, and let x(t,y) be a one dimensional state
variable, Let u(t,y) denote a control variable, and let
the state equation be
For t[0,T ] and y [0,h ]. We denote the region [0,T ]x
[0,h] by D, and welet its boundary D be split into two
parts and as shown in Figure 12.1. The initial
conditions will be stated on the part of the boundary
the spatial coordinate y. The function v(t) in (12.33) is
an exogenous breeding function at time t of x when
y=0. In the cattle ranching example in Section 12.2.2,
v(t) measures the number of newly born calves at time
Let F(t,y,x,u) denote the profit rate when x(t,y)=x,
u(t,y)=u at a point (t,y) in D. Let Q(t) be the value of
one unit of x(t,h) at time t and let S(y) be the value of
one unit of x(T,y) at time T.
where xt=x/t andxy= x/y.The boundary conditions
on are stated for the part of the boundary of D.
whichgives the consistency requirement in the sense
that the price and the salvage value of a unit x(T,h)
the discounted parameter maximum principle requires
For all (t,y) D and all u .
These general forms allow for the function F in (12.2)
to contain arguments such as x/ y, 2x/ y2,etc. It is
also possible to consider controls on the boundary. In
this case v(t) in (12.33) will become a control variable.
Let t denote time and y denote the age of an animal.
Let x(t,y) denote the number of cattle of age y on the
ranch at time t. Let h be the age at maturity at which
the cattle are slaughtered. Thus, the set [0,h] is the set
of all possible ages of the cattle. Let u(t,y) be the rate
at which y-aged cattle are bought at time t, where we
agree that a negative value of u denotes a sale.
Subtracting x(t,y) from both sides of (12.42), dividing
by t, and taking the limit as t 0,yields
given in (12.32)-(12.34). Here x0(y) denotes the initial
distribution of cattle at various ages, and v(t) is an
exogenously specified breeding rate.
Let P(t,y) be the purchase or sale price of a y-aged
animal at time t. Let P(t,h)=Q(t) be the slaughter value
at time t and let P(T,y)=S(y) be the salvage value of a
y-aged animal at the horizon time T. The functions Q
and S represent the proceeds of the cattle ranching
business. Let C(y) be the feeding and corralling costs
for a y-aged animal per unit of time. Let denote
the goal level purchase rate of y-aged cattle at time t.
where q is a constant.
subject to the boundary and consistency conditions
constant. We will use the boundary conditions to
determine g and k.
We substitute (12.38) into (12.48) and get
In the region D1 D2.
in (12.48) to obtain
is completely characterized by the initial distribution x0 .
Also the solution (12.49) in region D3 is the ending
game, because in this region the animals do not
mature, but must be sold at whatever their age is at
the terminal time T.
An animal at age y at time t, where (t,y) is in D1D2,
will mature at time t-y+h. Its slaughter value at that
time is Q(t-y+h). However, the total feeding and
corralling cost in keeping the animal from its age y until
it matures is given by Thus, (t,y) represents
the net benefit obtained from having an animal at age
y at time t.
Interpret the optimal control u* in (12.47). Whenever
(t,y) > P(t,y) , we buy more than the goal level
and when (t,y) < P(t,y), we buy less than the goal
Example of an oil producer who pumps oil from a
where 1 is the starting stock of a new oil well.
If t = ti , then , which means that we have
abandoned the old well and drilled a new well with
stock equal to v(ti).
where P is the unit price of oil and Q is the drilling cost
of drilling a well having an initial stock of 1.
An impulse control variable v , and two associated
functions. The first function is G(x,v,t), which represents
the cost of profit associated with the impulse control.
The sencond function is g(x,v,t), which represents the
instantaneous finite change in the state variable when
the impulse control is applied.
replaced by a sign when i = 1.
Note that condition (vii) involves the partial derivative
of HI with respect to t. Thus, in autonomous problems,
where condition (vii) means that the
Hamiltonian H is continuous at those times where an
impulse control is applied.
After the drilling, which is given by
shown in Figure 12.4, which represent the condition
prior to drilling. Figure 12.4 is
drawn under the assumption that
The curve is drawn in Figure 12.5.
BC of the curve lies in the no drilling region, which is
above thecurve as indicated in Figure 12.5. The part
AB of the curve is shown darkened and represents
the drilling curve for the problem. The optimal state
trajectory starts from x(0)=1 and decays exponentially
at rate b until it hits the drilling curve AB at point Y.
T = the given terminal or horizon time,
x(t) = the quality of the machine at time t, 0 x 1; a
higher value of x denotes a better quality,
u(t) = the ordinary control variable denoting the rate of
maintenance at time t ; 0 u G < b/g,
b = the constant rate at which quality deteriorates in
the absence of any maintenance,
g = the maintenance effectiveness coefficient,
= the production rate per unit time per unit quality of
K = the trade-in value per unit quality, i.e., the old machine provides only a credit against the price of the new machine and it has no terminal salvage value,
t1 = the replacementtime; for simplicity we assume at
most one replacement to be optimal in the given
horizon time; see Section 12.3.3,
= the replacement variable, 0 1; represents a
fraction of the old machine replaced by the same
fraction of a new machine. This interpretation will
make sense because we will show that v is either 0
or 1 .
quality x has a quality x . Furthermore, we note that
the solution of the state equation will always satisfy
0 x 1, because of the assumption that u Ub/g.
The switching point is given by solving –1+g=0.
provided the right-hand is in the interval (t1,T];
otherwise set We can graph the optimal
maintenance control in the interval (t1,T] as in Figure
12.7. Note that this is the optimal maintenance on the
new machine. To find the optimal maintenance on the
old machine, we need to obtain the value of (t) in the
interval (t1,T] .
In plotting Figure 12.8, we have assumed that 0<(0)
<1. This is certainly the case, if T is not too large so
(12.88). From (12.92), we have v*(t1)=1 and, therefore,
(t1) =K from (12.85) and from (12.83). Since
gK 1 from (12.96), we have
and, thus, u*(t1)=0 from (12.90). That is, zero
maintenance is optimal on the old machine just before
it is replaced. Since from (12.97), we have
from Figure 12.7. That is, full maintenance is
optimal on the new machine at the beginning.
trajectory x*(t) is shown by CDEFG under the
assumption that t1 > 0 and t1< t2, where t2 is the
intersection point of curves (t) and (t), as shown in
Figure 12.9 has been drawn for a choice of the
problem parameters such that t1= t2.
(12.84), we have
Using (12.100) in (12.90), we can get u*(t),t[0,t1].
and the switching point by solving –1+g=0.
If 0, then the policy of no maintenance is optimal
in the interval [0,t1]. If > 0, the optimal maintenance
policy for the old maintenance is
In plotting Figure 12.7 and 12.8,we have assumed