Chapter 12: Differential Games, Distributed System, and Impulse Control More than one decision maker, each having separate objective functions which each.

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,

Impulse control,

Allow to make discrete changes in the state variables

at selected instants of time in an optimal fashion.

12.1 Differential Games

Different types of solutions such as minimax, Nash,

Pareto-optimal, along with possibilities of cooperation

and bargaining.

12.1.1 Two Person Zero-Sum Differential Games

which player 1 wants to maximize and player 2 wants

to minimize.

is the minimax solution.

The necessary conditions for u* and v* ,

is a saddle point of the Hamiltonian function H.

when U=V=E1,

12.1.2 Nonzero-Sum Differential Games

N players, represent the control

variable for the ith player,

denote the objective function which the ith player

wants to maximize.

A Nash solution:

Open-Loop Nash Solution

for all

Closed-Loop Nash Solution

we must recognize the dependence of the other

player’s actions on the state variable x. Therefore,

Interpretation to the adjoint variable i . Any

perturbation x in the state vector causes them to

revise their controls by the amount

12.1.3 An Application to the Common-Property

Fishery Resource

Let denote the turn-pike (or optimal biomass) level

given by (10.12).

As shown in Exercise 10.2,

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

positive profits.

As far as producer 1 is concerned, he wants to attain

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

from fishing.

The Nash solution;

The direct verification involves defining a modified

growth function:

And using the Green’s theorem results of Section

10.1.2. Since by assumption, we have

for From (10.12) with g replaced by g1, it can

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 producer1 originally has sole possession

of the fishery, but anticipates a rival entry. Producer1

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.

Model for licensing of fishermen let the control variable

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),

Deal, Sethi, and Thompson(1979), Deal(1979),

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

Erickson(1991).

For applications of differential games to economics

and management science in general, see the book by

Dockner, Jrgensen , Long, Sorger(2000).

12.2 Distributed Parameter SystemsSystems in which the state and control variables are defined in terms of space as well as time dimensionsare called distributed parameter systems and are described by a set of partial differential or difference equations.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, andDerzko(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.

Let y denote a one dimensional spatial vector, let t

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 we let 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

D as

x0(y) gives the starting distribution of x with respect to

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

t.

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.

Figure 12.1: Region D with Boundaries and

12.2.1 The Distributed Parameter Maximum Principle

where xt=x/t and xy= x/y.The boundary conditions

on are stated for the part of the boundary of D.

which gives the consistency requirement in the sense

that the price and the salvage value of a unit x(T,h)

must agree.

We let u*(t,y) denote the optimal control function. Then

the discounted parameter maximum principle requires

that

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.

12.2.2 The Cattle Ranching ProblemLet 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

The boundary and consistency conditions for x are

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.

The deviation penalty cost is given by

where q is a constant.

subject to the boundary and consistency conditions

(12.38)-(12.40).

Figure 12.2: A Partition of Region D

where g is an arbitrary one-variable function and k is a

constant. We will use the boundary conditions to

determine g and k.

We substitute (12.38) into (12.48) and get

This gives

In the region D1 D2.

For region D3, we use the condition (12.39) on (T,y)

in (12.48) to obtain

(12.50) in the region D1 as the beginning game, which

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.

12.2.3 Interpretation of the Adjoint Function

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

level.

12.3 Impulse Control

Example of an oil producer who pumps oil from a

single well.

where 1 is the starting stock of a new oil well.

Figure 12.3: Solution of Equation 12.52

12.3.1 The Oil Driller’s Problem

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.

12.3.2 The Maximum Principle for Impulse Optimal ControlAn impulse control variable v , and two associatedfunctions. The first function is G(x,v,t), which representsthe 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.

Impluse Hamiltonian function

When t1 = 0 then the equality sign in (vii) should be

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.

12.3.3 Solution of the Oil Driller’s Problem

Optimal impulse control at t1 is

After the drilling, which is given by

(12.70).

shown in Figure 12.4, which represent the condition

prior to drilling. Figure 12.4 is

drawn under the assumption that

Figure 12.4: Boundary of No-Drilling Region

From (12.70),

The curve is drawn in Figure 12.5.

BC of the curve lies in the no drilling region, which is

above the curve 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.

Figure 12.5: Drilling Time

Figure 12.6: Value of

This intersection point Y determines the drilling time t1.

12.3.4: Machine Maintenance and ReplacementT = 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

the machine,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,

C = the cost of new machine per unit quality; C > K,

t1 = the replacement time; 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 .

We have assumed that a fraction of a machine with

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 U b/g.

12.3.5 Application of the Impulse Maximum Principle

The solution of (12.84) for t1 < t T is

The switching point is given by solving –1+g=0.Thus,

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] .

Figure 12.7: Optimal Maintenance Policy

and compute the time

which makes

In plotting Figure 12.8, we have assumed that 0<(0)

<1. This is certainly the case, if T is not too large so

that

As in the oil driller’s problem, we obtain (t) by using

(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.

Figure 12.8: Replacement Time t1 and Maintenance Policy

AB represents the replacement curve. The optimal

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 12.8.

Figure 12.9 has been drawn for a choice of the

problem parameters such that t1= t2 .

Figure 12.9: The Case t1= t2

Using (t1)=K obtained above and the adjoint equation(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 >0.