Keywords: uncertain disturbances, pursuer, evader, differential games
American Journal of Applied Mathematics and Statistics, 2013 1 (2),
pp 2126.
DOI: 10.12691/ajams121
Received January 18, 2013; Revised March 02, 2013; Accepted April 15, 2013
Copyright: © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction and System Description
Let and be Euclidean spaces and denote the Euclidean norm. We consider the uncertain dynamical system modelled by:
 (1) 
 (2) 
 (3) 
 (4) 
The subscripts and in equations (1)(4) stand for pursuer and evader respectively. Also, and . The vectors and are uncertain vectors.
2. The Problem
The pursuer uses control to attempt to capture the evader, while the evader uses control to avoid being captured. The problem of interest is the minimization and the maximization of the final miss by pursuer and evader respectively, in the presence of uncertainties and.
3. The Game
The final miss is defined as a weighted quadratic form:
 (5) 
To make the game meaningful, we shall impose the following limitations:
 (6) 
 (7) 
Where are positive definite matrices such that
T is the final time. Joining (6) and (7) to (5) , are the following payoff functional defined as:
 (8) 
the controller is the minimizer of and the controller is the maximizer. In equations (1) and (2), the uncertainty is acting against the wish of the controller and also acting against the wish of the maximizer in equations (3) and (4).
Subsequently, we make the following assumptions:
Assumption A1: Assuming that the uncertainties
where and are Lebesgue measurable and ranged within a compact set, then there exist constants such that
Assumption A2: There exist nonsingular matrices and of appropriate dimensions such that:
 (9) 
 (10) 
Also given any and there exist a unique solution such that the following matrix Lyapunov equations are satisfied.
 (11) 
4. Problem Formulation
Defining a new state variable
 (12) 
then, from equations (1) – (4) we have
 (13) 
where
 (14) 
We shall write equation (13) in a compact form as:
 (15) 
where
 (16) 
We also impose the condition that
On the basis of (15), the following problems arise:
Problem:1
Subject to
This problem would be solved under the assumption that the payoff functional defined by can be separated to be of the form:
 (17) 
Based on the aforementioned assumption and noting that
 (18) 
we arrived at the formulation of the following two optimal control problems define by Problems (2) and (3)
Problem (2):
Subject to
 (19) 
Problem (3):
Subject to
 (20) 
5. Solution
Necessary Condition for a Saddle Point
For problems (2) and (3) we introduce the following assumptions:
i. The matrix functions are constant matrices of appropriate dimensions.
ii. Control functions , and disturbances , in problems (2) and (3) are generated by strategies.
Now consider problem (2) and define the Hamiltonian for the problem as
 (21) 
The adjoint equation satisfies
 (22) 
Now,
Therefore
 (23) 
6. Deduction
From (23) we deduce the following three cases, namely:
then any admissible is an optimal solution
We knock off case (b) since cannot be negative as an adjoint vector.
Case (c) is a trivial solution. Assume that the solution is not trivial, we consider case (a).
Let be continuous and let be define as
 (24) 
where and are square matrices and is a solution to (19) . Differentiating (14) to get:
 (25) 
Hence,
 (26) 
Therefore on the basis of (12) we have
 (27) 
Substituting (14) in (17) we get
 (28) 
For (18) to hold, it is necessary and sufficient that
 (29) 
Let and be strategies for and respectively, we then summarize the saddle point solution as follows:
 (30) 
For problem (3) the Hamiltonian is defined as
 (31) 
Following the same procedure as explained in problem (2), the adjoint vector satisfies
 (32) 
Now,
 (33) 
Since is a maximizer, we choose such that
 (34) 
Therefore
 (35) 
The following three cases can be deduced from (25)
 (36) 
then any admissible is an optimal solution.
Since is a minimizer and from (26) we shall knock off case (e) and we shall also assume a nontrivial solution, hence we shall let be continuous and be defined as
 (37) 
Where and are square matrices and is a solution of (10). We observed that
 (38) 
From (22)
 (39) 
Therefore
 (40) 
becomes
 (41) 
On substituting (37) into (41) we have the following equation
 (42) 
For (42) to hold
 (43) 
Let and be strategies for control and disturbance respectively, then, the saddle point solution is summarized as follows:
 (44) 
7. Value of the Game
We now employ the results in (20) and (34) to compute the value of the objective functional defined in problems (2) and (3)
We recall that
 (45) 
 (46) 
 (47) 
Multiply (45) by and equation (46) by to get
 (48) 
 (49) 
adding equations (48) and (49) together to get
 (50) 
Integrating (50) we get the following:
We know from (47) that , therefore,
Hence,
and from
therefore
 (51) 
Similarly, we consider
 (52) 
 (53) 
 (54) 
From (52) and (53) we have
 (55) 
 (56) 
Combining (55) and (56) we have
 (57) 
Therefore
 (58) 
Now, given that , then
 (59) 
From (37) substitute for to get
 (60) 
Therefore
 (61) 
Combining (51) and (61) ,the value of the game is given as
 (62) 
8. Sufficient Condition for a Saddle Point
We shall employ the sufficiency theorem given by [Gutman, S. (1975)] to show that the solution obtained for each of the cases is indeed a saddle point.
Let be defined as
 (63) 
We assume that is continuous on and it is a function.
Define
 (64) 
By virtue of (23) and (24) we have
 (65) 
 (66) 
using (29), (66) is reduced to
 (67) 
In (66) we substitute for using (23) and (24) to get
 (68) 
Now
 (69) 
and
 (70) 
On the basis of (69) and (70), (30) is indeed a saddle point.
9. Conclusion
The idea of saddle point (min, max) controllers arises in engineering problems where extreme conditions are to be overcome (Gutman,1975) A natural example is the “boosted period” of missiles when high thrust acts on the body so that every small deviation from the designed specifications cause unpredictable (input) disturbances in three nodes.
In this work we have considered situations where the disturbances affect the motions of the pursuer and the evader respectively. In our subsequent paper we hope to apply the results in this work to problems arising from pricing of general insurance policies, particularly in a competitive and noncooperative market.
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