Global A differential Game Related to Terrorism : Stackelberg Differential Game of E-differentiable and E-convex function

In this work, the Stackelberg differential game of E-differentiable and E-convex function is studied in order to fight the terrorism taking into account the government’s procedures such as education quality, better job opportunity, social justice, religious awareness and security arrangements. We consider Stackelberg differential game. Firstly, the government is the leader and the terrorist organization is the follower. Secondly, the terrorist organization is the leader and the government is a follower. Furthermore, we apply the necessary conditions of the Stackelberg differential game for these cases to obtain the optimal strategy of this problem. 2010 Mathematics Subject Classifications: 49N70, 49N90, 91A23


Introduction
The government's tasks are very important for counter-terror, the government must be applied some subject of mathematics to get methods for combating terrorism, particularly Operations Research.
Counter-terror measures range comes from security arrangements and the government's procedures to freezing assets of a terrorist organization or even to invade their territories for assassinating the terrorists.Since any action against terrorists must be taken into account their reactions.In this paper, the Stackelberg approach is used for studying the interaction strategies of the government and the terrorist organization.Therefore, we use the concept of E-differentiable and E-convex functions to transform a non-differentiable and non-convex function to E-differentiable and E-convex function [1,2].In [3] it is imposed that the success of combating terrorism depends on public opinion, while in [4] the efficiency of "water" and "fire" strategies are compared.Hsia [5,6] introduced the fuzzy differential game to guard a territory movable and not movable, a Nash-collative differential game is presented in [7].A min-max fuzzy differential game with fuzzy on the objective and control, and the large-scale differential game are discussed in [8][9][10][11].The terrorism infighting by Stackelberg and Nash strategies are introduced in [12], the interactions strategics between R&D, defense and preemption is presented in [13], a minmax differential game approach is studied to fight the terrorism [14,15].The policies of anti-terrorism and economy of terrorism are introduced in [16,17].The study of terror support and recruitment (defense and peace economic) is presented in [18].Megahed [19] discussed that governments must be made some procedures for fighting the terrorism such as unemployment, justice social, religious awareness, improving the education with considering the security measurements The organization's power is measured by the terrorist's activities, the organization's resources such as weapons, financial capital, and technological expertise.The power of terror organizations changes with the time, the recruitment of terrorists is through existing terrorists.The decreasing rate of terrorists is affected by their own action and the antiterrorist actions of the government through of the education quality, increase the chances of labor, social justice, religious awareness and security arrangements.The objectives of the government drive his utility from the loss of the terrorist resources and their activities but incur costs for combating terror and dis-utility from the terror organizations, the later tries to maximize its power both by its size and its terrorist attacks.
In this work, we study how to help the governments to counterterrorism, the Stackelberg differential game plays the main role to combat the terrorism.

Problem Formulation
The differential game with the state variable z(t) which describe the resource of the terror organization (TO).It may also include weapons, financial capital, the network of supporters, etc. and another state variable M (t) which describe the government's activities, the education quality, increasing the chances of labor, social justice, religious awareness and security arrangements, t ∈ [0, ∞) is the time.The two players are the government and (TO) with non-negative strategy v 1 (t), v 2 (t) respectively.The stock of resources of (TO) grows according to the growth of a linear function g(z) i.e., g(z) = rz, r > 0, and the government's procedures grow according to the function A(M ) = µM , where µ > 0 is the growth rate of the government's procedures.Carrying out attacks make a reduction the growth of the resource stock as it affects negatively the number of terrorists (e.g.suicide bombing or caught and killed terrorists).Furthermore, weapons and financial means, it may even include a reduction of the network of supporters.However, the growth reduction of resources stocks does not only depend on the strength of attacks v 2 (t) but it is also influenced by fighting the terrorists v 1 (t).This effectiveness of the control variables of the two players on the growth of the resource may be denoted as "harvest function h(v 1 , v 2 ).As a consequence, the dynamic equations of the resources stocks and government's tasks z(t), M (t) respectively can be written as )is non-differentiable and not convex, z 0 denotes the initial stock of terrorist's resources, M 0 is the initial government's procedures and a, b are positive constants.Moreover, we assume that z(t) ≥ 0, M (t) ≥ 0 for all t ≥ 0 (1) Now, we define an operator E : R n → R n such that (h Since the increase rate of counter-terror measure and attacks leads to a reduction of growth, we assume that the partial derivatives are greater than zero: The counter-terror measures exhibit marginally decreasing efficiency Moreover the increasing rate of attacks induces disproportional higher losses of resources i.e. (h Finally, the instruments reinforce each other, i.e. (h • E) v 1 v 2 > 0, which makes economical sense.This positive interaction means that the minor efficiency of fighting terrorism increases with the strength of terrorist's attacks.In addition to, the Inada conditions of the economy are assumed to be fulfilled lim This gives that the optimal strategies are nonnegative v 1 (t) ≥ 0, and v 2 (t) ≥ 0, t > 0 Player 1 (the government) derives its benefit from the loss of the terrorist resources and their activitiesM (t) besides showing the dis-utility of these terrorist organization but incur costs for combating terror.For simplicity, all these terms must be linear.Thus, the objective of the government is max where ω, c, k, q and α are positive constants.
The second player (TO) derives its benefit from the resource stock z(t) and the terrorist actions at intensity v 2 (t).Then the objective of (To) problem is max where σ, β and γ are positive constants.
The decreasing rates η i , i = 1, 2 are assumed to be greater than the growth and activity rates, r, µ In this paper, we will derive the Stackelberg equilibria.The solution procedures rely on Pontryagin's maximum [11] 3

. The stackelberg differential game
The Stackelberg approach is a hierarchical solution concept, since one of the players, the leader, has a stronger position in the decision process.This solution is the result of sequential decisions: the leader announces his strategy firstly so that the other players, the followers, can only react to the leader strategy.In this paper, we consider the government plays the leader's role and the terrorist's organizations (TO) play the follower's role, and another case is vice versa.

Stackelberg game with the government is the leader and (TO) is the follower
The government chooses the rate of counter-terror measures before (TO) decides on the rate of attacks.Consider the following problem, where the government and the terrorist organizations are is the leader and the follower respectively

The follower Problem
Firstly, we find the optimization of the follower (To) which consider the action of a leader is given The Hamiltonian function of the follower (To), H 2 , is defined by 2 , where τ, δ are positive integers Consider the following operator (10) and the harvest function The adjoint variable λ 1 satisfy the following differential equation Remark 1.Since m > 1 (10) and (11) implies to any increase of combating measures leads to a more cautious behavior of the terrorists as well as a lower harvest of the terrorists' resources.

The leader Problem (The government)
To obtain the optimal strategy v 1 for the government, the government must be taken into account the optimal strategy of the follower which is represented by an additional costate equation.
The Hamiltonian function of the leader from the necessary conditions, The adjoint variables satisfy the following differential equations Remark 2. The adjoint variable ψ 3 of the leader with respect to the adjoint variable λ 1 of the follower has no influence on the optimization of the leader Proposition 1.A feasible solution of the game with the government is the leader and the (TO) is the follower is exists if and only if Proof. .The optimal solution of the leader and follower is feasible if and only if The optimal strategies are given by (18) and the harvest function with the operator E is

Proposition 2. The values of the steady state for the inventory resources and the governments's procedures are given by
where c 1 is the constant, for t → ∞, then c 1 = 0, and For t → ∞ then c 0 = 0 and

Proposition 3. (i) The government as the leader is more active but the (TO) as the follower is more cautiously if and only if
(ii) The government as the leader is more cautiously and the (TO) as the follower is more aggressively if and only if Proof.
(i) The government as the leader is more active if and only if M (t) > 0, from Proposition 2 we have The proof of 2 is similar to 1 with less than sign instead of greater than sign Lemma 1.The objectives of the government and (TO) are given by Proof. .The solution of the differential equation ,and thus similarly the solution of the differential equation ż from ( 20) and ( 21) in the objectives of the leader and follower,and by integration we have where v 1 , v 2 and h(v 1 .v 2 ) are defined in ( 17), (18), and ( Remark 3. As shown in ( 20), ( 21) and according to the condition (1), we must be assume that, M 0 > av 1 −bv 2 µ , and The Stackelberg with the (TO) is the leader and the government is the follower In this case, the (TO) attacks the government before the government makes counterterror measures.Consider the following problem, where the(TO) is the leader and government is the follower.

The follower problem (The government problem)
Firstly, we consider the optimization of the follower ( government) which consider action of leader is given The Hamiltonian function of the follower H 1 is defined by 2 , where τ, δ are positive integers Consider the following operator and the harvest function Remark 4. : The follower (government) reacts with a higher strength of counter-terror measures in case (TO) intensifies its rate attacks, as shown in ( 23 1 Fig. 3 (The government is more aggressively with any incrossing of (TO) attacks).
The adjoint variables λ 1 , λ 2 satisfy the following differential equations The leader problem (TO) To obtain on the optimal strategy v 2 for the (TO), it must be taken into account the optimal strategy of the follower which is represented by an additional costate equation The Hamiltonian function of the leader From the necessary conditions and thus The adjoint variables satisfy the differential equations Proposition 4. The optimal strategies, the harvest function and the steady state values of state variables z(t) and M (t) with the (TO) is the leader and the government is follower are given by Proof. from (23), (24), and (27) we get on v 1 , v 2 , and (h Similarly the solution of the differential equation ż 23),( 24) and (27), we have The objectives functional of the leader (TO), J 2 , and the follower, J 1 , are given by Proof. .From (20) and (21) in the follower's objective, J 1 , and the leader's objective, J 2 , then  (ii) The proof of 2 is similar to 1 with greater than sign instead of less than sign ) which implies to a higher harvest (i) If m 1−n = 1 see Fig.1(An increasing of counter-terror measures and the terrorists is more cautious), (ii) If m 1−n < 1 see Fig.2 (The (TO) is more aggressively when the government increases their counter-terror measures), (iii) If m

α
+ aλ 2 n(ω − λ 1 ) av 1 − bv 2 ) Proposition 5. (i) The government as the follower is more cautiously and the (TO) as leader is more aggressively if and only if The government as the follower is more cautiously and the (TO) as leader is more aggressively if M (t) < 0, Since M (t) = av 1 −bv 2 µ then, av 1 < bv 2 .From the values v 1 and v 2 of Proposition 4, we find that