Elsevier

Applied Mathematics and Computation

Volume 347, 15 April 2019, Pages 477-491
Applied Mathematics and Computation

Optimal control of counter-terrorism tactics

https://doi.org/10.1016/j.amc.2018.11.022Get rights and content

Abstract

This paper presents an optimal control problem to analyze the efficacy of counter-terrorism tactics. We present an algorithm that efficiently combines the Minimum Principle of Pontryagin, the shooting method and the cyclic descent of coordinates. We also present a result that allows us to know a priori the steady state solutions. Using this technique we are able to choose parameters that reach a specific solution, of which there are two. Numerical examples are presented to illustrate the possibilities of the method. Finally, we study the sufficient conditions for optimality and suggest an improvement on the functional which also guarantees local optimality.

Introduction

Modeling a “stock” of terrorists, is not common, but has precedents, especially after September 11, 2001 [1]. In this sense [2] presents an intelligent ecological metaphor to analyze actions by Governments and citizens against terror. In [3] a model for the transmission dynamics of extreme ideologies in vulnerable populations is presented. In [4] the authors propose a terror-stock model that treats the suicide bombing attacks in Israel. In other countries like, for example, Spain or Ireland, the problem has also been analyzed.

Several papers develop dynamical models of terrorism. In [5] the authors incorporate the effects of both military/police and nonviolent/persuasive intervention to reduce the terrorist population. This idea is widely developed in [6] where the controls are two types of counter-terror tactics: “water” and “fire”, which is the model we shall consider in this paper. The effect of intelligence (water tactics) in counter-terrorism is analyzed also in [7]. Nowadays, it is agreed that counter-terrorism policies have the potential to generate positive support for terrorism [8]. Recently, in [9] a model with two-states (undetected and detected terrorists) and only one control variable (the number of undercover intelligence agents) is considered.

In this context we present in this work a new approach to analyze the efficacy of counter-terrorism tactics. We state an optimal control problem that attempts to minimize the total cost of terrorism. An excellent summary of optimal control application in these issues can be consulted in [10] and its economic implications in [11].

The optimization criterion is to minimize the discounted damages created by terror attacks plus the costs of counter-terror efforts. The underlying mathematical problem is complicated. It constitutes a multi-control, constrained problem where the optimization interval is infinite. An important feature is that the time t is not explicitly present in the problem (hence, it is a time-autonomous problem), except in the discount factor. Using Pontryagin’s Minimum Principle, the shooting method and the cyclic descent of coordinates we develop an optimization algorithm. We also present a method (based upon [12]) for computing the optimal steady-states in multi-control, infinite-horizon, autonomous models. This method does not require the solution of the dynamic optimization problem. Using it, we can choose parameters that reach a desirable steady-state solution. The problem presents two steady-states, albeit one of them in a region where it becomes effectively one-dimensional. We focus mainly on the multi-control problem.

The paper is organized as follows. Section 2 presents the mathematical model. The optimization algorithm is developed in Section 3, and the method for computing the optimal steady-states is analyzed in Section 4. Section 5 presents several numerical examples which illustrate the performance of the algorithm under different conditions. In Section 6 we discuss Arrow’s sufficient conditions for optimality in our problem. We also suggest an improvement on the functional in which the cost function is convex in the number of terrorists (due to the value added by information sharing, interactions, etc.) and show how the solution found by our method in this case satisfies the sufficient conditions locally. Finally, the main conclusions of our work are discussed in Section 7.

Section snippets

Mathematical model

We use the excellent model provided by [6], which classifies counter-terrorism tactics into two categories:

  • “Fire” strategies are tactics that involve significant collateral damage. They include, for example, the killing of terrorists through drones, the use of indiscriminate checkpoints or the aggressive blockade of roads.

  • “Water” strategies, on the other hand, are counter-measures that do not affect innocent people, like intelligence arrests against suspect individuals.

The fire and water

Optimization algorithm

The above problem (68), is an Optimal Control Problem (OCP) where the total costs have to be minimized, given the state dynamics and the control constraints. Denoting u(t)=(u(t),v(t))=(u1(t),u2(t)) we have:minu(t)J=0F(t,x(t),u(t))dtsubject to satisfying:x˙(t)=f(t,x(t),u(t)),0t<;x(0)=x0u(t)U(t),0t<The problem presents several noteworthy features. First, the optimization interval is infinite. Second, the time t is not explicitly present in the problem (time-autonomous problem), except in

Steady-state solutions

In [12] a method for computing the optimal steady-state in infinite-horizon one-dimensional problems is presented which does not require the solution of the dynamic optimization problem, in which the bounds U(t) do not play any role. Tsur considers a one-dimensional version of our problem:minu(t)J=0G(x(t),u(t))ertdtx˙(t)=f(x(t),u(t)),x(0)=x0We propose another adaptation of the CCD method. Beginning with some admissible u0, we construct a sequence (uj) and at each stage, we compute the

Base case

We examine now the behavior of our approach in several examples. For the sake of comparison, we use the (carefully chosen) parameters used in [6]. The discount rate is a typical r=0.05. The outflow rate is assumed to be 5% and the constant inflow rate term is small τ=105. The parameter k is chosen such that the steady state is normalized to 1 in the absence of counter-terrorism tactics, and neglecting τ. This way, x is measured as a percentage of the steady-state size of the terrorist

Sufficient conditions

We consider now the multi-control OCP in its general Bolza form:minu(t)J=0TF(x(t),u(t),t)dt+B[T,x(T)]subject to:x˙(t)=f(x(t),u(t),t);x(0)=x0u(t)U(t),0tTwhere x(t)=(x1(t),,xn(t))Rn is the state vector and u(t)=(u1(t),,un(t))Rn the control vector. We assume the following:

  • (i)

    F and f=(f1(t),,fn(t)) are continuous.

  • (ii)

    F and f have continuous second derivatives with respect to t and x but their second derivatives with respect to u may be discontinuous.

  • (iii)

    The control variable u(t) needs only be

Conclusions

We consider ways for a government to optimally employ “water” and “fire” strategies for fighting terrorism. The model tries to balance the costs of terror attacks with the cost of terror control. We present two main contributions. First, a new effective algorithm for computing the dynamical solution whose cyclic nature allows its use in models of greater dimension (say with more controls or more state variables) without any conceptual modification. Secondly, we present a method for computing

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