Optimal control methodology for the counter-terrorism strategies: the relaxation based approach. (English) Zbl 1496.91042

Summary: This paper deals with a novel application of the advanced optimal control techniques to a counter-terrorism optimal decision-making problem. The recently developed model of the counter-terrorism security policies (see [L. Armijo, Pac. J. Math. 16, 1–3 (1966; Zbl 0202.46105)]) naturally incorporates a competing interest group dynamics. The optimal dynamics can be found in this case by considering a specific optimal control problem. in our work we propose an essential formal improvement of the above applied optimal control problem and its numerical treatment. We formulate a suitable phase-constrained optimal control problem for the initial counter-terrorism model and study a specific relaxation of this model. The proposed \(\beta\)-relaxation approach makes it possible to eliminate some model inconsistencies and to design an optimal counter-terrorism strategy. We develop a new numeric algorithm for the improved dynamic optimization problem. We next study some analytic properties of this algorithm. The resulting computational technique involves a gradient based method in combination with the proposed relaxation scheme (see [V. Azhmyakov and W. Schmidt, J. Optim. Theory Appl. 130, No. 1, 61–77 (2006; Zbl 1135.49020)]). The obtained numerical approach is finally applied to an illustrative example.


91B06 Decision theory
49N90 Applications of optimal control and differential games
Full Text: DOI


[1] Armijo, L., Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J. Math., 16, 1-3 (1966) · Zbl 0202.46105
[2] Azhmyakov, V.; Schmidt, W., Approximations of relaxed optimal control problems, J. Optim. Theory Appl., 130, 61-77 (2006) · Zbl 1135.49020
[3] Azhmyakov, V., A gradient type algorithm for a class of optimal control processes governed by hybrid dynamical systems, IMA J. Math. Control Inf., 28, 3, 291-307 (2011) · Zbl 1228.49029
[4] Azhmyakov, V.; Basin, M. V.; Raisch, J., Proximal point based approach to optimal control of affine switched systems, Discrete Event Dyn. Syst., 22, 1, 61-81 (2012) · Zbl 1242.49080
[5] Azhmyakov, V.; Cabrera, J.; Poznyak, A., Optimal fixed-levels control for non-linear systems with quadratic costs functional, Optim. Control Appl. Methods, 37, 1035-1055 (2016) · Zbl 1348.49023
[6] Azhmyakov, V.; Juarez, R., A first-order numerical approach to switched-mode systems optimization, Nonlinear Anal. Hybrid Syst, 25, 126-137 (2017) · Zbl 1376.49015
[7] Azhmyakov, V., A Relaxation Based Approach to Optimal Control of Switched Systems (2019), Elsevier: Elsevier Oxford, UK · Zbl 1408.34001
[8] Azhmyakov, V.; Egerstedt, M.; Verriest, E. I., On the optimal control of Volterra integro-differential equations, 58th IEEE Conference on Decision and Control, 3340-3345 (2019), Nice, France
[9] Bayón, L.; Ayuso, P.; García, P. J.; Grau, J. M.; Ruiz, M. M., Optimal control of counter-terrorism tactics, Appl. Math. Comput., 347, 477-491 (2019) · Zbl 1428.49031
[10] Berkovitz, L. D., Optimal Control Theory (2013), Springer: Springer New York · Zbl 0295.49001
[11] Betts, J., Practical Methods for Optimal Control Problems Using Nonlinear Programming (2001), SIAM: SIAM Philadelphia, USA
[12] Blanchini, F.; Miami, S., Set Theoretic Methods in Control (2008), Springer: Springer Boston
[13] Boltyanski, V.; Martini, H.; Soltan, V., Geometric Methods and Optimization Problems (1999), Kluver Academic Publishers: Kluver Academic Publishers Dordrecht · Zbl 0933.90002
[14] Castillo, C.; Song, B., Models for the transmission dynamics of fanatic behaviors, Bioterrorism: Mathematical Modeling Applications in Homeland Security, vol. 2003, 155-172 (2003), SIAM
[15] Caulkins, J. P.; Grass, D.; Feichtinger, G.; Tragler, G., Optimizing counter-terror operations: should one fight fire with ‘fire’ or ‘water’?, Comput. Oper. Res., 35, 1874-1885 (2008) · Zbl 1139.91029
[16] Caulkins, J. P.; Feichtinger, G.; Grass, D.; Tragler, G., Optimal control of terrorism and global reputation: a case study with novel threshold behavior, Oper. Res. Lett., 37, 387-391 (2009) · Zbl 1193.93063
[17] Fattorini, H. O., Infinite Dimensional Optimization and Control Theory (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0931.49001
[18] Feichtinger, G.; Novak, A. J., Terror and counterterror operations: differential game with cyclical Nash solution, J. Optim. Theory Appl., 139, 541-556 (2008) · Zbl 1159.91322
[19] Feichtinger, G.; Novak, A.; Wrzaczek, S., Optimizing counter-terroristic operations in an asymmetric Lanchester model, IFAC Proc. Vol., 45, 27-32 (2012)
[20] Gill, P. E.; Murray, W.; Wright, M. H., Practical Optimization (1981), Academic Press: Academic Press New York, USA · Zbl 0503.90062
[21] Goldstein, A. A., Convex programming in hilbert space, Bull. Am. Math. Soc., 70, 709-710 (1964) · Zbl 0142.17101
[22] Grass, D.; Caulkins, J. P.; Feichtinger, G.; Tragler, G.; Behrens, D. A., Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption, and Terror (2008), Springer Berlin Heidelberg · Zbl 1149.49001
[23] Ioffe, A. D.; Tichomirov, V. M., Theory of Extremal Problems (1979), North Holland: North Holland Amsterdam
[24] Kaplan, E. H., Staffing models for covert counterterrorism agencies, Socioecon. Plann. Sci., 47, 2-8 (2013)
[25] Kaplan, E. H., Socially efficient detection of terror plots, Oxf. Econ. Pap., 67, 104-115 (2014)
[26] Kurzhanski, A.; Veliov, V., Modeling Techniques and Uncertain Systems (1994), Springer: Springer New York
[27] Lum, C.; Kennedy, L. V., Evidence-based counterterrorism policy, vol. 3, 3-9 (2012), Springer: Springer NY
[28] Polak, E., Optimization (1997), Springer: Springer New York · Zbl 0886.90140
[29] Poznyak, A.; Polyakov, A.; Azhmyakov, V., Attractive Ellipsoids in Robust Control (2014), Springer: Springer Basel, Switzerland · Zbl 1314.93006
[30] Roxin, E., The existence of optimal controls, Mich. Math. J., 9, 109-119 (1962) · Zbl 0105.07801
[31] Seidl, A.; Kaplan, E. H.; Caulkins, J. P.; Wrzaczek, S.; Feichtinger, G., Optimal control of a terror queue, Eur. J. Oper. Res., 248, 246-256 (2016) · Zbl 1347.90025
[32] Teo, K. L.; Goh, C. J.; Wong, K. H., A Unified Computational Approach to Optimal Control Problems (1991), Wiley: Wiley New York · Zbl 0747.49005
[33] Udwadia, F.; Leitmann, G.; Lambertini, L., A dynamical model of terrorism, Discrete Dyn. Nat. Soc. (2006) · Zbl 1211.91215
[34] Wardi, Y., Optimal control of switched-mode dynamical systems, Proceedings of the 11th International Workshop on Discrete Event Systems, 4-8 (2012), Guadalajara
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.