On a class of hybrid differential games. (English) Zbl 1391.91049
Summary: This paper is intended to present a systematic application of the hybrid systems framework to differential games. A special class of bimodal linear-quadratic differential games is presented and illustrated with examples; two particular classes of switching rules, time-dependent and state-dependent switches are discussed. The main contribution of the paper consists in formulating necessary optimality conditions for determining optimal strategies in both cooperative and non-cooperative cases. A practically relevant hybrid differential game of pollution reduction is considered to illustrate the developed framework.
MSC:
| 91A23 | Differential games (aspects of game theory) |
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |
References:
| [1] | Antsaklis P (2000) A brief introduction to the theory and applications of hybrid systems. Proc IEEE Spec Issue Hybrid Syst Theory Appl 88(7):879-886 |
| [2] | Azhmyakov, V.; Attia, S.; Gromov, D.; Raisch, J.; Bemporad, A. (ed.); Bicchi, A. (ed.); Butazzo, G. (ed.), Necessary optimality conditions for a class of hybrid optimal control problems, 637-640 (2007), Berlin · Zbl 1221.49052 |
| [3] | Başar T (1976) On the uniqueness of the Nash solution in linear-quadratic differential games. Int J Game Theory 5(2-3):65-90 · Zbl 0354.90101 |
| [4] | Başar T, Olsder GJ (1999) Dynamic noncooperative game theory, vol 200, 2nd edn. SIAM, New York · Zbl 0946.91001 |
| [5] | Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser · Zbl 0890.49011 |
| [6] | Bellman R (1957) Dynamic programming. Princeton University Press, Princeton · Zbl 0077.13605 |
| [7] | Bellman R (1970) Introduction to matrix analysis, 2nd edn. McGraw-Hill Book Co., New York-Düsseldorf-London · Zbl 0216.06101 |
| [8] | Bernhard, P.; Farouq, N.; Thiery, S., An impulsive differential game arising in finance with interesting singularities, 335-363 (2006), Berlin · Zbl 1274.91068 · doi:10.1007/0-8176-4501-2_18 |
| [9] | Bonneuil N, Boucekkine R (2016) Optimal transition to renewable energy with threshold of irreversible pollution. Eur J Oper Res 248(1):257-262 · Zbl 1346.91173 · doi:10.1016/j.ejor.2015.05.060 |
| [10] | Branicky M, Borkar V, Mitter S (1998) A unified framework for hybrid control: model and optimal control theory. IEEE Trans Autom Control 43(1):31-45 · Zbl 0951.93002 · doi:10.1109/9.654885 |
| [11] | Breton M, Zaccour G, Zahaf M (2005) A differential game of joint implementation of environmental projects. Automatica 41(10):1737-1749 · Zbl 1125.91309 · doi:10.1016/j.automatica.2005.05.004 |
| [12] | Caines, PE; Egerstedt, M.; Malhamé, R.; Schoellig, A.; Bemporad, A. (ed.); Bicchi, A. (ed.); Butazzo, G. (ed.), A hybrid Bellman equation for bimodal systems, 656-659 (2007), Berlin · Zbl 1221.49054 |
| [13] | Dockner E, Jorgensen S, Van Long N, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, Cambridge · Zbl 0996.91001 · doi:10.1017/CBO9780511805127 |
| [14] | El Ouardighi F, Benchekroun H, Grass D (2014) Controlling pollution and environmental absorption capacity. Ann Oper Res 220(1):111-133 · Zbl 1301.91033 · doi:10.1007/s10479-011-0982-4 |
| [15] | Engwerda J (2005) LQ dynamic optimization and differential games. Wiley, Hoboken |
| [16] | Fabra N, García A (2015) Dynamic price competition with switching costs. Dyn Games Appl 5(4):540-567 · Zbl 1348.91113 · doi:10.1007/s13235-015-0157-z |
| [17] | Fleming W, Rishel R (1975) Deterministic and stochastic optimal control, applications of mathematics, vol 1. Springer, New York · Zbl 0323.49001 · doi:10.1007/978-1-4612-6380-7 |
| [18] | Goebel R, Sanfelice R, Teel A (2009) Hybrid dynamical systems. IEEE Control Syst Mag 29(2):28-93 · Zbl 1395.93001 · doi:10.1109/MCS.2008.931718 |
| [19] | Grass D, Caulkins JP, Feichtinger G, Tragler G, Behrens DA (2008) Optimal control of nonlinear processes. With applications in drugs, corruption, and terror. Springer, Berlin · Zbl 1149.49001 · doi:10.1007/978-3-540-77647-5 |
| [20] | Gromov D, Gromova E (2014) Differential games with random duration: a hybrid systems formulation. In: Petrosyan LA, Zenkevich NA (eds) Contributions to game theory and management, vol VII. Graduate School of Management SPbU, St. Petersburg, Russia, pp 104-119 · Zbl 1347.91060 |
| [21] | Hull DG (2003) Optimal control theory for applications. Springer, Berlin · Zbl 1113.49001 · doi:10.1007/978-1-4757-4180-3 |
| [22] | Jørgensen S, Gromova E (2016) Sustaining cooperation in a differential game of advertising goodwill accumulation. Eur J Oper Res. doi:10.1016/j.ejor.2016.03.029 · Zbl 1346.90461 |
| [23] | Kononenko A (1980) The structure of the optimal strategy in controlled dynamic systems. USSR Comput Math Math Phys 20(5):13-24 · Zbl 0478.49013 · doi:10.1016/0041-5553(80)90085-3 |
| [24] | Kort PM, Wrzaczek S (2015) Optimal firm growth under the threat of entry. Eur J Oper Res 246(1):281-292 · Zbl 1346.91145 · doi:10.1016/j.ejor.2015.04.030 |
| [25] | Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, Hoboken · Zbl 0159.13201 |
| [26] | Long NV, Prieur F, Puzon K, Tidball M (2014) Markov perfect equilibria in differential games with regime switching strategies. CESifo Working Paper Series No. 4662 · Zbl 1401.91030 |
| [27] | Lunze J, Lamnabhi-Lagarrigue F (2009) Handbook of hybrid systems control: theory, tools, applications. Cambridge University Press, Cambridge · Zbl 1180.93001 · doi:10.1017/CBO9780511807930 |
| [28] | Lygeros J, Johansson K, Simic S, Zhang J, Sastry S (2003) Dynamical properties of hybrid automata. IEEE Trans Autom Control 48(1):2-17 · Zbl 1364.93503 · doi:10.1109/TAC.2002.806650 |
| [29] | Malafeev O (1977) Stationary strategies in differential games. USSR Comput Math Math Phys 17(1):37-46 · Zbl 0383.90118 · doi:10.1016/0041-5553(77)90067-2 |
| [30] | Marín-Solano J, Shevkoplyas EV (2011) Non-constant discounting and differential games with random time horizon. Automatica 47(12):2626-2638 · Zbl 1235.49074 · doi:10.1016/j.automatica.2011.09.010 |
| [31] | Masoudi N, Zaccour G (2013) A differential game of international pollution control with evolving environmental costs. Environ Dev Econ 18(06):680-700 · doi:10.1017/S1355770X13000399 |
| [32] | Morari M, Baotic M, Borrelli F (2003) Hybrid systems modeling and control. Eur J Control 9(2):177-189 · Zbl 1293.93421 · doi:10.3166/ejc.9.177-189 |
| [33] | Petrosjan L, Zaccour G (2003) Time-consistent Shapley value allocation of pollution cost reduction. J Econ Dyn Control 27(3):381-398 · Zbl 1027.91005 · doi:10.1016/S0165-1889(01)00053-7 |
| [34] | Reddy P, Engwerda JC (2014) Necessary and sufficient conditions for Pareto optimality in infinite horizon cooperative differential games. IEEE Trans Autom Control 59(9):2536-2542 · Zbl 1360.49030 · doi:10.1109/TAC.2014.2305933 |
| [35] | Reddy P, Zaccour G (2015) Open-loop Nash equilibria in a class of linear-quadratic difference games with constraints. IEEE Trans Autom Control 60(9):2559-2564 · Zbl 1360.91036 · doi:10.1109/TAC.2015.2394873 |
| [36] | Reddy, P.; Schumacher, J.; Engwerda, J., Optimal management with hybrid dynamics—the shallow lake problem, 111-136 (2015), Berlin · Zbl 1403.49015 · doi:10.1007/978-3-319-20988-3_7 |
| [37] | Riedinger P, Iung C, Kratz F (2003) An optimal control approach for hybrid systems. Eur J Control 9(5):449-458 · Zbl 1293.49059 · doi:10.3166/ejc.9.449-458 |
| [38] | Seierstad A, Sydsæter K (1987) Optimal control theory with economic applications. Elsevier, North-Holland Inc, Amsterdam · Zbl 0613.49001 |
| [39] | Shaikh MS, Caines PE (2007) On the hybrid optimal control problem: theory and algorithms. IEEE Trans Autom Control 52(9):1587-1603 corrigendum: IEEE TAC, Vol. 54 (6), 2009, p 1428 · Zbl 1366.93061 |
| [40] | Shinar J, Glizer VY, Turetsky V (2007) A pursuit-evasion game with hybrid pursuer dynamics. In: Control Conference (ECC), 2007 European, IEEE, pp 1306-1313 · Zbl 1195.91009 |
| [41] | Shinar J, Glizer VY, Turetsky V (2009) A pursuit-evasion game with hybrid evader dynamics. In: Control Conference (ECC), 2009 European, IEEE, pp 121-126 · Zbl 1195.91009 |
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