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Uncertain optimal control of linear quadratic models with jump. (English) Zbl 1286.93202
Summary: Based on the uncertain optimal control with jump, in this paper, we study a kind of special uncertain optimal control problem: linear-quadratic $$(LQ)$$ uncertain optimal control problem with jump which has a quadratic objective function for a linear uncertain control system with jump. We obtain a necessary and sufficient condition for the existence of optimal control. As an application, we discuss an uncertain $$LQ$$ optimal control problem for the enterprize’s investment decisions.

##### MSC:
 93E20 Optimal stochastic control 49N10 Linear-quadratic optimal control problems
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##### References:
 [1] Kalman, R. E., Contribution to the theory of optimal control, Boletin Sociedad Matemática Mexicana, 5, 1, 102-119, (1960) · Zbl 0112.06303 [2] Wonham, W. M., On a matrix Riccati equation of stochastic control, SIAM Journal on Control and Optimization, 6, 1, 681-697, (1968) · Zbl 0182.20803 [3] Bismut, J. M., Linear quadratic optimal stochastic control with random coefficients, SIAM Journal on Control and Optimization, 14, 3, 419-444, (1976) · Zbl 0331.93086 [4] Bonsoussan, A., A stochastic control of partially observed systems, (1992), Cambridge Univ. Press Cambridge, UK [5] Davis, M. H.A., Linear estimation and stochastic control, (1977), Chapman and Hall London. UK · Zbl 0437.60001 [6] Chen, S.; Li, X.; Zhou, X., Stochastic linear quadratic regulators with indefinite control weigh costs, SIAM Journal on Control and Optimization, 36, 5, 1685-1702, (1998) · Zbl 0916.93084 [7] Wu, H.; Zhou, X., Characterizing all optimal controls for an indefinite stochastic linear quadratic control problem, IEEE Transactions on Automatic Control, 47, 7, 1119-1122, (2002) · Zbl 1364.49044 [8] Zhou, X.; Li, D., Continuous-time mean-variance portfolio selection: a stochastic LQ framework, Applied Mathematics and Optimization, 42, 1, 19-33, (2000) · Zbl 0998.91023 [9] Kohlmann, M.; Tang, S., Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging, Stochastic Processes and their Applications, 97, 2, 255-288, (2002) · Zbl 1064.93050 [10] Kohlmann, M.; Zhou, X., Relationship between backward stochastic differential equations and stochastic controls: a linear-quadratic approach, SIAM Journal on Control and Optimization, 38, 5, 1392-1407, (2000) · Zbl 0960.60052 [11] Lim, A. E.B., Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Mathematics of Operations Research, 29, 1, 132-161, (2004) · Zbl 1082.91050 [12] Liu, B., Uncertainty theory: an introduction to its axiomatic foundations, (2004), Springer-Verlag Berlin · Zbl 1072.28012 [13] Zhao, Y.; Zhu, Y., Fuzzy optimal control of linear quadratic models, Computers and Mathematics with Applications, 60, 1, 67-73, (2010) · Zbl 1198.93239 [14] Qin, Z.; Bai, M.; Ralescu, D., A fuzzy control system with application to production planning problems, Information Sciences, 181, 5, 1018-1027, (2011) · Zbl 1208.93051 [15] Liu, B., Uncertainty theory, (2007), Springer-Verlag Berlin [16] Liu, B., Uncertainty theory: A branch of mathematics for modeling human uncertainty, (2010), Springer-Verlag Berlin [17] Zhu, Y., Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems, 41, 7, 535-547, (2010) · Zbl 1225.93121 [18] Deng, L.; Zhu, Y., Uncertain optimal control with jump, ICIC Express Letters, Part B: Applications, 3, 2, 419-424, (2012) [19] Liu, B., Some research problems in uncertainty theory, Journal of Uncertain Systems, 3, 1, 3-10, (2009) [20] Liu, B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, 2, 1, 3-16, (2008) [21] Liu, B., Theory and practice of uncertain programming, (2009), Springer-Verlag Berlin · Zbl 1158.90010 [22] Solow, R. M., A contribution to the theory of economic growth, Quarterly Journal of Economics, 70, 1, 65-94, (1956) [23] Merton, R., An asymptotic theory of growth under uncertainty, Review of Economic Studies, 42, 3, 375-393, (1975)
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