American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential
  • MCRF Home
  • This Issue
  • Next Article
    Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain
March  2011, 1(1): 83-118. doi: 10.3934/mcrf.2011.1.83

A deterministic linear quadratic time-inconsistent optimal control problem

Jiongmin Yong 1,

1. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816

Received  October 2010 Revised  November 2010 Published  March 2011

A time-inconsistent optimal control problem is formulated and studied for a controlled linear ordinary differential equation with a quadratic cost functional. A notion of time-consistent equilibrium strategy is introduced for the original time-inconsistent problem. Under certain conditions, we construct an equilibrium strategy which can be represented via a Riccati--Volterra integral equation system. Our approach is based on a study of multi-person hierarchical differential games.
Keywords: linear-quadratic optimal control problem, Time-inconsistency, multi-person hierarchical differential games, time-consistent equilibrium strategy, Riccati--Volterra integral equation system..
Mathematics Subject Classification: Primary: 49L20, 49N10, 49N70; Secondary: 91A23, 91A6.
Citation: Jiongmin Yong. A deterministic linear quadratic time-inconsistent optimal control problem. Mathematical Control & Related Fields, 2011, 1 (1) : 83-118. doi: 10.3934/mcrf.2011.1.83
References:
[1]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation,, Preprint, (2008).   Google Scholar

[2]

L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (1974).   Google Scholar

[3]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem,, working paper., ().   Google Scholar

[4]

E. V. Böhm-Bawerk, "The Positive Theory of Capital,", Books for Libraries Press, (1891).   Google Scholar

[5]

I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time,, preprint, (2008).   Google Scholar

[6]

I. Ekeland and T. Privu, Investment and consumption without commitment,, preprint, (2007).   Google Scholar

[7]

S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533.  doi: 10.2307/2297304.  Google Scholar

[8]

S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences,, preprint., ().   Google Scholar

[9]

P. J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibriub model,, preprint., ().   Google Scholar

[10]

D. Hume, "A Treatise of Human Nature,", First Edition, (1739).   Google Scholar

[11]

W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871).   Google Scholar

[12]

P. Krusell and A. A. Smith, Jr., Consumption and saving decisions with quasi-geometric discounting,, Econometrica, 71 (2003), 366.  doi: 10.1111/1468-0262.00400.  Google Scholar

[13]

D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443.  doi: 10.1162/003355397555253.  Google Scholar

[14]

A. Malthus, An essay on the principle of population, 1826,, in, 2 (1986).   Google Scholar

[15]

J. Marin-Solano and J. Navas, Non-constant discounting in finite horizon: The free terminal time case,, J. Economic Dynamics and Control, 33 (2009), 666.  doi: 10.1016/j.jedc.2008.08.008.  Google Scholar

[16]

A. Marshall, "Principles of Economics,", 1st ed., (1890).   Google Scholar

[17]

M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics,, The Economic Journal, 95 (1985), 124.  doi: 10.2307/2232876.  Google Scholar

[18]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and Davis Hume,, History of Political Economy, 35 (2003), 241.  doi: 10.1215/00182702-35-2-241.  Google Scholar

[19]

V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (1909).   Google Scholar

[20]

B. Peleg and M. E. Yaari, On the existence of a consistent course of action when tastes are changing,, Review of Economic Studies, 40 (1973), 391.  doi: 10.2307/2296458.  Google Scholar

[21]

R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185.  doi: 10.2307/2296548.  Google Scholar

[22]

A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759).   Google Scholar

[23]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165.  doi: 10.2307/2295722.  Google Scholar

[24]

L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25.  doi: 10.1016/0047-2727(86)90070-8.  Google Scholar

[25]

J. Yong, A deterministic time-inconsistent optimal control problem -- An essentially cooperative approach,, Acta Appl. Math. Sinica, ().   Google Scholar

[26]

J. Yong, and X. Y. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations,", Applications of Mathematics (New York), (1999).   Google Scholar

show all references

References:
[1]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation,, Preprint, (2008).   Google Scholar

[2]

L. D. Berkovitz, "Optimal Control Theory,", Applied Mathematical Sciences, 12 (1974).   Google Scholar

[3]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochasitc control problem,, working paper., ().   Google Scholar

[4]

E. V. Böhm-Bawerk, "The Positive Theory of Capital,", Books for Libraries Press, (1891).   Google Scholar

[5]

I. Ekeland and A. Lazrak, Being serious about non-commitment: subgame perfect equilibrium in continuous time,, preprint, (2008).   Google Scholar

[6]

I. Ekeland and T. Privu, Investment and consumption without commitment,, preprint, (2007).   Google Scholar

[7]

S. M. Goldman, Consistent plans,, Review of Economic Studies, 47 (1980), 533.  doi: 10.2307/2297304.  Google Scholar

[8]

S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences,, preprint., ().   Google Scholar

[9]

P. J. Herings and K. I. M. Rohde, Time-inconsistent preferences in a general equilibriub model,, preprint., ().   Google Scholar

[10]

D. Hume, "A Treatise of Human Nature,", First Edition, (1739).   Google Scholar

[11]

W. S. Jevons, "Theory of Political Economy,", Mcmillan, (1871).   Google Scholar

[12]

P. Krusell and A. A. Smith, Jr., Consumption and saving decisions with quasi-geometric discounting,, Econometrica, 71 (2003), 366.  doi: 10.1111/1468-0262.00400.  Google Scholar

[13]

D. Laibson, Golden eggs and hyperbolic discounting,, Quarterly J. Econ., 112 (1997), 443.  doi: 10.1162/003355397555253.  Google Scholar

[14]

A. Malthus, An essay on the principle of population, 1826,, in, 2 (1986).   Google Scholar

[15]

J. Marin-Solano and J. Navas, Non-constant discounting in finite horizon: The free terminal time case,, J. Economic Dynamics and Control, 33 (2009), 666.  doi: 10.1016/j.jedc.2008.08.008.  Google Scholar

[16]

A. Marshall, "Principles of Economics,", 1st ed., (1890).   Google Scholar

[17]

M. Miller and M. Salmon, Dynamic games and the time inconsistency of optimal policy in open economics,, The Economic Journal, 95 (1985), 124.  doi: 10.2307/2232876.  Google Scholar

[18]

I. Palacios-Huerta, Time-inconsistent preferences in Adam Smith and Davis Hume,, History of Political Economy, 35 (2003), 241.  doi: 10.1215/00182702-35-2-241.  Google Scholar

[19]

V. Pareto, "Manuel d'économie Politique,", Girard and Brieve, (1909).   Google Scholar

[20]

B. Peleg and M. E. Yaari, On the existence of a consistent course of action when tastes are changing,, Review of Economic Studies, 40 (1973), 391.  doi: 10.2307/2296458.  Google Scholar

[21]

R. A. Pollak, Consistent planning,, Review of Economic Studies, 35 (1968), 185.  doi: 10.2307/2296548.  Google Scholar

[22]

A. Smith, "The Theory of Moral Sentiments,", First Edition, (1759).   Google Scholar

[23]

R. H. Strotz, Myopia and inconsistency in dynamic utility maximization,, Review of Econ. Studies, 23 (1955), 165.  doi: 10.2307/2295722.  Google Scholar

[24]

L. Tesfatsion, Time inconsistency of benevolent government economics,, J. Public Economics, 31 (1986), 25.  doi: 10.1016/0047-2727(86)90070-8.  Google Scholar

[25]

J. Yong, A deterministic time-inconsistent optimal control problem -- An essentially cooperative approach,, Acta Appl. Math. Sinica, ().   Google Scholar

[26]

J. Yong, and X. Y. Zhou, "Stochastic Controls: Hamiltonian Systems and HJB Equations,", Applications of Mathematics (New York), (1999).   Google Scholar

[1]

Lihua Bian, Zhongfei Li, Haixiang Yao. Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020026
[2]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97
[3]

Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016
[4]

Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193
[5]

Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial & Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188
[6]

Shigeaki Koike, Hiroaki Morimoto, Shigeru Sakaguchi. A linear-quadratic control problem with discretionary stopping. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 261-277. doi: 10.3934/dcdsb.2007.8.261
[7]

Libin Mou, Jiongmin Yong. Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method. Journal of Industrial & Management Optimization, 2006, 2 (1) : 95-117. doi: 10.3934/jimo.2006.2.95
[8]

Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764
[9]

Peng Zhang, Yongquan Zeng, Guotai Chi. Time-consistent multiperiod mean semivariance portfolio selection with the real constraints. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020039
[10]

Henri Bonnel, Ngoc Sang Pham. Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions. Journal of Industrial & Management Optimization, 2011, 7 (4) : 789-809. doi: 10.3934/jimo.2011.7.789
[11]

Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control & Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651
[12]

Tyrone E. Duncan. Some linear-quadratic stochastic differential games for equations in Hilbert spaces with fractional Brownian motions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5435-5445. doi: 10.3934/dcds.2015.35.5435
[13]

Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control & Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271
[14]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001
[15]

Russell Johnson, Carmen Núñez. Remarks on linear-quadratic dissipative control systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 889-914. doi: 10.3934/dcdsb.2015.20.889
[16]

Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243
[17]

Hanxiao Wang, Jingrui Sun, Jiongmin Yong. Weak closed-loop solvability of stochastic linear-quadratic optimal control problems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2785-2805. doi: 10.3934/dcds.2019117
[18]

Georg Vossen, Stefan Volkwein. Model reduction techniques with a-posteriori error analysis for linear-quadratic optimal control problems. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 465-485. doi: 10.3934/naco.2012.2.465
[19]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020026
[20]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020032

2019 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences