Infinite Horizon Optimal Control

Carlson, Dean A.; Haurie, Alain; Leizarowitz, Arie

doi:10.1007/978-3-642-76755-5

Cited by 300 publications

(71 citation statements)

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“…where g(·) is the same as in (14) (and the same as in (2)- (7)). It can be easily seen that the set W is convex and compact.…”

Section: Occupational Measures Formulationmentioning

confidence: 99%

“…Infinite horizon problems of optimal control have been studied intensively in both deterministic and stochastic settings (see Anderson and Kokotovic [3], Arisawa, Ishii, and Lions [5], Bardi and Capuzzo-Dolcetta [10], Bensoussan [12], Carlson, Haurie, and Leizarowitz [14], Colonius and Kliemann [17], Fleming and Soner [21], Grüne [29], Kushner [34], Kushner and Dupuis [35], Vigodner [46], and references therein). In the stochastic setting, the linear programming formulation is a common tool for treating the problems (see, e.g., Basak, Borkar, and Ghosh [11], Borkar [13], Hernandez-Lerma and Lasserre [31], Stockbridge [44], Yin and Zhang [48]).…”

Section: Introduction and Description Of The Problemsmentioning

confidence: 99%

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Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

Gaitsgory¹,

Quincampoix²

2009

SIAM J. Control Optim.

153

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Abstract. We establish that deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finite-dimensional LPPs, the solutions of which can be used for construction of the optimal controls. General results are illustrated with numerical examples.Key words. long run average optimal control, singularly perturbed control systems, occupational measures, averaging method, linear programming AMS subject classifications. 34E15, 34C29, 34A60, 93C70 DOI. 10.1137/0406168021. Introduction and description of the problems. In this paper we show that, under some conditions, deterministic long run average problems of optimal control are "asymptotically equivalent" to infinite-dimensional linear programming problems (LPPs) and we establish that these LPPs can be approximated by finitedimensional LPPs, the solutions of which can be used for numerical construction of the optimal controls.Infinite horizon problems of optimal control have been studied intensively in both deterministic and stochastic settings (see Anderson and Kokotovic [3] Mendiondo and Stockbridge [38]. A linear programming approach to long run average optimal control problems in the deterministic setting appears to be new and, to the best of our knowledge, there are no publications devoted to this topic (under different assumptions and for a different problem, a linear programming formulation was discussed in Evans and Gomes [20]). A linear programming approach to deterministic optimal control problems on a finite time interval has been studied in Rubio [42].Let us introduce the problems that we will be dealing with. Consider the control systemẏ

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“…where g(·) is the same as in (14) (and the same as in (2)- (7)). It can be easily seen that the set W is convex and compact.…”

Section: Occupational Measures Formulationmentioning

confidence: 99%

Section: Introduction and Description Of The Problemsmentioning

confidence: 99%

Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

Gaitsgory¹,

Quincampoix²

2009

SIAM J. Control Optim.

153

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“…Therefore the unique turnpike defines also an equilibrium in the limit game G 0 . It is established, in the theory of turnpikes for infinite horizon control or open-loop equilibrium problems under the overtaking optimality criterion that the uniform ς-reachability condition is satisfied by the trajectories converging toward their respective attractors; see the book [5] for a complete discussion of these topics. In order to link these trajectories with a ς-equilibrium of the G ε game we need this last assumption.…”

Section: Lemma 52 the Turnpike Attractor For The Iholdg Defined In mentioning

confidence: 99%

Singularly Perturbed Piecewise Deterministic Games

Haurie¹,

Moresino²

2008

SIAM J. Control Optim.

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Abstract. In this paper we consider a class of hybrid stochastic games with the piecewise openloop information structure. These games are indexed over a parameter ε which represents the time scale ratio between the stochastic (jump process) and the deterministic (differential state equation) parts of the dynamical system. We study the limit behavior of Nash equilibrium solutions to the hybrid stochastic games when the time scale ratio tends to 0. We also establish that an approximate equilibrium can be obtained for the hybrid stochastic games using a Nash equilibrium solution of a reduced order sequential discrete state stochastic game and a family of local deterministic infinite horizon open-loop differential games defined in the stretched out time scale. A numerical illustration of this approximation scheme is also developed.

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“…Although conditions for the optimal control of partial differential equations have been derived either in abstract settings (e.g. Lions 1971) or for specific problems, 7 our derivation not only makes the paper self contained, but it is also close to the optimal control formalism used by economists, so it can be used for analyzing other types of economic problems, where state variables are governed by diffusion processes. Furthermore, the Pontryagin principle developed in this paper allows for an extension of the Turing mechanism for generation of spatial patterns, to the optimal control of systems under diffusion.…”

Section: Introductionmentioning

confidence: 99%

“…A new -to our knowledge -characteristic of our continuous spacetime approach is that we are able to embed Turing analysis in an optimal control recursive infinite horizon approach in a way that allows us to 7 See for example Lenhart and Bhat (1992); Lenhart et al (1999); Bhat et al (1999); Raymond andZidani (1998, 1999). locate sufficient conditions on parameters of the system (for example, the discount rate on the future, and interaction terms in the dynamics) for diffusive instability to emerge even in systems that are being optimally controlled.…”

Section: Introductionmentioning

confidence: 99%

Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

Brock

Xepapadeas

2005

SSRN Journal

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This paper extends Turing analysis to standard recursive optimal control frameworks in economics and applies it to dynamic bioeconomic problems where the interaction of coupled economic and ecological dynamics under optimal control over space creates (or destroys) spatial heterogeneity. We show how our approach reduces the analysis to a tractable extension of linearization methods applied to the spatial analog of the well known costate/state dynamics. We explicitly show the existence of a non-empty Turing space of diffusive instability by developing a linear-quadratic approximation of the original non-linear problem. We apply our method to a bioeconomic problem, but the method has more general economic applications where spatial considerations and pattern formation are important. We believe that the extension of Turing analysis and the theory associated with the dispersion relationship to recursive infinite horizon optimal control settings is new. JEL Classification Q2, C6

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Infinite Horizon Optimal Control

Abstract: The first edition was published in 1987 as Volume 290 of the series "Lecture Notes in Economics and Mathematical Systems".

Cited by 300 publications

References 0 publications

Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting

Singularly Perturbed Piecewise Deterministic Games

Optimal Control and Spatial Heterogeneity: Pattern Formation in Economic-Ecological Models

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