Duality in Linear Programming Problems Related to Deterministic Long Run Average Problems of Optimal Control

Finlay, Luke; Gaitsgory, Vladimir; Lebedev, Ivan

doi:10.1137/060676398

Cited by 25 publications

(46 citation statements)

References 35 publications

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“…The optimal values of the problems (3.17) and (3.18) are equal, this being one of the duality relationships between these two problems (see Theorem 4.1 in [35]). The problem (3.18) will be referred to as associated dual problem.…”

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confidence: 99%

“…Note that the optimal value of the problem (3.13) is equal to the optimal value of the averaged IDLP problem (2.29). The former is in fact dual with respect to the later, the equality of the optimal values being one of the duality relationships between the two (see Theorem 4.1 in [35]). For brevity, (3.13) will be referred to as just averaged dual problem.…”

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confidence: 99%

“…As in [48], our consideration is based on earlier results on averaging of SP control systems obtained in [39], [41], [42], [44], [47] (see also [6], [7], [8], [10], [51], [52], [67]) and on results obtained in [35], [45], [46], [47] that establish the equivalence of optimal control problems to certain infinite dimensional (ID) linear programming (LP) problems (related results on IDLP formulations of optimal control problems in both deterministic and stochastic settings can be found in [5], [15], [16], [21], [31], [37], [50], [53], [57], [60], [69], [71], [72] and [76]). In contrast to [48], where mostly optimal control problems with time discounting criteria were dealt with, this paper is devoted to the consideration of the problems with long run average optimality criteria.…”

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confidence: 99%

“…Namely, the optimal values of these two problems are related by the inequality 9) and also, under certain conditions (see [35] and [47]),…”

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confidence: 99%

“…the duality relationships including the equality of the optimal values (see Theorem 5.2(ii) in [35]). A function 20) will be called a solution of the (N, M )-approximating associated dual problem if…”

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confidence: 99%

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On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

Gaitsgory

Manic

Rossomakhine

2014

Set-Valued Var. Anal

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Abstract. The paper aims at the development of tools for analysis and construction of near optimal solutions of singularly perturbed (SP) optimal controls problems with long run average optimality criteria. The idea that we exploit is to first asymptotically approximate a given problem of optimal control of the SP system by a certain averaged optimal control problem, then reformulate this averaged problem as an infinite-dimensional (ID) linear programming (LP) problem, and then approximate the latter by semi-infinite LP problems. We show that the optimal solution of these semi-infinite LP problems and their duals (that can be found with the help of a modification of an available LP software) allow one to construct near optimal controls of the SP system. We demonstrate the construction with a numerical example.Key words. Singularly perturbed optimal control problems, Averaging and linear programming, Occupational measures, Numerical solution AMS subject classifications. 34E15, 34C29, 34A60, 93C701. Introduction and preliminaries. Problems of optimal control of singularly perturbed (SP) systems have been studied intensively in both deterministic and stochastic settings (see [2]

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confidence: 99%

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confidence: 99%

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confidence: 99%

“…Namely, the optimal values of these two problems are related by the inequality 9) and also, under certain conditions (see [35] and [47]),…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

Gaitsgory

Manic

Rossomakhine

2014

Set-Valued Var. Anal

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Analysis of economic model predictive control with terminal penalty functions on generalized optimal regimes of operation

Dong

Angeli

2018

Intl J Robust & Nonlinear

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Summary In this paper, we propose the novel concept of “control storage function” and introduce upper and lower bounds to the best asymptotic average performance for nonlinear control systems based on suitable notions of dissipativity and controlled dissipativity. This allows to extend and unify the formulation and analysis of economic model predictive control for general optimal operation regimes, including, in particular, steady‐state or periodic operation. A closed‐loop economic performance and stability analysis are carried out within this generalized framework. As a special case, when the optimal operation is periodic, we present a new approach to formulate terminal cost functions. Finally, several examples and counterexamples are proposed and discussed to show the merits of the proposed approach.

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Economic Model Predictive Control: Some Design Tools and Analysis Techniques

Angeli

Müller

2018

Handbook of Model Predictive Control

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In recent years, Economic Model Predictive Control has emerged as a variant of traditional (tracking) MPC which aims at maximizing economic profitability by directly minimizing, over a receding prediction horizon, the costs incurred in system's operation. Several design alternatives have been proposed in the literature, as well as suitable tools for the analysis of stability and performance of the different design methods. This note provides the authors' perpective on some of the most relevant results that appeared within this framework as well as, for interested readers, references to the technical papers where such results have first been discussed.

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Duality in Linear Programming Problems Related to Deterministic Long Run Average Problems of Optimal Control

Cited by 25 publications

References 35 publications

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria

Analysis of economic model predictive control with terminal penalty functions on generalized optimal regimes of operation

Economic Model Predictive Control: Some Design Tools and Analysis Techniques

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