Pareto optimality in infinite horizon linear quadratic differential games

Reddy, Puduru Viswanadha; Engwerda, Jacob

doi:10.1016/j.automatica.2013.03.004

Cited by 44 publications

(21 citation statements)

References 19 publications

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“…Example 2 illustrates this point. 1+3(x * (N)) 2 , then (6)- (7) in Theorem 1 hold for an arbitrary choice of {(u 1 (k), u 2 (k)), k ∈ T}. However, as shown in Example 1, although any choice of {(u 1 (k), u 2 (k)), k ∈ T} does provide a Pareto solution, the weighted sum minimization problem is not necessarily solvable.…”

Section: Necessary Conditionsmentioning

confidence: 99%

“…Engwerda put forward necessary and sufficient conditions for the existence of Pareto solutions in a finite horizon cooperative differential game. Reddy and Engwerda studied the relationship between the Pareto optimality and the weighted sum minimization for the infinite horizon cooperative differential game. Reddy and Engwerda generalized the existing results in the work of Engwerda from finite horizon to infinite horizon.…”

Section: Introductionmentioning

confidence: 99%

“…1,3,21,22 Pareto optimum has been deeply studied for the deterministic systems. 6,7,23,24 Engwerda 23 determined the set of Pareto-efficient strategies for the regular convex cooperative differential game characterized by linear affine systems and quadratic cost functionals. Engwerda 6 put forward necessary and sufficient conditions for the existence of Pareto solutions in a finite horizon cooperative differential game.…”

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confidence: 99%

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Necessary/sufficient conditions for Pareto optimum in cooperative difference game

Lin

Zhang

2017

Optim Control Appl Methods

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Summary This paper is concerned with the necessary/sufficient conditions for the Pareto optimum of the cooperative difference game in finite horizon. Utilizing the necessary and sufficient characterization of the Pareto optimum, the problem is transformed into a set of constrained optimal control problems with a special structure. Employing the discrete version of Pontryagin's maximum principle, the necessary conditions for the existence of the Pareto solutions are derived. Under certain convex assumptions, it is shown that the necessary conditions are sufficient too. Next, the obtained results are extended to the linear‐quadratic case. For a fixed initial state, the necessary conditions resulting from the maximum principle and the convexity condition on the cost functional provide the necessary and sufficient description of the well‐posedness of the weighted sum optimal control problem. For an arbitrary initial state, the solvability of the related difference Riccati equation provides a sufficient condition under which the Pareto‐efficient strategies are equivalent to the weighted sum optimal controls. In addition, all Pareto solutions are derived based on the solutions of a set of difference equations. Two examples show the effectiveness of the proposed results.

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Section: Necessary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Necessary/sufficient conditions for Pareto optimum in cooperative difference game

Lin

Zhang

2017

Optim Control Appl Methods

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“…For instance, a Nash equilibrium strategy or other equilibrium type strategy with its finite support probability measure might have been practiced only with an infinite number of recurring support pure strategies [3], [4]. In the case of the infinite support probability measure, the majority of the support pure strategies cannot be recurred even once as a number of the recurred ones constitute the zero-measure set [5], [6]. However, for most classes of infinite twoperson non-cooperative games their solutions are unknown or at least are non-effectively computable [1], [2], [7].…”

Section: Infinite Two-person Non-cooperative Games Isomorphic To mentioning

confidence: 99%

“…However, now at or the game (6) by 7still is not bimatrix inasmuch as the players' payoff values 7are not arranged as ordinary flat matrices. The sampling on hypercube (1) down to lattice (8) with (4) and (5) is just the primary and incomplete step in converting the infinite game into the bimatrix game.…”

Section: Sampling the Players' Payoff Functions Variouslymentioning

confidence: 99%

Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

Romanuke

Kamburg

2016

Applied Computer Systems

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Approximation in solving the infinite two-person non-cooperative games is studied in the paper. An approximation approach with conversion of infinite game into finite one is suggested. The conversion is fulfilled in three stages. Primarily the players’ payoff functions are sampled variously according to the stated requirements to the sampling. These functions are defined on unit hypercube of the appropriate Euclidean finite-dimensional space. The sampling step along each of hypercube dimensions is constant. At the second stage, the players’ payoff multidimensional matrices are reshaped into ordinary two-dimensional matrices, using the reversible index-to-index reshaping. Thus, a bimatrix game as an initial infinite game approximation is obtained. At the third stage of the conversion, the player’s finite equilibrium strategy support is checked out for its weak consistency, defined by five types of inequalities within minimal neighbourhood of every specified sampling step. If necessary, the weakly consistent solution of the bimatrix game is checked out for its consistency, strengthened in that the cardinality of every player’s equilibrium strategy support and their densities shall be non-decreasing within minimal neighbourhood of the sampling steps. Eventually, the consistent solution certifies the game approximation acceptability, letting solve even games without any equilibrium situations, including isomorphic ones to the unit hypercube game. A case of the consistency light check is stated for the completely mixed Nash equilibrium situation.

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H∞$$ {H}_{\infty } $$ constraint Pareto optimal control for discrete‐time Markov jump linear stochastic systems in finite horizon

Jiang

Cui

Zhao

2022

Asian Journal of Control

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This paper mainly focuses on discrete‐time Markov jump linear stochastic systems (DMJLSSs) with external disturbances and multiplicative noise. Due to the multi‐player in the systems and multi‐objective cost function, the strategy design for these systems is more difficult than the traditional Markov jump stochastic systems. We combine H∞$$ {H}_{\infty } $$ theory with Pareto control and use the weighted sum method of the objective functional of each player to solve the problem of Pareto cooperative game. Four coupled difference matrix‐valued recursion equations (CDMREs) are derived by stochastic bounded real lemma (SBRL). Then, a sufficient and necessary condition for the existence of the Pareto optimal strategy is obtained, and the Pareto solution for every controller is derived under this strategy. A simulation example is given to verify the correctness of the theory.

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Pareto optimality in infinite horizon linear quadratic differential games

Cited by 44 publications

References 19 publications

Necessary/sufficient conditions for Pareto optimum in cooperative difference game

Necessary/sufficient conditions for Pareto optimum in cooperative difference game

Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

H∞$$ {H}_{\infty } $$ constraint Pareto optimal control for discrete‐time Markov jump linear stochastic systems in finite horizon

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