Infinite horizon differential games for abstract evolution equations

Shaiju, A. J.

doi:10.1590/s0101-82052003000300003

Cited by 3 publications

(5 citation statements)

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“…The other approach is based on the theory developed by Berkovitz [7] for differential games in Euclidean spaces, wherein the definition of strategy is a combination of "K strategies" discussed by Isaacs [22] and Friedman's lower strategy (e.g., see [19,20]) and the definition of payoff and saddle point follows that of Krasovskii and Subbotin [25]. For example, infinite-dimensional differential games with strategies and payoff in the sense of Berkovitz were studied in [21] and [35] for finite horizon and infinite horizon, respectively, and the value function is characterized as the unique viscosity solution of Hamilton-Jacobi-Isaacs equation. It should be noted that the principle result of all of these investigations is that if the so-called Isaacs condition holds then the differential game has a value.…”

Section: Evolution Of the Expected Value Of Intensity Of Reflected Simentioning

confidence: 99%

“…Specifically we will use some of the results in [21,35] to show that our differential game has a value and this value function is the unique viscosity solution of Hamilton-JacobinIsaacs equation.…”

Section: Value Of Differential Gamementioning

confidence: 99%

“…In [21,35], Berkovitz's approach of constructing optimal strategies [7] was shown to be applicable to the infinite-dimensional differential game as well. It should be noted that the Berkovitz method involves using some feedback maps to construct the saddle point for the game, where these feedback maps are obtained by using some appropriate level sets related to the value function.…”

Section: <= Tnmentioning

confidence: 99%

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A zero-sum electromagnetic evader–interrogator differential game with uncertainty

Banks

2012

Applicable Analysis

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We consider dynamic electromagnetic evasion-interrogation games in which the evader can use ferroelectric material coatings to attempt to avoid detection while the interrogator can manipulate the interrogating frequencies to enhance detection. The resulting problem is formulated as a two-player zero-sum dynamic differential game in which the cost functional is based on the expected value of the intensity of the reflected signal. We show that there exists a saddle point for the relaxed form of this dynamic differential game in which the relaxed controls appear bilinearly in the dynamics governed by a partial differential equation. We also present a computational framework for construction of approximate saddle point strategies in feedback form for a special case of this relaxed differential game with strategies and payoff in the sense of Berkovitz. AMS subject classifications: 83C50, 91A23, 49N70, 49N90, 65M32, 68T37, 60J70.

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Section: Evolution Of the Expected Value Of Intensity Of Reflected Simentioning

confidence: 99%

Section: Value Of Differential Gamementioning

confidence: 99%

Section: <= Tnmentioning

confidence: 99%

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A zero-sum electromagnetic evader–interrogator differential game with uncertainty

Banks

2012

Applicable Analysis

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“…There are several papers in which viscosity solutions and Isaacs equations have been used to construct a saddle (and an approximate saddle) point strategies for infinite dimensional differential games in Hilbert spaces [12,17,19]. In these papers Berkovitz's notion of strategy and pay-off [1] is employed.…”

Section: ∆ (T) := {β : M(t) → N(t) Nonanticipating}mentioning

confidence: 99%

“…where o( 1 n ) is independent of Z and W n by (19). Therefore we can define a nonanticipative strategy α n ∈ Γ (t) by setting α n [Z](τ) = W n (τ).…”

Section: Remark 1 Condition (19) Says That Trajectories Starting At mentioning

confidence: 99%

Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Święch

2010

Annals of the International Society of Dynamic Games

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Path-dependent Hamilton–Jacobi equations in infinite dimensions

Bayraktar

Keller

2018

Journal of Functional Analysis

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We propose notions of minimax and viscosity solutions for a class of fully nonlinear path-dependent PDEs with nonlinear, monotone, and coercive operators on Hilbert space. Our main result is well-posedness (existence, uniqueness, and stability) for minimax solutions. A particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle. One of the main difficulties, the lack of compactness in infinite-dimensional Hilbert spaces, is circumvented by working with suitable compact subsets of our path space. As an application, our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of (delay) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiȋ-Subbotin approach similarly as in finite dimensions. 8. Applications to differential games 33 Appendix A. Properties of solution sets of evolution equations 40 Appendix B. Other notions of solutions 45 B.1. Classical solutions 45 B.2. Viscosity solutions 47 References 48 C(S) := {u : S → R continuous with respect to d ∞ }, USC(S) := {u : S → R upper semi-continuous with respect to d ∞ }, LSC(S) := {u : S → R lower semi-continuous with respect to d ∞ }.Clearly, the elements of those function spaces are non-anticipating.2.2. The operator A, related trajectory spaces, and evolution equations. Throughout this work, we fix an operator A : [0, T ] × V → V * and assume that the following hypotheses hold. H(A): (i) The mappings t → A(t, x), v , v ∈ V , are measurable. (ii) Monotonicity: For a.e. t ∈ (0, T ) and every x, y ∈ V , A(t, x) − A(t, y), x − y ≥ 0. (iii) Hemicontinuity: For a.e. t ∈ (0, T ) and every x, y, v ∈ V , the mappings s → A(t, x + sy), v , [0, 1] → R, are continuous.

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Infinite horizon differential games for abstract evolution equations

Cited by 3 publications

References 8 publications

A zero-sum electromagnetic evader–interrogator differential game with uncertainty

A zero-sum electromagnetic evader–interrogator differential game with uncertainty

Sub- and Super-optimality Principles and Construction of Almost Optimal Strategies for Differential Games in Hilbert Spaces

Path-dependent Hamilton–Jacobi equations in infinite dimensions

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