Sampled-Data Nash Equilibria in Differential Games with Impulse Controls

Sadana, Utsav; Reddy, Puduru Viswanadha; Başar, Tamer; Zaccour, Georges

doi:10.1007/s10957-021-01920-0

Cited by 9 publications

(6 citation statements)

References 21 publications

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“…From the third equation of game (15), it shows that the initial state ȳ2,0 is required to ensure the completeness condition [25,26]…”

Section: Uncertain Random Two-person Nonzero-sum Gamementioning

confidence: 99%

“…Subsequently, Nash [8,9] demonstrated the existence of equilibrium points for a game utilizing Kakutani's generalized fixed point theorem. The Nash equilibrium points for a series of two-person nonzero-sum games were then found [10][11][12][13][14][15]. The Nash equilibrium and saddle-point equilibrium are solutions to different kinds of games.…”

Section: Introductionmentioning

confidence: 99%

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Two‐person games for uncertain random singular dynamic systems

Chen

Zhu

Park

2022

IET Control Theory & Appl

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A complex system exhibiting uncertainty as well as randomness may be portrayed by an uncertain random singular difference equation. This paper investigates two-person nonzero-sum and zero-sum games based on uncertain random singular difference equations. First, an approach is proposed to translate the two-person nonzero-sum game into an equivalent game for a standard uncertain random dynamic system. The relevant recursive equations are developed to search the Nash equilibrium for the converted game. Solving the related recursive equations yields the solution to such a game. Following that, a Max-Min Theorem is provided for finding the saddle-point equilibrium of an uncertain random two-person zero-sum game. Finally, a numerical example is offered to demonstrate the validity of the findings.

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“…From the third equation of game (15), it shows that the initial state ȳ2,0 is required to ensure the completeness condition [25,26]…”

Section: Uncertain Random Two-person Nonzero-sum Gamementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Two‐person games for uncertain random singular dynamic systems

Chen

Zhu

Park

2022

IET Control Theory & Appl

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“…Recent papers by Cosso [21] and El Asri and Mazid [30] consider dynamic programming approach for zero-sum stochastic DGs where both players use only impulse control. Works by Aïd et al [1], Basei et al [8], Campi and De Santis [18] and Sadana et al [54,55] study some nonzero-sum DGs with impulse controls. We mention that in [1] authors studied a DG between two notions that have different targets for the currency exchange rate, and provided a system of QVIs that needs to be solved in order to compute the NE.…”

Section: Below By (H) and (H Cmentioning

confidence: 99%

A Differential Game Control Problem in Finite Horizon with an Application to Portfolio Optimization

Asri¹,

Lalioui²

2022

Preprint

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This paper considers a new class of deterministic finite-time horizon, two-player, zero-sum differential games (DGs) in which the maximizing player is allowed to take continuous and impulse controls whereas the minimizing player is allowed to take impulse control only. We seek to approximate the value function, and to provide a verification theorem for this class of DGs. We first, by means of dynamic programming principle (DPP) in viscosity solution (VS) framework, characterize the value function as the unique VS to the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) double-obstacle equation. Next, we prove that an approximate value function exists, that it is the unique solution to an approximate HJBI double-obstacle equation, and converges locally uniformly towards the value function of each player when the time discretization step goes to zero. Moreover, we provide a verification theorem which characterizes a Nash-equilibrium for the DG control problem considered. Finally, by applying our results, we derive a new continuous-time portfolio optimization model, and we provide related computational algorithms.

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“…In contrast, the literature in differential games with impulse controls has been very limited, and has predominantly dealt with zero-sum games (see, e.g., [23] and [24]). With the exception of our previous papers [2,25,26], the equilibrium solutions in nonzero-sum differential games with impulse controls have been obtained under the assumption that the impulse timing is known a priori [27].…”

Section: Literature Reviewmentioning

confidence: 99%

“…In [2], we provided an algorithm for computing the open-loop Nash equilibrium in linear-quadratic dynamic games with impulse control. Reference [25] characterized the sampled-data Nash equilibrium for the class of games introduced in [2]. Further, [26] determined the FNE for a specialized case of linear-state differential games (LSDGs) with impulse controls, and showed, contrary to the case with ordinary controls, that the FNE and OLNE do not coincide when linear value functions are used to determine the FNE.…”

Section: Literature Reviewmentioning

confidence: 99%