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Feedback Nash Equilibria in Differential Games with Impulse Control

Utsav Sadana, Puduru Viswanadha Reddy, Georges Zaccour

TL;DR

This work addresses a finite-horizon, two-player nonzero-sum differential game with asymmetric controls: Player 1 uses piecewise-continuous controls while Player 2 employs impulse actions. It develops a verification framework that couples the Hamilton-Jacobi-Bellman equation for the continuous player with quasivariational inequalities for the impulse player to obtain feedback Nash equilibria, and proves a finite upper bound on the number of impulses. A key contribution is the complete analytical characterization of the feedback Nash equilibrium in a scalar linear-quadratic differential game with impulse control, including explicit forms for the optimal feedback laws and impulse rules under threshold-based continuation sets. Numerical examples illustrate the equilibria, state trajectories, and the time-consistency of the strategies, highlighting practical implications for regulation and security applications where discrete interventions interact with continuous dynamics.

Abstract

We study a class of deterministic finite-horizon two-player nonzero-sum differential games where players are endowed with different kinds of controls. We assume that Player 1 uses piecewise-continuous controls, while Player 2 uses impulse controls. For this class of games, we seek to derive conditions for the existence of feedback Nash equilibrium strategies for the players. More specifically, we provide a verification theorem for identifying such equilibrium strategies, using the Hamilton-Jacobi-Bellman (HJB) equations for Player 1 and the quasi-variational inequalities (QVIs) for Player 2. Further, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results to a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable towards a specific target value, and Player 2 has a similar objective with a different target value. We provide, for the first time, an analytical characterization of the feedback Nash equilibrium in a linear-quadratic differential game with impulse control. We illustrate our results using numerical experiments.

Feedback Nash Equilibria in Differential Games with Impulse Control

TL;DR

This work addresses a finite-horizon, two-player nonzero-sum differential game with asymmetric controls: Player 1 uses piecewise-continuous controls while Player 2 employs impulse actions. It develops a verification framework that couples the Hamilton-Jacobi-Bellman equation for the continuous player with quasivariational inequalities for the impulse player to obtain feedback Nash equilibria, and proves a finite upper bound on the number of impulses. A key contribution is the complete analytical characterization of the feedback Nash equilibrium in a scalar linear-quadratic differential game with impulse control, including explicit forms for the optimal feedback laws and impulse rules under threshold-based continuation sets. Numerical examples illustrate the equilibria, state trajectories, and the time-consistency of the strategies, highlighting practical implications for regulation and security applications where discrete interventions interact with continuous dynamics.

Abstract

We study a class of deterministic finite-horizon two-player nonzero-sum differential games where players are endowed with different kinds of controls. We assume that Player 1 uses piecewise-continuous controls, while Player 2 uses impulse controls. For this class of games, we seek to derive conditions for the existence of feedback Nash equilibrium strategies for the players. More specifically, we provide a verification theorem for identifying such equilibrium strategies, using the Hamilton-Jacobi-Bellman (HJB) equations for Player 1 and the quasi-variational inequalities (QVIs) for Player 2. Further, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results to a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable towards a specific target value, and Player 2 has a similar objective with a different target value. We provide, for the first time, an analytical characterization of the feedback Nash equilibrium in a linear-quadratic differential game with impulse control. We illustrate our results using numerical experiments.

Paper Structure

This paper contains 17 sections, 4 theorems, 91 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review
  3. Model
  4. Feedback Nash equilibrium
  5. Strategy of Player 1 and Player 2
  6. Verification theorem
  7. Continuation and intervention set
  8. A scalar linear-quadratic differential game with targets
  9. Optimal control problem of Player 1
  10. Impulse control problem of Player 2
  11. Intervention set and continuation set
  12. Numerical examples
  13. Conclusions
  14. Appendix
  15. Proof of Proposition \ref{['FBNEprop_bound_number']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let Assumptions assum:Vi-FBNEassum:lipschitz hold. Suppose there exist functions $V_i:[0,T]\times \mathbb{R}^n\rightarrow \mathbb{R} (i=1,2)$ such that $V_1(t,x)$ satisfies the HJB equations HJB_all and $V_2(t,x)$ satisfies the QVIs eq:QVI_all for all $(t,x)\in \Sigma$. Suppose there exist strategie Then, $(\gamma^*,\delta^*)$ is a FNE of the differential game described by (FBNEcont_state:strategi

Figures (4)

  • Figure 1: Evolution of the intervention region for the parameters in Table \ref{['FBNEtab:Parameters']}.
  • Figure 2: Value function for the parameters in Table \ref{['FBNEtab:Parameters']}.
  • Figure 3: Evolution of the intervention region for the parameters in Table \ref{['FBNEtab:Parameters']} with $w_2=1$.
  • Figure 4: Value function for the parameters in Table \ref{['FBNEtab:Parameters']} with $w_2=1$.

Theorems & Definitions (19)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Definition 2
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Definition 3
  • ...and 9 more