Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

FlipDyn with Control: Resource Takeover Games with Dynamics

Sandeep Banik, Shaunak D. Bopardikar

TL;DR

FlipDyn-Con models a finite-horizon, two-player zero-sum game of dynamic resource takeovers in a discrete-time hybrid system, where defender and adversary alternate control through takeover actions. It develops a DP-like recursion for the saddle-point value with cost-to-go matrices and derives NE takeover strategies for general systems. In the LQ setting, it provides parameterized linear state-feedback policies coupled via a scalar η_k, with exact scalar solutions and constructive bounds for higher dimensions to obtain approximate NE values. Numerical experiments on adversarial control of linear systems demonstrate the benefits of the synthesized NE policies over LQR and threshold-based baselines, and illustrate how system stability affects takeover incentives and policy switching.

Abstract

We introduce FlipDyn with control, a finite-horizon zero-sum resource takeover game, where a defender and an adversary decide when to takeover and how to control a common resource. At each discrete-time step, the players can take over or retain control, incurring state and control-dependent costs. The system is modeled as a hybrid dynamical system, with a discrete \texttt{FlipDyn} state determining control authority. Our contributions are: (i) For arbitrary non-negative costs, we derive the saddle-point value of the \texttt{FlipDyn} game and the corresponding Nash equilibria (NE) takeover strategies. (ii) For linear dynamical systems with quadratic costs, we establish sufficient conditions under which the game admits an NE. (iii) For scalar linear dynamical systems with quadratic costs, we derive parameterized NE takeover strategies and saddle-point values independent of the continuous state. (iv) For higher-dimensional linear dynamical systems with quadratic costs, we derive approximate NE takeover strategies and control policies, and compute bounds on the saddle-point values. We validate our results through a numerical study on adversarial control of a linear system.

FlipDyn with Control: Resource Takeover Games with Dynamics

TL;DR

FlipDyn-Con models a finite-horizon, two-player zero-sum game of dynamic resource takeovers in a discrete-time hybrid system, where defender and adversary alternate control through takeover actions. It develops a DP-like recursion for the saddle-point value with cost-to-go matrices and derives NE takeover strategies for general systems. In the LQ setting, it provides parameterized linear state-feedback policies coupled via a scalar η_k, with exact scalar solutions and constructive bounds for higher dimensions to obtain approximate NE values. Numerical experiments on adversarial control of linear systems demonstrate the benefits of the synthesized NE policies over LQR and threshold-based baselines, and illustrate how system stability affects takeover incentives and policy switching.

Abstract

We introduce FlipDyn with control, a finite-horizon zero-sum resource takeover game, where a defender and an adversary decide when to takeover and how to control a common resource. At each discrete-time step, the players can take over or retain control, incurring state and control-dependent costs. The system is modeled as a hybrid dynamical system, with a discrete \texttt{FlipDyn} state determining control authority. Our contributions are: (i) For arbitrary non-negative costs, we derive the saddle-point value of the \texttt{FlipDyn} game and the corresponding Nash equilibria (NE) takeover strategies. (ii) For linear dynamical systems with quadratic costs, we establish sufficient conditions under which the game admits an NE. (iii) For scalar linear dynamical systems with quadratic costs, we derive parameterized NE takeover strategies and saddle-point values independent of the continuous state. (iv) For higher-dimensional linear dynamical systems with quadratic costs, we derive approximate NE takeover strategies and control policies, and compute bounds on the saddle-point values. We validate our results through a numerical study on adversarial control of a linear system.
Paper Structure (18 sections, 9 theorems, 131 equations, 4 figures)

This paper contains 18 sections, 9 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. FlipDyn-Con for General Systems
  4. Saddle-point value
  5. NE takeover strategies of the FlipDyn game
  6. FlipDyn-Con for LQ Problems
  7. Control policy for the FlipDyn-Con LQ Problem
  8. Scalar dynamical system
  9. $n-$dimensional system
  10. Conclusion and Future Directions
  11. Proof of Theorem 1
  12. Proof of Theorem 2
  13. Proof of Theorem 3
  14. Proof of Proposition 1
  15. Proof of Corollary 1
  16. ...and 3 more sections

Key Result

Theorem 1

(Case $\alpha_k = 0$) Under Assumption ast:general_costs, for a fixed pair of control policies, $u_{\mathbf{L}}$ and $w_{\mathbf{L}}$, the FlipDyn-Con game eq:opti_E_cost governed by the continuous state dynamics eq:Flip_dynamics and FlipDyn dynamics eq:FlipDyn_compact, admits a unique pair of NE ta where ${\check{\Xi}_{k+1}} := V^{1}_{k+1}(F_k^{1}(x,w_k),w_{k+1},\Xi_{k+2}^{1}) - V^{0}_{k+1}(F_k^{

Figures (4)

  • Figure 1: Saddle-point value $V^{0}$ and $V^{0}_\text{LQR}$ (Defender LQR control law) for state transition coefficient (a) $E = 0.85$, (b) $E = 1.0$ starting with $\texttt{FlipDyn}$ state $\alpha_{0} = 0$.
  • Figure 2: Saddle-point value $V^{0}$ and $V^{0}_{\delta}$ ($\delta = 0.1,0.2,0.4$ and $0.6$) for state transition coefficient (a) $E = 0.85$, (b) $E = 1.0$ starting with $\texttt{FlipDyn}$ state $\alpha_{0} = 0$.
  • Figure 3: Saddle-point value parameters $\mathbf{p}_{k}^{i}, k \in \{1,2,\dots,L\}, i \in \{0,1\}$ for state transition constant (a) $E = 0.85$, (c) $E = 1.0$. The parameters $\mathbf{p}_{k}^{i},$M-NE corresponds to the parameters of the saddle-point under a mixed NE takeover over the entire time horizon. Defender takeover strategies $\beta_{k}$ and adversary takeover strategies $\gamma_{k}$ for state transition (b) $E = 0.85$ and (d) $E = 1.0$. M-NE corresponds to the mixed NE policy.
  • Figure 4: Maximum eigenvalues ($\lambda_{1}(\overline{P}_{k}^{\alpha})$) of saddle-point value parameters $\overline{P}_{k}^{\alpha}, k \in \{0,1,\dots,L+1\}, \alpha \in \{0,1\}$ for state transition constant (a) $e = 0.85$, (c) $e = 1.0$. The parameters $P_{k}^{i},$M-NE corresponds to saddle-point value parameter recursion under a mixed NE takeover over the entire time horizon. Defender takeover strategy $\beta_{k}$ and adversary takeover strategy $\gamma_{k}$ for state transition (b) $e = 0.85$ and (d) $e = 1.0$. M-NE corresponds to the mixed NE policy.

Theorems & Definitions (24)

  • Definition 1: Nash Equilibrium bacsar1998dynamic
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Corollary 1
  • Remark 2
  • Remark 3
  • Corollary 2
  • ...and 14 more