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Semidefinite Programming Duality in Infinite-Horizon Linear Quadratic Differential Games

Yuto Watanabe, Chih-Fan Pai, Yang Zheng

TL;DR

The paper develops a primal–dual SDP framework for infinite-horizon continuous-time LQ differential games, using Gramian representations to convert trajectory energies into finite-dimensional variables. Under a stabilizing solution to the ARE and a regularity condition, it proves that a saddle point is attained by linear static policies, and extends the approach to $\mathcal{H}_\infty$ suboptimal control. A key insight is that the ARE arises as the dual of the supporting SDPs, enabling a clean KKT-based construction of the Nash equilibrium and revealing the role of duality in noncooperative settings. The work provides a novel, constructive SDP-based proof of the saddle-point property and offers a versatile analytical tool for both LQ games and robust $\mathcal{H}_\infty$ control with potential practical implications for synthesis and verification.

Abstract

Semidefinite programs (SDPs) play a crucial role in control theory, traditionally as a computational tool. Beyond computation, the duality theory in convex optimization also provides valuable analytical insights and new proofs of classical results in control. In this work, we extend this analytical use of SDPs to study the infinite-horizon linear-quadratic (LQ) differential game in continuous time. Under standard assumptions, we introduce a new SDP-based primal-dual approach to establish the saddle point characterized by linear static policies in LQ games. For this, we leverage the Gramian representation technique, which elegantly transforms linear quadratic control problems into tractable convex programs. We also extend this duality-based proof to the $\mathcal{H}_\infty$ suboptimal control problem. To our knowledge, this work provides the first primal-dual analysis using Gramian representations for the LQ game and $\mathcal{H}_\infty$ control beyond LQ optimal control and $\mathcal{H}_\infty$ analysis.

Semidefinite Programming Duality in Infinite-Horizon Linear Quadratic Differential Games

TL;DR

The paper develops a primal–dual SDP framework for infinite-horizon continuous-time LQ differential games, using Gramian representations to convert trajectory energies into finite-dimensional variables. Under a stabilizing solution to the ARE and a regularity condition, it proves that a saddle point is attained by linear static policies, and extends the approach to suboptimal control. A key insight is that the ARE arises as the dual of the supporting SDPs, enabling a clean KKT-based construction of the Nash equilibrium and revealing the role of duality in noncooperative settings. The work provides a novel, constructive SDP-based proof of the saddle-point property and offers a versatile analytical tool for both LQ games and robust control with potential practical implications for synthesis and verification.

Abstract

Semidefinite programs (SDPs) play a crucial role in control theory, traditionally as a computational tool. Beyond computation, the duality theory in convex optimization also provides valuable analytical insights and new proofs of classical results in control. In this work, we extend this analytical use of SDPs to study the infinite-horizon linear-quadratic (LQ) differential game in continuous time. Under standard assumptions, we introduce a new SDP-based primal-dual approach to establish the saddle point characterized by linear static policies in LQ games. For this, we leverage the Gramian representation technique, which elegantly transforms linear quadratic control problems into tractable convex programs. We also extend this duality-based proof to the suboptimal control problem. To our knowledge, this work provides the first primal-dual analysis using Gramian representations for the LQ game and control beyond LQ optimal control and analysis.

Paper Structure

This paper contains 17 sections, 6 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem formulation
  3. LQ differential game problem
  4. Algebraic Riccati equation and saddle point
  5. Problem statement: SDP duality in LQ games
  6. SDP Duality and Gramian Representation in Control
  7. Duality in SDPs
  8. Gramian representation in control problems
  9. Nash Equilibrium via SDP duality
  10. Step (a): SDPs as upper and lower bounds
  11. Step (b): Equal upper and lower values by SDP duality
  12. Step (c): Establishing saddle point \ref{['eq:u-w-pair']} via KKT analysis
  13. Application to $\mathcal{H}_\infty$ control
  14. $\mathcal{H}_\infty$ control as an LQ game
  15. SDP-based proof for \ref{['theorem:Hinf-control']}
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the zero-sum LQ game eq:LQ-game-value with an initial condition $x(0)\in\mathbb{R}^n$. With assumption:value-game, we have ${\underline{p}}^\star = {\overline{p}}^\star = x_0^{{\mathsf T}} P^\star$ and the pair of linear static policies constitutes a saddle point of the LQ game eq:LQ-game-value.

Figures (1)

  • Figure 1: (a) The landscape of $J_\gamma$ with $q=10$ and the saddle point $(K^*,L^*)$ (denoted by the blue dot); (b) the landscape of $J_\gamma$ with $q=10$ over a broader region; (c) the landscape of $J_\gamma$ with $q=1$. The red and blue dashed lines represent $J_\gamma$ with $L=L^\star$ and $K=K^\star$, respectively. Values are clipped to $200$ wherever $|J_\gamma|>200$. In (c), we can observe that $\min_{K}J_\gamma(Kx,L^\star x)=-\infty <J_\gamma(K^\star x,L^\star x)$ and thus $(K^*,L^*)$ does not constitute a saddle point.

Theorems & Definitions (12)

  • Remark 1: Open- vs closed-loop information structures
  • Theorem 1
  • Lemma III.1
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Remark 2: Role of the ARE in LQ control
  • Remark 3
  • Theorem 2
  • ...and 2 more