Turnpike property of nonzero-sum linear-quadratic differential games

¹ Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, PR China
² Guangzhou Railway Polytechnic, Guangzhou, Guangdong, 511300, PR China
³ School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, PR China
⁴ Center for Applied Mathematics of Guangxi (GUET), Guilin 541004, PR China

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Received: 8 November 2024
Accepted: 29 December 2025

Abstract

This paper investigates the turnpike properties of deterministic nonzero-sum linearquadratic (LQ) differential games. Under certain assumptions on the Hamiltonian matrix of the nonzero-sum LQ differential game, we establish the solvability of both the coupled non-symmetric differential Riccati equation (DRE) and the algebraic Riccati equation (ARE). Moreover, we identify the convergence relationship between the DRE and ARE, which is essential for understanding the turnpike properties. Over a finite but sufficiently long time horizon, the open-loop Nash equilibrium is shown to remain exponentially close to the solution of a two-objective optimization problem for the majority of the time horizon.