Suboptimal open-loop solution of a Stackelberg linear-quadratic differential game with cheap control of a follower: analytical/numerical study

A two-player finite horizon linear-quadratic Stackelberg differential game is considered. The feature of this game is that the control cost of a follower in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. The open-loop solution of this game is studied. Using the game’s solvability conditions, obtaining such a game’s solution is reduced to the solution of a proper boundary-value problem. Due to the smallness of the follower’s control cost, this boundary-value problem is singularly perturbed. The asymptotic behaviour of the solution to this problem is analysed. Based on this analysis, the asymptotic behaviour of the open-loop optimal players’ controls and the optimal values of the cost functionals is studied. Using these results, asymptotically suboptimal players’ controls are designed. An illustrative example of a supply chain problem with a small control cost of a retailer is presented.

💡 Research Summary

This paper investigates a finite‑horizon, two‑player linear‑quadratic (LQ) Stackelberg differential game in which the follower’s control effort is penalized by a very small weight ε². The state dynamics are linear, (\dot Z(t)=A(t)Z(t)+B_u(t)u(t)+B_v(t)v(t)), with the leader controlling u(t)∈ℝʳ and the follower controlling v(t)∈ℝˢ. The leader’s cost functional contains a standard quadratic state term, a quadratic leader‑control term, and a term ε² vᵀG_{uv}v that couples the follower’s control. The follower’s cost contains a quadratic state term, a quadratic follower‑control term weighted by ε², and a coupling term uᵀG_{vu}u. Because ε is assumed to be very small, the game belongs to the class of “cheap‑control” problems, which are known to generate singularly perturbed structures.

The authors first apply a state transformation Z=R_v(t)z that diagonalizes the follower’s input matrix, turning B_v into (\begin{bmatrix}0\I_s\end{bmatrix}) and simplifying the cost matrices. After this change of variables the game is expressed by equations (9)–(11). The Stackelberg open‑loop equilibrium is defined in the usual hierarchical way: for any leader trajectory u(t) the follower solves a standard LQ optimal control problem, yielding a best‑response v₀