One Class of Stackelberg Linear–Quadratic Differential Games with Cheap Control of a Leader: Asymptotic Analysis of an Open-Loop Solution

We consider a two-player finite horizon linear–quadratic Stackelberg differential game. For this game, we study the case where the control cost of a leader in the cost functionals of both players is small, which means that the game under consideration is a cheap control game. We look for open-loop optimal players’ controls of this game. Using the game’s solvability conditions, the obtaining such controls is reduced to the solution to a proper boundary-value problem. Due to the smallness of the leader’s control cost, this boundary-value problem is singularly perturbed. Asymptotic behavior of the solution to this problem is analyzed. Based on this analysis, the asymptotic behavior 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 is presented.