###### Abstract  
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  
As was mentioned, in this paper, we consider a cheap control differential game. Cheap control differential games were extensively studied in the literature. Thus, in [12; 13; 14; 25; 26; 29; 30; 31; 32; 36] various zero-sum cheap control games were analyzed. Different cheap control Nash equilibrium games were studied in [33; 37; 38]. However, to the best of our knowledge, a cheap control Stackelberg equilibrium differential game has not yet been studied in the literature. It should be noted that, like a zero-sum differential game and a Nash equilibrium differential game, a Stackelberg equilibrium differential game is not a cooperative game. Moreover, like a Nash equilibrium differential game, a Stackelberg equilibrium differential game is a nonzero-sum differential game. In spite of this, a Stackelberg equilibrium differential game differs considerably from a zero-sum differential game and from a Nash equilibrium differential game. Namely, in contrast to a zero-sum differential game and a Nash equilibrium differential game, players in a Stackelberg equilibrium differential game do not act simultaneously. A Stackelberg equilibrium differential game is a hierarchical game. In this game, one of the players (the leader) makes its best decision based on the sets of possible best decisions of the other players (the followers). Then, each of the followers chooses its best decision (from the aforementioned set), which corresponds to the best decision of the leader. Thus, the leader has an advantage in comparison to the followers. More details on a Stackelberg equilibrium differential game can be found in the book [39]. There are different real-life applications of a Stackelberg equilibrium differential game. Thus, such a game is used in analysis and optimization of supply and marketing channels (see, e.g., the work [40] and references therein). In [41; 42], the application of a Stackelberg equilibrium differential game to some production-and-sale problems is considered. Application of a Stackelberg equilibrium differential game to a security (defenders-attackers) problem is presented in [43] (see also the references therein). In [44], the problem of a transponder's communication channel with multiple-user terminals is modeled by a Stackelberg equilibrium differential game. Application of a Stackelberg equilibrium differential game to analysis and optimization of power systems is proposed in [45]. A small cost of the leader's control in its cost functional (a cheap control of the leader) can considerably increase the aforementioned advantage of this player. Opposite, a small cost of the follower's control in its cost functional (a cheap control of the follower) can considerably decrease the leader's advantage. This circumstance can be important; for instance, in a pursuit-evasion Stackelberg equilibrium differential game where the optimization of a state-dependent part of the leader/follower cost functional is much more essential than the optimization of the control energy expenditure part of the cost functional. Such results in the cases of zero-sum differential games and Nash equilibrium differential games were obtained in [13; 25; 29; 31; 32; 33]. The main motivation of this (purely theoretical) paper is to study a novel nontrivial class of cheap control differential games.