###### Abstract
More precisely, in this paper, we consider a Stackelberg finite horizon linear-quadratic differential game with two players (a leader and a follower), and with a small cost of the leader's control in both cost functionals. We study an asymptotic behavior of an open-loop solution to this game. For this purpose, we consider the boundary-value problem associated with the Stackelberg game by the solvability conditions. Due to the small cost of the leader's control, the aforementioned boundary-value problem is singularly perturbed in the conditionally stable case. An asymptotic solution to this problem is formally constructed and justified. Based on this solution, asymptotic expansions of the optimal controls of the leader and the follower, as well as asymptotic expansions of the optimal values of the cost functionals, are constructed. Using these results, suboptimal controls of the players are formally designed and justified. Note that a cheap control problem is closely related with a control problem for a singularly perturbed system. Namely, the former can be converted to the latter by a proper change in the control variable with the small (cheap) cost (see, e.g., [46; 47; 48; 49]). Linear-quadratic Stackelberg differential games for singularly perturbed systems were studied in the works [50; 51; 52; 53; 54; 55]. However, in these works (in contrast with the present paper), the infinite horizon version of the game was considered, and a suboptimal closed-loop (feedback) solution to the considered game was derived. It should also be noted that the asymptotic method of solution for the considered cheap control Stackelberg differential game is a kind of an iterations' method. Another iterations' methods and their applications to approximate solutions to optimal control problems and differential games can be found in [32; 56; 57; 58; 59; 60; 61; 62; 63] and the references therein. This paper is organized as follows. In Section 2, the Stackelberg cheap control game is rigorously formulated. The main notions and assumptions are introduced. The objectives of the paper are stated. In Section 3, the solvability conditions of the considered game are presented. An asymptotic solution to the singularly perturbed boundary-value problem, appearing in these solvability conditions, is formally constructed, and justified in Section 4. Section 5 is devoted to obtaining the main results of the paper. In this section, the asymptotic expansion of the game's solution is formally derived and justified. Also, the suboptimal players' controls are formally designed and justified. In Section 6, an illustrative example is presented. Conclusions are given in Section 7. The following main notations are applied in the paper:  
1. \(E^{n}\) denotes the \(n\)-dimensional real Euclidean space;
2. \(\|\cdot\|\) denotes the Euclidean norm either of a vector (\(\|z\|\)) or of a matrix (\(\|A\|\));
3. The superscript \"\(T\)\" denotes the transposition either of a vector (\(z^{T}\)) or of a matrix (\(A^{T}\));
4. \(I_{n}\) denotes the identity matrix of dimension \(n\);
5. \(\text{col}\left(x_{1},x_{2},\ldots,x_{k}\right)\), where \(x_{1}\in E^{n_{1}}\), \(x_{2}\in E^{n_{2}}\),..., \(x_{k}\in E^{n_{k}}\), denotes the column block-vector of the dimension \(n_{1}+n_{2}+\cdots+n_{k}\) with the upper block \(x_{1}\), the next block \(x_{2}\), and so on, and the lower block \(x_{k}\);
6. \(\text{Re}(\mu)\) denotes the real part of a complex number \(\mu\).