## 2 Problem Statement  
### Initial Game Formulation  
The following differential equation describes the dynamics of the game:  
\[\frac{dZ(t)}{dt}=\mathcal{A}(t)Z(t)+\mathcal{B}_{\text{u}}(t)u(t)+\mathcal{B} _{\text{v}}(t)v(t),\ \ t\in[0,t_{f}],\ \ \ Z(0)=Z_{0}, \tag{1}\]  
where \(Z(t)\in E^{n}\) is the state vector, \(u(t)\in E^{r}\), (\(r<n\)), \(v(t)\in E^{s}\), (\(s\leq n\)) are the players' controls; \(t_{f}>0\) is a given time-instant; \(\mathcal{A}(t)\) and \(\mathcal{B}_{i}(t)\), (\(i=\text{u},\text{v}\)), are given matrix-valued functions of corresponding dimensions continuous for \(t\in[0,t_{f}]\); \(Z_{0}\in E^{n}\) is a given vector. In what follows, we assume:  
**(A1)** For any \(t\in[0,t_{f}]\), the matrix \(\mathcal{B}_{\text{u}}(t)\) has full column rank \(r\). The cost functionals of the first player (the player with the control \(u(t)\) is called a leader) and the second player (the player with the control \(v(t)\) is called a follower) are, respectively,  
\[\mathcal{J}_{\text{u}}(u,v)=\frac{1}{2}\int_{0}^{t_{f}}\big{[}Z^{T}(t)\mathcal{ D}_{\text{u}}(t)Z(t)+\epsilon^{2}u^{T}(t)G_{\text{u},\text{u}}(t)u(t)+v^{T}(t)G_{ \text{u},\text{v}}(t)v(t)\big{]}dt, \tag{2}\]  
and  
\[\mathcal{J}_{\text{v}}(u,v)=\frac{1}{2}\int_{0}^{t_{f}}\big{[}Z^{T}(t)\mathcal{ D}_{\text{v}}(t)Z(t)+v^{T}(t)G_{\text{v},\text{v}}(t)v(t)+\epsilon^{2}u^{T}(t)G_{ \text{v},\text{u}}(t)u(t)\big{]}dt, \tag{3}\]  
where, for any \(t\in[0,t_{f}]\), \(\mathcal{D}_{i}(t)\), \(G_{i,j}(t)\), (\(i=\text{u},\text{v}\); \(j=\text{u},\text{v}\)) are given symmetric matrices of corresponding dimensions; the matrix-valued functions \(\mathcal{D}_{i}(t)\), \(G_{i,j}(t)\), (\(i=\text{u},\text{v}\); \(j=\text{u},\text{v}\)) are continuous for \(t\in[0,t_{f}]\); \(\epsilon>0\) is a small parameter. We assume that:  
**(A2)** For any \(t\in[0,t_{f}]\), the matrices \(\mathcal{D}_{i}(t)\), (\(i=\text{u},\text{v}\)) are positive semi-definite. **(A3)** For any \(t\in[0,t_{f}]\), the matrices \(G_{\text{u},\text{u}}(t)\), \(G_{\text{v},\text{v}}(t)\), \(G_{\text{u},\text{v}}(t)\) and \(G_{\text{v},\text{u}}(t)\) are positive definite. **Remark 1**. : _In what follows, for the sake of technical simplicity (but without loss of generality), we set \(G_{\mathrm{u,u}}(t)\equiv I_{r}\), \(G_{\mathrm{v,v}}(t)\equiv I_{s}\), \(t\in[0,t_{f}]\)._  
We are going to solve the game, given by differential Equation (1) and the cost Functionals (2) and (3), with respect to the open-loop Stackelberg (hierarchical) equilibrium (solution) \(\big{(}u^{*}(t),v^{*}(t)\big{)}\). This equilibrium is defined as follows (for more details see, e.g., [39; 64; 65]). First of all, let us note that we look for the Stackelberg solution to the game (1)-(3) in the set of all pairs \(\big{(}u(t),v(t)\big{)}\), \(t\in[0,t_{f}]\), \(u(t)\in E^{r}\), \(v(t)\in E^{r}\), and the vector-valued functions \(u(t)\) and \(v(t)\) are continuous in the interval \([0,t_{f}]\). Now, for any aforementioned \(u(t)\), let  
\[v^{0}\big{(}t;u(t)\big{)}\stackrel{{\triangle}}{{=}}\mathrm{ argmin}_{v(t)}\mathcal{J}_{\mathrm{v}}\big{(}u(t),v(t)\big{)} \tag{4}\]  
along the trajectories of (1). Furthermore, let  
\[u^{*}(t)\stackrel{{\triangle}}{{=}}\mathrm{ argmin}_{u(t)}\mathcal{J}_{\mathrm{u}}\Big{(}u(t),v^{0}\big{(}t;u(t)\big{)}\Big{)} \tag{5}\]  
along the trajectories of (1) with \(v(t)=v^{0}\big{(}t;u(t)\big{)}\). Moreover, let  
\[v^{*}(t)\stackrel{{\triangle}}{{=}}v^{0}\big{(}t;u^{*}(t)\big{)}. \tag{6}\]  
Finally, the pair \(\big{(}u^{*}(t),v^{*}(t)\big{)}\) is the open-loop Stackelberg solution to the game (1)-(3). We call the trajectory of differential Equation (1), generated by the pair of controls \(\big{(}u(t)=u^{*}(t),v(t)=v^{*}(t)\big{)}\), the Stackelberg optimal trajectory of the game (1)-(3).