The linear pulse system \[ \begin{cases} \dot{x} = A(t)x+C(t)u(t)+f(t),\;t\neq t_{i},\\ x(t_{i})-x(t_{i}+0)=B_{i}x(t_{i})+D_{i}v_{i}+I_{i},\end{cases} \tag{1} \] with the boundary conditions \[ x(\alpha)=a,\quad x(\beta)=b, \tag{2} \] and controls \(u(t),\;\{v_{i}\}_{i=1}^{p}\) is considered. Let \(L_{2}([\alpha,\beta], {\mathbb{R}}^{r})\) be the space of square integrable mappings from \([\alpha,\beta]\subset {\mathbb{R}}\) to \({\mathbb{R}}^{r}\) and \(D^{r}[1,p]\) be the set of sequences \(\{\xi _{i}\in {\mathbb{R}}^{r}\}_{i=\overline{1,p}}\). The scalar product in \(\Pi _{p}^{r}:= L_{2}([\alpha,\beta], {\mathbb{R}}^{r})\times D^{r}[1,p]\) is defined by \[ \left(\{g,\xi\},\{h,\eta\}\right)=\int_{\alpha }^{\beta }(g,h) dt+\sum_{i=1}^{p}(\xi_{i},\eta_{i}). \] The control problem is said to be solvable if, for any \(\{f,I\}\in\Pi _{p}^{n},\;a,b\in{\mathbb{R}}^{n}\), there exists \(\{u,v\}\in\Pi _{p}^{m}\) such that the boundary value problem (1), (2) has a solution. Under the assumption \(\det (Id+B_{i})\neq 0, \;i=\overline{1,p}\), it is proved that the control problem is solvable iff the fundamental matrix \(Y(t)\) of the conjugate system \[ \dot{y}=-A^{T}(t)y,\;t\neq t_{i}, \quad y(t_{i})-y(t_{i}+0)=-(Id+B_{i}^{T})^{-1}B_{i}^{T}y(t_{i}), \] satisfies the conditions \(\det (C^{T}(t)Y(t))\neq 0, \;\det(D_{I}^{T}Y(t_{i}))\neq 0.\) By means of \(Y(t)\) an explicit formula for the control \(\{u,v\}\) is established. The authors also consider the following problem: for a given \(\{f,I\}\) find a control \(\{u,v\}\) which solve (1), (2) with minimal \(\beta >\alpha \).