Necessary optimality conditions in the one boundary control problem for Qoursat–Darboux systems

In this paper, a boundary optimal control problem described by the Goursat–Darboux system is considered under the assumption that the control domain is open.
We consider the problem of minimizing of the functional
$$ I(u)=\varphi(a(t_1))+G(z(t_1,x_1)), $$
under constraints
\begin{gather*} u(t)\in U\subset R^r, \quad t\in T=[t_0,t_1],\\ z_{tx}=B(t,x)z_t+f(t, x, z, z_x), \quad(t, x)\in D=[t_0, t_1]\times[x_0, x_1],\\ z(t,x_0)=a(t), \quad t\in T=[t_0, t_1],\\ z(t_0, x)=b(x), \quad x\in X=[x_0,x_1],\\ a(t_0)=b(x_0)=a_0,\\ \dot{a}=g(t,a,u),\quad t\in T,\\ a(t_0)=a_0. \end{gather*}

Here, $f(t,x,z,z_x)$ is a given $n$-dimensional vector-function which is continuous with respect to set of variables, together with partial derivatives with respect to $z,z_x$ up to second order, $B(t,x)$ is a given measurable and bounded matrix function, $b(x)$ is a given $n$-dimensional absolute continuous vector-valued function, $t_0, t_1, x_0, x_1$ ($t_0<t_1; x_0<x_1$) are given, $a_0$ a is a given constant vector, $g(t,a,u)$ given $n$-dimensional vector-function which is continuous with respect to the set of variables together with partial derivatives with respect to $(a,u)$ up to second order, $\varphi(a)$ and $G(z)$ are given twice continuously differentiable scalar functions, $U$ is a given nonempty, bounded, and open set, and $u(t)$ is a measurable and bounded $r$-dimensional control vector-function.
The first and second order necessary conditions of optimality are established.