On optimality of singular controls in an optimal control problem

In this paper, a Moskalenko type optimal control problem is considered. We consider the optimal control problem of minimizing the terminal type functional
$$ \mathrm{S(u,v)}=\varphi(y(x_1))+\int_{x_0}^{x_1}G(x,z(t_1,x))dx, $$
under constraints
\begin{gather*} u(t,x)\in U\subset R^r, \quad (t,x)\in D=[t_0,t_1]\times[x_0,x_1],\\ v(x)\in V\subset R^q,\quad x\in X=[x_0,x_1],\\ z_t(t,x)=f(t,x,z(t,x),u(t,x)),\quad (t,x)\in D,\\ z(t_0,x)=y(x),\quad x\in X,\\ y(x_0)=y_0. \end{gather*}

Here, $f (t,x,z,u)$ ($g (x,y,v)$) is an $n$-dimensional vector function which is continuous on the set of variables, together with partial derivatives with respect to $z (y)$ up to second order, $t_0, t_1, x_0, x_1$ ($t_0<t_1$, $x_0<x_1$) are given, $\varphi(y)$ ($G(x,z)$) is a given twice continuously differentiable with respect to $y(z)$ scalar function, $U (V)$ is a given nonempty bounded set, and $u(t, x)$ is an $r$-dimensional control vector function piecewise continuous with respect to $t$ and continuous with respect to $x$, $v(x)$ is a $q$-dimensional piecewise continuous vector of control actions.
The necessary optimality conditions for singular controls in the sense of the Pontryagin maximum principle have been obtained.