The authors generalize a projected Newton method of Bertsekas to the optimal control problem \[ \text{minimize } \int^T_0 L(x(t), u(t), t)dt \quad \text{over } u\in U= \{u\in L^\infty[0, T]\mid u_{\min}(t)\leq u(t)\leq u_{\max}(t)\} \] such that \(x\in W^{1,\infty}_N[0, T]\) solves \(\dot x(t)= f(x(t), u(t), t)\), \(x(0)= x_0\). Some numerical examples are given.