  In this article we study optimal control problems for systems that are affine
with respect to some of the control variables and nonlinear in relation to the
others. We consider finitely many equality and inequality constraints on the
initial and final values of the state. We investigate singular optimal
solutions for this class of problems, for which we obtain second order
necessary and sufficient conditions for weak optimality in integral form. We
also derive Goh pointwise necessary optimality conditions. We show an example
to illustrate the results.
