The following optimal control problem is considered: \[ (P)\quad \min imize\quad C(z)\quad subject\quad to\quad M(z)=0,\quad z\in {\mathcal Z}, \] where \({\mathcal Z}\) denotes the set of pairs \(z=(x,u)\) such that x: [0,1]\(\to {\mathbb{R}}^ n\) is absolutely continuous and u: [0,1]\(\to {\mathbb{R}}^ m\) is summable, and \(M(x,u)(t)=x'(t)-A(t)x(t)-B(t)u(t)\) where A and B are appropriate matrix-valued functions. It is assumed that all other possible constraints are embedded in the cost functional by setting \(C(z)=\infty\) when the constraint is violated. Next, the dual problem of (P) is considered: \[ (D)\quad \max imize\quad L(p)\quad subject\quad to\quad p\in {\mathcal L}^{\infty} \] where \({\mathcal L}^{\infty}\) is the space of essentially bounded functions from [0,1] into \({\mathbb{R}}^ n\), and \(L(p)=\inf \{C(z)+:\quad z\in {\mathcal Z}\}\) where \(\) stands for \(\int^{1}_{0}f(t)g(t)dt\). If C is convex and satisfies a certain boundedness hypothesis, then there exists a solution p to (D) and \(L(p)=\inf \{C(z):\quad z\in {\mathcal Z},\quad M(z)=0\}.\) Under a certain ''interiority assumption'' on C, it is proved that each \(p\in {\mathcal L}^{\infty}\) satisfying \(L(p)>-\infty\) is equivalent (i.e. equal almost everywhere) to a function with bounded variation. Moreover, for some control problems without state constraints, any such p is absolutely continuous provided it is continuous at 0 and 1. For state constrained problems with convex cost functional, one can establish the absolute continuity of \(p+\omega\) where \(\omega\) is a multiplier associated with the state constraint. Further, the following approximation to the dual problem is introduced: \[ (D_ h)\quad \max imize\quad L(p)\quad subject\quad to\quad p\in {\mathcal S}_ h \] where \({\mathcal S}_ h\) is a closed subset of a finite dimensional subspace of \({\mathcal L}^{\infty}\). It is shown that problem \((D_ h)\) has solutions if C satisfies the boundedness assumption (but without the convexity). If \(p_ h\) solves \((D_ h)\), one can take as an approximation to a solution to (P) any \(z_ h\in {\mathcal Z}\) for which \[ (1)\quad L(p_ h)=C(z_ h)+. \] In the further part of the paper the authors consider the case when \[ C(x,u)=e(x(0),x(1))+\int^{1}_{0}f(x(t),u(t),t)dt \] where \(e: {\mathbb{R}}^{2n}\to {\mathbb{R}}\cup \{\infty \}\) and f is a normal integrand (in the sense of R. T. Rockafellar). The evaluation of the dual functional L in terms of conjugate functions \(e^*\) and \(f^*\) is given and the relations between solutions to (P) and (D) are examined. Under the assumption that f(\(\cdot,t)\) is uniformly convex, some estimates for \(\| z-z_ h\|^ 2\) are proved (here \(\| \cdot \|\) is the \({\mathcal L}^ 2\) norm, z is a solution to (P) and \(z_ h\) satisfies (1)). Especially, the case of piecewise polynomial approximations of degree 1 and 2 is considered. Finally, an algorithm for solving \((D_ h)\) is constructed and a local quadratic convergence theorem for this algorithm is proved.