The authors consider the control system: \(\overset{.}{x}=f(x,u,t)\), \( x(t_{0})=x_{0}\), \(u(t)\in U(t)\), \(t\in \lbrack t_{0},t_{1}]\), where \(t\) is the time, \(x\in \mathbb{R}^{n}\) a state variable, \(x_{0}\) is fixed, and the control \(u(t)\in U(t)\) for a.a. \(t\in \lbrack t_{0},t_{1}]\). The set-valued mapping \(U:\mathbb{R}\rightrightarrows \mathbb{R}^{s}\) is measurable and essentially bounded, and the mapping \(f:\mathbb{R}^{n}\times \mathbb{R} ^{s}\times \lbrack t_{0},t_{1}]\rightarrow \mathbb{R}^{n}\) is continuous, the mapping \(f(\cdot ,u,t)\) being continuously differentiable for every \(u\) and \(t\). A pair of functions \((x(\cdot ),u(\cdot ))\) is an admissible process if \(x(t)\) is a solution to the Cauchy problem and \(u(\cdot )\) is an admissible control, that is \(u\) is measurable and essentially bounded on \( [t_{0},t_{1}]\), and \(u(t)\in U(t)\) for a.a. \(t\in \lbrack t_{0},t_{1}]\). The authors consider the optimization problem for the functional: \( J(u)=\int_{t_{0}}^{t_{1}}f_{0}(x(t),u(t),t)dt+\psi _{0}(x_{1})\rightarrow \mathrm{extr}\), over the set of all admissible pairs \((x(\cdot ),u(\cdot ))\) satisfying the transversality condition \(\psi _{1}(x(t_{1}))=0\), where \(\psi _{1}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{k_{1}}\) is a continuously differentiable mapping and \(k_{1}\geq 0\) a non-negative integer. In the expression of \(J\), the function \(f_{0}\) satisfies the same smoothness conditions as \(f\) and the function \(\psi _{0}\) is continuously differentiable. The admissible process \((\widehat{x}(\cdot ),\widehat{u} (\cdot ))\) satisfies the controllability condition if there exist \(\delta ,C>0\) such that for every \(e=(e_{1},e_{0})\in \mathbb{R}^{k_{1}}\times \mathbb{R}\) satisfying the inequality \(\left\vert e_{1}\right\vert +\left\vert e_{0}-J(\widehat{u})\right\vert \leq \delta \) there exists an admissible process \((x(\cdot ),u(\cdot ))\) such that \(\psi _{1}(x(t_{1}))=e_{1}\), \(J(u)=e_{0}\), and \(\rho (u,\widehat{u})=\mathrm{meas}\{t\in \lbrack t_{0},t_{1}]:\widehat{u}(t)\neq u(t)\}\leq C(\left\vert e_{1}\right\vert +\left\vert e_{0}-J(\widehat{u})\right\vert )\). The admissible process \((\widehat{x}(\cdot ),\widehat{u}(\cdot ))\) satisfies the maximum principle if there exists \((\lambda _{0},\lambda _{1})\neq 0\) in \( \mathbb{R}\times \mathbb{R}^{k_{1}}\) such that the transversality condition \( p(t_{1})=-\frac{\partial l}{\partial x_{1}}(\lambda _{0},\lambda _{1}, \widehat{x}(t_{1}))\), holds and the condition of maximum of the Hamiltonian with respect to \(u\): \(H(\lambda _{0},p(t),\widehat{x}(t),\widehat{u} (t),t)=\max_{u\in U(t)}H(\lambda _{0},p(t),\widehat{x}(t),u,t)\) for a.a. \( t\in \lbrack t_{0},t_{1}]\) holds, the Hamiltonian \(H\) being defined by: \( H(\lambda _{0},p,x,u,t)=\lambda _{0}f_{0}(x,u,t)+\left\langle p,f(x,u,t)\right\rangle \). Here \(l(\lambda _{0},\lambda _{1},x_{1})=\lambda _{0}\psi _{0}(x_{1})+\left\langle \lambda _{1},\lambda \psi _{1}(x_{1})\right\rangle \), \(p(t)\) is an absolutely continuous solution to the linear (with respect to \(p\)) and nonhomogeneous equation: \(\overset{.}{p} =-\frac{\partial H}{\partial x}(\lambda _{0},p(t),\widehat{x}(t),\widehat{u} (t),t)=-p(t)\frac{\partial f}{\partial x}(\widehat{x}(t),\widehat{u} (t),t)+\lambda _{0}\frac{\partial f_{0}}{\partial x}(\widehat{x}(t),\widehat{ u}(t),t)\). The main result of the paper proves that if the admissible process \((\widehat{x}(\cdot ),\widehat{u}(\cdot ))\) does not satisfy the maximum principle, i.e. for every \(\lambda =(\lambda _{0},\lambda _{1})\neq 0 \) and for the corresponding solution \(p\) to the adjoint system, the condition of maximum of the Hamiltonian fails over a set of positive measure, then this process satisfies the controllability condition. For the proof, the authors introduce a finite-dimensional approximation of the optimal control problem, a countable set of admissible controls \( \{u_{i}(\cdot )\}\) such that the set \(\{u_{1}(t),u_{2}(t),\ldots \}\) is everywhere dense in \(U(t)\) for a.a. \(t\in \lbrack t_{0},t_{1}]\). They recall the definition of an approximative continuity point for a given function \(\varphi :[t_{0},t_{1}]\rightarrow \mathbb{R}^{n}\) and they draw computations in finite-dimensional spaces. The paper ends with the description of two examples.