Solving a continuous-time optimal control problem under state and control constraints is known to be a very hard task. In this note, we propose a suboptimal solution based on the Euler auxiliary system (EAS). We show that we can determine a continuous-time stabilizing control whose cost not only converges to the optimal as the EAS time parameter vanishes, but it is also upper bounded by the discrete-time cost, no matter how such a parameter is chosen. In particular, continuous-time linear problems with convex cost can be solved by considering a fictitious receding-horizon scheme. Both stability and constraints satisfaction are guaranteed for the continuous-time system. This scheme turns out to be very useful when, due to unstable or poorly damped dynamics, the digital implementation of the control requires a very small (virtually zero) sampling time, since the "time parameter" of the EAS can be much greater than the sampling time, without compromising stability, with a strongly reduced computational burden.