This series of notes have their origin into accelerated course in optimal control given to multidisciplinary Phd students in Mathematics, Physics or Control Engineering and the purpose is to present the seminal results of this theory in view of applications to concrete problems. For that we choose to make the presentation using two real life applications which are the core of current research projects on the academic side and motivated by industrial applications. The two applications concern the swimming at low Reynolds number and the contrast problem in Magnetic resonance Imaging.