Optimal control of the inversion of two spins in Nuclear Magnetic Resonance

Highlights: ► We investigate the simultaneous optimal control of the inversion of two spins. ► We examine the energy minimum solution. ► We compare this solution with the time-minimum one. ► Experimental implementation using techniques of Nuclear Magnetic Resonance. -- Abstract: We investigate the optimal control of the inversion of two spin 1/2 particles in Nuclear Magnetic Resonance. The two spins, which differ by their resonance offset, are controlled by the same radio frequency magnetic field. Using the Pontryagin Maximum Principle, we compute the optimal control sequence which allows to reach the target state in a given time, while minimizing the energy of the magnetic field. A comparison with the time-optimal solution for bounded control amplitude realizing the same control in the same time is made. An experimental illustration is done using techniques of Nuclear Magnetic Resonance.