In the context of future Permanent Magnet Synchronous Machines (PMSMs) with a high number of phases (>7) in integrated drives, this paper proposes several control strategies when multiphase PMSMs with non-sinusoidal back electromotive forces (back-EMFs) operate in healthy and open-circuit faults. In all operation modes, the considered constraint on current is related to the maximum root mean square (RMS) current allowable in one phase of the machine. The constraint on voltage limits the maximum peak value of the phase voltage determined by the DC-bus voltage of the converter. When one or two phases are open-circuited, to maximize torque and respect the constraints, new current references obtained by several proposed methods in rotating and natural frames are imposed to the machine. Due to the non-sinusoidal waveform of back-EMFs and the considered constraints, numerical computations based on analytical formulations are required to obtain maximal torque-speed characteristics, including the flux-weakening operation. The usefulness of the proposed strategies is verified by numerical and experimental results.