This paper presents the use of the memristor as a new element for designing passivity-based controllers. From the port-Hamiltonian description of the electrical circuits with memristors, a target dynamics is assigned to the matching equation proposed by the methodology known as Interconnection and Damping Assignment-Passivity-Based Control. The inclusion of the memristor element extends the closed-loop dynamics and it results in an extra term in the control algorithm that can be seen as a state-modulated gain. Two mechanical examples, in the form of position control systems, are included to show possible applications.