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The modular multilevel converter (MMC) has become one of the most promising candidates for medium-/high-power applications, specifically for high-voltage direct current transmission systems. This paper proposes a control strategy for the MMC based on the sum-of-squares decomposition method. A bilinear mathematical discrete model of the MMC is derived in the rotating reference frame by including the dynamics of ac-side currents, circulating currents, dc-side current, and stored energy of the converter. The coordinates of the developed model are transformed to the origin, and the steady-state operating point of the MMC is calculated based on the desired reference values of the states. A quadratic Lyapunov function is defined and by converting the Lyapunov difference inequality to an SOS problem, a stabilizing controller is designed. The designed controller not only guarantees the Lyapunov stability of the states, but also controls the external and internal dynamics of the MMC with only one controller, instead of using several proportional-integral controllers. To improve the region of stability of the MMC, an optimization problem is formulated and solved, so the controller can stabilize the MMC from an initial operating point far from the steady-state operating point. The performance of the proposed strategy for a 20-level MMC is evaluated based on the time-domain simulation studies and is compared with the existing control methods. [ABSTRACT FROM PUBLISHER]