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This paper studies two optimal control problems for a fractional-order pendulum in the case when admissible control actions belong to the class of square integrable functions on a segment. The first problem is to find control actions transferring a system to a given state with the minimum control norm under a fixed control time. The second problem is to find control actions transferring the system to a given state within the minimum time under a given constraint on the control norm. The authors demonstrate that the problem can be reduced to the problem of moments, as well as derive the feasible statement and solvability conditions for the latter. Solution of the problem is obtained analytically in the form of quadratures. A series of computing experiments are conducted and the qualitative features of system dynamics are discussed.