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This note generalizes constrained optimization methods in a finite-dimensional space into Hilbert spaces and investigates computational methods for optimal control problems with functional inequality constraints. Two methods are proposed by applying the feasible direction method and the constrained quasi-Newton method. Subsidiary problems for direction-finding that are originally linear-quadratic programming in a Hilbert space can be transformed into linear-quadratic ones in Rn. Thus, the control problem can be solved by a series of finite-dimensional programming problems.