Can a control verb simultaneously be a raising verb?

Question

For example, take believes, which is a raising to object verb. However, employing various selection tests, you can observe that believes selects an experiencer subject:

#the cat believes to be out of the bag
*there believes to be a lot of hens here
*it believes to be raining

In these tests believes seems to select a specific subject because it rejects instances of tight selection (idioms) and existential there. This behavior is much like subject control verbs, like wants.

However, given the tests in object position:

?# Harold believes the cat to be out of the bag
Harold believes there to be a lot of hens here
Harold believes it to be raining

We see that believes doesn't seem to select for a specific object, which would leave us to presume it is a raising to object verb.

Does this mean that believes is both subject control and object raising?

NOTE: This is assuming the VP internal subject hypothesis.

Answer 1

Probably not, or at least not believe in English. Control is a relation between an argument of the matrix clause and a (big) PRO in the embedded one, which supplies the PRO with reference.

In examples with believe, the trace left after raising to object seems to be the only (relevant) empty category in the sentence:

(1) I [believe [the cat]_i [t_i to be out of the bag].

Answer 2

Well, not simultaneously, no. Not in the same sentence.

But as these exercises makes clear, it can happen that a particular English verb, like want,

governs Equi (aka "Equivalent Noun Phrase Deletion" and "Control") in one context,

• Bill wants ___to leave.
  Bill wants to be examined by the doctor. ≠ The doctor wants to examine Bill.

and Raising (aka "Subject-Raising") in another.

• Bill wants Max to leave.
  Bill wants [it to rain/there to be a party].
  Bill wants Max to be examined by the doctor. = Bill wants the doctor to examine Max.
  [NB: want does not passivize Raised objects: *Max is wanted to leave by Bill. ]

Essentially, English want governs A-Equi when it meets the conditions
  (coreferential subjects of both want and its complement clause)
but governs B-Raising when it doesn't meet the conditions for A-Equi
  (complement subject not coreferential to the subject of want)

As for the classification of believe, the syntax lab reports from these same exercises list believe as

2-place with B-Raising (mental predicate, experiencer subject, propositional object).
  [NB: believe may also passivize Raised objects on the next cycle up, producing a rule sandwich:
          e.g, The men are believed to have been seen by the women ]

As @Ivan has pointed out, if a verb selects a particular type of subject, that is not the same as,
and makes no prediction about, whether that verb governs Equivalent Noun Phrase Deletion.

By the way, this is not a question about syntax so much it is about English syntax --
be aware that things can work very differently in different languages.