Can we give access to controlled functionality in Fully homomorphic encryption schemes? Homomorphic encryption schemes are PKE schemes with an additional special method Evaluate. Evaluate method takes input any function (as boolean circuit) and encrypted inputs of the function and evaluates the circuit. The results of such function are still encrypted under the public key PK.

Sometimes, it would be too wide of a scope to allow execution of any arbitrary function.

How can one limit FHE schemes to allow only fewer circuits?
This concept is called targeted malleability:

http://crypto.stanford.edu/~dabo/pubs/abstracts/reshom.html

The abstract and introduction of that paper give a good overview of the ideas. In brief, the goal is to ensure that a homomorphic evaluator can only produce a ciphertext by evaluating a function from an "approved" class of functions. It is trivial to achieve this by attaching to the ciphertext a noninteractive proof/argument that it was computed correctly, but this yields a ciphertext whose length is at least linear in the length of the chain of homomorphic operations performed. (This may be acceptable if only a small number of such "hops" are needed.)

The solutions given in the paper instead use succinct noninteractive arguments of correct computation relative to the previous ciphertext in the chain.  This keeps the ciphertext length from growing much, because only one proof is attached to the ciphertext. However, the argument system needs a stronger property of "extractability" (of the previous statement in the chain to be proved) to make this work.