The short answer is yes. Many states, including Florida, have adopted "pet trust statutes" which specifically allow people to name their pets as the beneficiaries of a trust. Prior to these lawsuits, there was some question on whether a pet could even be the beneficiary of a trust. The beneficiary of a trust owns an interest in the trust and animals cannot own "things." Thus, prior to these statutes, people had to leave a power of attorney with a person they trust to care for their pets but that person could do whatever the wanted because no term naming the pets as the beneficiary were valid or binding.
These new statutes allow you to set-up a trust that cares for your pet. But because the trust primarily benefits an animal, it is incapable of defending itself. Your pets are unable to hire trustees to defend their interests in the trust. Thus, pet trusts are still dependent on humans to operate and protect them.
Florida requires Trustees to act in the best interests of the trust, including if the trust is for the benefit of animals. Most statutes terminate the trust upon the death of the pet. The trust instrument must include a provision on what to do with the assets once the pet is dead, if it does not, the trustee is generally allowed to do what is best for the trust (including dissolving it).
If you believe that your loved one's trust was formed under the undue influence of another, then you may want to contact a lawyer. Undue influence can serve as the grounds to contest the validity of a trust. An attorney can review the pros and cons of contesting the trust's validity, including the cost and potential recovery. You should consider the likelihood that you are draining the trust's resources with a lawsuit. A lawyer can help you make the best decision for you and your family.