Claiming my over 18 year old son who has Asperger’s

At age 19 or over, a child can be a qualifying relative dependent if you provide more than half their support and if they have less than $4700 of taxable income.

The child can be a qualifying child dependent if they are a full-time student, or they are disabled, and they live in your home more than half the year. For income tax purposes, “disabled“ means the child cannot perform gainful work, due to a condition that is permanent or will last at least one year or will lead to death. Because of the gainful work requirement, many people who meet a medical definition of disability are not disabled for income tax purposes. Being able to work a minimum wage job without major accommodations is generally considered to be proof that the person can perform gainful work. However, work in a sheltered workshop for disabled people is not considered gainful work even if the person is paid. 

Q. Can I claim my over 18 year old son who has Asperger’s

A. Yes, if he is is considered totally & permanently disabled and still lives with you.  Then he meets the qualifying child rules and his age and income don't matter. 

 A child's disability doesn't affect tax filing until he turns 19 (24 if a full time student). Then his disability allows him to continue to be considered a "Qualifying Child" for tax purposes regardless of age. 

No specific medical diagnosis constitutes disabled to tax purposes.

“The term ‘disability’ means, with respect to an individual –

            (a)        a physical or mental impairment that substantially limits one or more of the major  life activities of such individual;

            (b)        a record of such impairment; or

            (c)        being regarded as having such an impairment.”

For the IRS, disabled means -

"an individual shall be considered to be disabled if he is unable to engage in any substantial gainful activity by reason of any medically determinable physical or mental impairment which can be expected to result in death or to be of long-continued and indefinite duration. An individual shall not be considered to be disabled unless he furnishes proof of the existence thereof in such form and manner as the Secretary may require. "

I believe the substance of that paragraph is that it is your Doctor's decision

1. He or she cannot engage in any substantial gainful activity because of a physical or mental condition and
2. A doctor determines the condition has lasted or can be expected to last continuously for at least a year or can lead to death.

However, the IRS defines Substantial gainful activity this way so if they have part time jobs then they are engaging in any substantial gainful activity so it would not meet the requirements.

"Substantial gainful activity is the performance of significant duties over a reasonable period of time while working for pay or profit, or in work generally done for pay or profit. Full-time work (or part-time work done at your employer's convenience) in a competitive work situation for at least the minimum wage conclusively shows that you are able to engage in substantial gainful activity.