How long the appian JDBC keeps the database connection alive? and How long it will be cached ?

This might be helpful:  KB-1673 General FAQs related to Appian and RDBMS 

I think this only way to control this is via the tomcatResources.xml

initialSize="5"
maxActive="200"
defaultTransactionIsolation="READ_COMMITTED"
maxWait="30000"
minIdle="5"
minEvictableIdleTimeMillis="90000"
timeBetweenEvictionRunsMillis="450000"

AFAIK, it is determined by database connection pool’s settings