Your best bet may be do a file test before you try to connect. Something like:

sub connectdb {
    unless (-r "as2con.db")
    {
        print "Database file as2con.db does not exist.\n";
        return;
    }
    my $dbase = DBI->connect("dbi:SQLite:dbname=as2con.db", "", "", {P
+rintWarn => 0, PrintError => 0});
    return $dbase;    
}
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Then your code would check the return status of the connectdb method to see if it was undefined, and handle it appropriately.