Thanks a lot for the clarification, I have finally decided to implement the redirection myself and skip the intermediate shells:

 #child 
  print "This is child $$\n";
  open STDERR, '>&STDOUT';
  open STDOUT, '> /pathlogs/mylog.log';
  my $command='sh /path/process.sh';  
  exec($command);
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This way, killing the pid of the child, we kill the command process and I also get the redirections.

It also seems to work with

 #child 
  print "This is child $$\n";
  open STDERR, '>&STDOUT';
  open STDOUT, '> /pathlogs/mylog.log';
  my $script='/soft/perl-version/bin/perl /path/process.pl';  
  exec($script);
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I will clearly have to research more to use correctly exec and pid managing :)