You're doing a lot of shell programming... the problem is with your system call, shells can be very different accross systems (their syntax can vary), as can ls/cp/mv... and their options

If you were to use File::Find::Rule with Path::Tiny your program gets a little simpler and more portable , like this, using the other AMs renaming idea

#!/usr/bin/perl --
use strict;
use warnings;
use POSIX();
use File::Find::Rule qw/ find rule /;
use Path::Tiny qw/ path /;

my $dir_to_search  = 'goner';
print join ' ', "Before move: ", find( file => in => $dir_to_search ,)
+, "\n";

my_move_log_files($dir_to_search);

print join ' ', "After move: ",  find( file => in => $dir_to_search ,)
+, "\n";


exit( 0 );

sub my_move_log_files {
    my @log_dirs_to_move = @_;
    for my $src_dir (@log_dirs_to_move) {
        my $dated =  "logs_dir".POSIX::strftime('%Y%m%d%H%M%S', localt
+ime );
        my $final = path( $src_dir, $dated);
        my $temp  = path( $src_dir, '..', $dated);
        path( $src_dir )->move( $temp );
        path( $src_dir )->mkpath;
        path( $temp )->move( $final );
    }
}
__END__
$ perl file-find-rule-path-tiny-move-rename-goner-assubdir.pl
Before move: goner/touchme.txt
After move: goner/logs_dir20150528161733/touchme.txt
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