I can, using a method devised below, do it in
Starting with a simple case,
1_0, it's pretty obvious the optimal steps are:
_10 or 10_
both of which take three moves (we know we can always invert solutions like the above).
At each point there are at most four choices, move the sheep that is two to the left of the space, one to the left of the space, one to the right of the space, or two to the right of the space. Let's call these moves
R; now the simple solutions above are
rLr (both palindromes).
Now lets look at the next puzzle,
00_11 l -> lRrLLrRl = lR + rLLr + Rl
You can see that this solution,
lRrLLrRl is a palindrome again and that it starts and ends just as the solution to
1_0. If we do the same for
111_000 we should start with
lRrLL, do something to reverse the line up of sheep and space then do
11_1000 l << this stage is the
1101_00 R << same instructions
11010_0 r << as the first half
110_010 L << of the previous
1_01010 L << plus one instruction: lRrLL
... now we want to get to 01010_1 so,...
_101010 l ... keep going left as fast as we can to the edge...
01_1010 R ... then right as fast as we can...
010101_ R ... and then make the centre a palindrome...
01010_1 l ... with a single last instruction, to find us at the right place!...
... then just use the reverse instructions from before...
010_011 L << these ones
0_01011 L << are just
00_1011 r << the reverse
0001_11 R << of the beginning
000_111 l << which is LLrl
i.e. lRrLL + lRRRl + LLrRl = lRrLLlRRRlLLrRl
Note that what was performed in the middle is exactly like what was performed in the middle of the previous puzzle (move as fast as possible in the same direction until the space hits the edge it was travelling toward, then go as fast as possible in the other direction, then palindrome-ise the centre instructions so as to get to the reverse of what had been reached before the process, then add the reverse of the start to complete a palindromic instruction set.
If we repeat this procedure we will find a solutions of length
where $p$ is the count of pairs of sheep.
1111_0000 we get:
0000_1111 l -> lRrLLlRRR + rLLLLr + RRRlLLrRl = lRrLLlRRRrLLLLrRRRlLLrRl (24)
Finally, for the puzzle in question,
11111_00000 (five pairs) we find this solution (and of course its inverse):
lRrLLlRRRrLLL + LlRRRRRlL + LLLrRRRlLLrRl:
1_010101010 L << centre portion starts with this move
0101010_011 L << and ends with this one