What are Linearly Solvable MDPs? Markov Decision Process (MDP) is a formalism mainly used in artificial intelligence on the structure of decision making of a learner/agent. The aim is to find a suitable policy that maximizes the expected discounted reward that an agent gets. (Or in control theory, minimize the cost.) 
I have seen this paper written by Todorov and Dvijotham (Inverse Optimal Control with Linear Solvable MDPs). 
I am not sure what the words 'linearly solvable MDP' means. Do you have any insights? Any examples? 
 A: Linearly solvable MDPs are a continuous relaxation of normal MDPs.  The differ from normal MDPs in that they don't use a traditional notion of an "action" variable and replace "action" with a next-state distribution u(x'|x).  Normally in MDPs there is an action variable that indexes a distribution over next states. In LMDPs, however, the optimal policy is not a mapping of states to action variables, it is a next-state distribution which minimizes the accumulated state costs, q(x), of the agent traversing the state-space while minimizing a divergence cost between the distributions of the optimal policy, u(x'|x), and the passive uncontrolled dynamics of the agent p(x'|x) through the state space.
The linear bellman equation looks like this
$v(x) = \min_{u(x'|x)} \Big[q(x) + KL(u||p) + \mathop{\mathbb{E}}_{x'\sim u}[v(x')]\Big]$
which can be solved in polynomial time as a largest eigenvector problem.  It has many other nice properties.  You can also do inverse control with them or embed traditional MDPs inside of them.
http://www.pnas.org/content/106/28/11478.full
https://homes.cs.washington.edu/~todorov/papers/TodorovNIPS06.pdf
