The interpolation relation in dynamic programming in optimal control for a discrete constrained LQR system is discussed in the article entitled:"On infinite horizon switched LQR problems with state and control constraints" by Maximilian Balandat,  Wei Zhang, and  Alessandro Abate. 

Abstract: This paper studies the Discrete-Time Switched LQR problem over an infinite time horizon, subject to

polyhedral constraints on state and control inputs. Specifically, we aim to find an infinite-horizon hybridcontrol

sequence, i.e., a sequence of continuous and discrete (switching) control inputs, that minimizes

an infinite-horizon quadratic cost function, subject to polyhedral constraints on state and (continuous)

control input. The overall constrained, infinite-horizon problem is split into two subproblems: (i) an

unconstrained, infinite-horizon problem and (ii) a constrained, finite-horizon one. We derive a stationary

suboptimal policy for problem (i) with analytical bounds on its optimality, and develop a novel

formulation of problem (ii) as a Mixed-Integer Quadratic Program. By introducing the concept of a safe

set, the solutions of the two subproblems are combined to achieve the overall control objective. Through

the connection between (i) and (ii) it is shown that, by proper choice of the design parameters, the

error of the overall suboptimal solution can be made arbitrarily small. The approach is tested on a

numerical example

See the paper on the link:

http://hybrid.eecs.berkeley.edu/~max/pubs/pdf/SCL12_IHCDSLQR.pdf