An explicit linear solution for the quadratic dynamic programming problem (Q1093562)

For a given vector \(x_ 0\), the sequence \(\{x_ t\}\) which optimizes the sum of discounted rewards \(r(x_ t,x_{t+1})\), where r is a quadratic function, is shown to be generated by a linear decision rule \(x_{t+1}=Sx_ t+R\). Moreover, the coefficients R, S are given by explicit formulas in terms of the coefficients of the reward function r. A unique steady-state is shown to exist (except for a degenerate case), and its stability is discussed.