Convex approximation of the solution of the matrix Riccati equation

Cooperative control problems in game theory and optimal control problems where a set of objectives is combined into one performance criterion in the linear quadratic case lead to a criterion in which the state and control weighting matrices, and , respectively are linear functions of the weighting vector μ. The solution is characterized by the matrix Riccati differential equation with the solution implicitly depending on μ. Approximations are utilized to explicitly reveal the dependence of the Riccati solution on μ. Lower and upper bound of the solution for all μ of interest are derived and a convex approximation of the Riccati solution, which is in between the bounds, is then defined.