Optimal control of turbulent flow as measure solutions

We use the concept of relaxed or measure-valued solutions for control probl ems of turbulent flow related to Navier-Stokes equation. Sufficient conditi ons guaranteeing the existence of measure solutions are presented. Results on existence of optimal controls for Blow up time and Bolza problems for su ch systems are also presented. New results on relaxed necessary conditions of optimality are proved. Further it is shown that the relaxed necessary co nditions reduce to classical Pontryagin type necessary conditions if measur e solutions degenerate into Dirac structure. The paper is concluded with an algorithm based on the new necessary conditions for computing optimal cont rols.