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We study the necessary optimality conditions for the general deterministicmean field type free-endpoint optimal control problem. Our study relies on theLagrangian approach that treats the mean field type control system as a crowdof infinitely many agents who are labeled by elements of some probabilityspace. Within the framework of the Lagrangian approach, we derive thePontryagin maximum principle. In particular, it implies that the costatevariables are also labeled by the elements of the probability space.Furthermore, we consider the Kantorovich and the Eulerian formalizations thosedescribe the mean field type control system via distribution on the set oftrajectories and nonlocal continuity equation respectively. We prove that eachlocal minimizer in the Kantorovich or Eulerian formulation corresponds to alocal minimizer within the Lagrangian approach. Using this, we deduce thePontryagin maximum principle in the Kantorovich and Eulerian forms. Toillustrate the general theory, we examine the model system of mean field typelinear quadratic regulator. For this system, we show that the optimal strategyis determined by the linear feedback.

Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach