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We consider infinite horizon optimal control problems with time averaging and time discounting criteria and derive linear programming-related representations of Cesàro and Abel limits of their optimal values in the case when they depend on the initial conditions. We show that Cesàro and Abel limits are equal if they are continuous functions of the initial condition, strengthening previous results that require uniform convergence to ensure this equality. The obtained representations of the limits of value functions are used to derive optimality conditions for the long-run average optimal control problem.

LP-related representations of Cesàro and Abel limits of optimal value functions