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We consider infinite horizon optimal control problems with time averaging andtime discounting criteria and give estimates for the Cesaro and Abel limits oftheir optimal values in the case when they depend on the initial conditions. Weestablish that these limits are bounded from above by the optimal value of acertain infinite dimensional (ID) linear programming (LP) problem and that theyare bounded from below by the optimal value of the corresponding dual problem.(These estimates imply, in particular, that the Cesaro and Abel limits existand are equal to each other if there is no duality gap). In addition, we obtainIDLP-based optimality conditions for the long run average optimal controlproblem, and we illustrate these conditions by an example.

Linear Programming Estimates for Cesaro and Abel Limits of Optimal Values in Optimal Control Problems