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We introduce an alternative approach for the analysis and numericalapproximation of the optimal feedback control mapping. It consists in lookingat a typical optimal control problem in such a way that feasible controls aremappings depending both in time and space. In this way, the feedback form ofthe problem is built-in from the very beginning. Optimality conditions arederived for one such optimal mapping, which by construction is the optimalfeedback mapping of the problem. In formulating optimality conditions, costatesin feedback form are solutions of linear, first-order transport systems, whileoptimal descent directions are solutions of appropriate obstacle problems. Wetreat situations with no constraint-sets for control and state, as well as themore general case where a constraint-set is considered for the controlvariable. Constraints for the state variable are deferred to a comingcontribution.

Optimal feedback control, linear first-order PDE systems, and obstacle problems