This talk is devoted to an in?nite horizon optimal control problem V (t0,x0) = infZ8 t0 L(t,x(t),u(t))dt over all trajectory-control pairs (x,u) of the control system ( x0(t) = f(t,x(t),u(t)), u(t) ? U(t) for a.e. t = t0 x(t0) = x0,
that may be also subject to a state constraint. Its history goes back to F.P. Ramsey, 1928. The value function V may be discontinuous and may admit in?nite values. In?nite horizon problems exhibit many phenomena not arising in the context of ?nite horizon ones. Among such phenomena let us recall that already in 1970ies it was observed that in the necessary optimality conditions it may happen that the co-state is di?erent from zero at in?nity and that only abnormal conditions hold true (even for problems without state constraints). A way to avoid it is to impose assumptions guaranteeing the local Lipschitz continuity of V (t,·) with the Lipschitz constant becoming zero at in?nity. Then V (t,·) has locally bounded superdi?erentials and this is crucial for getting normal necessary optimality conditions. However, such regularity of the value function requests very strong assumptions on the Lagrangian L, much stronger than in the case of ?nite horizon problems. In general, superdi?erentials are not bounded and may become ”horizontal” leading to abnormal optimality conditions. The value function solves a Hamilton-Jacobi equation in a generalized sense, that may, in turn, involve an abnormal Hamiltonian because of these horizontal sub/superdi?erentials. Despite these facts both necessary optimality conditions and uniqueness of solutions to the Hamilton-Jacobi equation hold true with such extended notions of super/subdi?erentials.
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