Problems with relaxed shifted controls and the pointwise maximum principle

In previous work [J. Warga, A proper relaxation of controls with variable shifts, J. Math Anal. Appl. 196 (1995) 783–793], it has been shown that controls with kk variable and unrelated time shifts form a set ℜℜ whose closure ℜ¯ in the weak star topology of L1(T,C(Ωk+1))∗L1(T,C(Ωk+1))∗ is a convex and compact space. This provides a basis for establishing the existence of an optimal solution of the shifted control problem by essentially copying the assumptions and arguments used for control problems without shifted controls. Similarly, the necessary optimality conditions, for original and relaxed minima, that define an extremal, previously studied for differential and functional equations, apply directly to such problems with shifted controls except that the maximum principle follows directly in integral form only. In the present note we first establish the form of dependence of the relaxed shifted controls on the shifts and then use the integral form of the maximum principle to derive a corresponding pointwise maximum principle for optimal (ordinary or relaxed) shifted controls acting through ordinary differential and certain functional–differential equations.