Periodic solutions of a control problem via marginal maps

We investigate the existence of periodic solutions to the control problem x˙=f(t,x,u)+g(t),x∈Rn,u∈Rm, (1) with g and f periodic in t with period 1. We form the associated quantities s(t,x)=supu∈Ω(x,f(t,x,u)),i(t,x)=infu∈Ω(x,f(t,x,u)) where (·,·) denotes the inner product inRn and Ω is a nonempty compact set in Rn. If us(t, x), ui(t, x) denote the (in general multivalued) controls for which s(t, x), i(t, x) are respectively attained, then we can form the family of marginal problems x˙∈λ(t)co¯¯¯¯¯f(t,x,us(t,x))+(1−λ(t))co¯¯¯¯¯f(t,x,ui(t,x))+g(t),λ(⋅)∈L∞([0,1],[0,1]). (2) We give sufficient conditions for the existence of a periodic solution of certain marginal problems, stated in terms of lim inf|x|→∞ and lim sup|x|→∞ of s(t,»)/¦x¦2 and i(t, x)j¦x¦2. Finally we state the relationship between the periodic solutions of the marginal problems and those of the original problem (1).