The authors deal with sequences of optimal control problems of the form \[ ({\mathfrak P}_ h)\quad \min \{\int^{1}_{0}f_ h(t,y,u)dt:\quad y'=g_ h(t,y,u),\quad y(0)=y^ 0_ h\}, \] where the state variable y belongs to the Sobolev space \(Y=W^{1,1}(0,1;{\mathbb{R}}^ n)\) and the control variable u is in \(U=L^ 1(0,1;{\mathbb{R}}^ m)\). A “limit problem” (\({\mathfrak P}_{\infty})\) is constructed such that - if \((u_ h,y_ h)\) is an optimal pair for (\({\mathfrak P}_ h)\) and if \((u_ h,y_ h)\) tends to \((u_{\infty},y_{\infty})\) in the topology \(wL^ 1(0,1;{\mathbb{R}}^ m)\times L^{\infty}(0,1;{\mathbb{R}}^ n)\), then \((u_{\infty},y_{\infty})\) is an optimal pair for (\({\mathfrak P}_{\infty}).\)
It is shown that, when the functions \(g_ h(t,y,u)\) appearing in the state equations are rapidly oscillating, it may arrive that the domain of the limit problem (\({\mathfrak P}_{\infty})\) is not given by a state equation \(y'=g_{\infty}(t,y,u)\), and, in some situations, it may coincide with the whole product space \(U\times Y.\)
The basic tool used for constructing the limit problem (\({\mathfrak P}_{\infty})\) is the \(\Gamma\)-convergence theory for functionals defined on a product space.