\(\Gamma\)-convergence and chattering limits in optimal control theory (Q2718684)

The paper deals with sequences of optimal control problems of the form NEWLINE\[NEWLINEP_h=\min_{(u,y)}\Big\{\int_0^1f_h(t,y(t),u(t)) dt: y'(t)=g_h(t,y(t),u(t)),\;y(0)=y_0\Big\},NEWLINE\]NEWLINE where the state variable \(y\) varies in the Sobolev space \(W^{1,1}(0,1;{\mathbb R}^n)\), the control variable \(u\) varies in a space \(U\) of measurable functions, and the \(f_h\) and \(g_h\) satisfy suitable conditions. The variational convergence problem for \(\{P_h\}\) consists in finding a limit problem \(P_\infty\) such that, if \((u_h,y_h)\) is an optimal pair of \(P_h\), up to subsequences it results that \((u_h,y_h)\to(u_\infty,y_\infty)\) in a suitable sense, where \((u_\infty,y_\infty)\) is an optimal pair of \(P_\infty\). This has been done in literature by using two major approaches. The first based on \(\Gamma\)-convergence theory, and the second on the notions of chattering parameter functions and of relaxed controls as parametrized measures in the sense of L.C. Young. In the paper, the two methods are first recalled and described. Then the relationships between the two kinds of variational limits produced by the two methods are investigated when \(U\) is the space of \(L^\infty\) functions. When the \(\{P_h\}\) is independent on \(h\), the two methods provide two apparently different, but closely connected, relaxed formulations. Then, a similar result is proved in the more general setting of variational convergences. A formula is first obtained relating the two variational limits each other. Then, a representation result for the variational limit in the second approach is given describing it in the form of a new control problem where parametrized measures do not appear. Finally, an example is discussed where, thanks to the representation results obtained, the \(\Gamma\)-limit of a sequence of problems with fully nonlinear state equations depending on highly oscillating parameters is explicitly written.