Lie Algebraic Obstructions to Γ-convergence of Optimal Control Problems∗

We investigate the possibility of describing the “limit problem” of a sequence of optimal control problems (P)(bn), each of which is characterized by the presence of a time dependent vector valued coefficient bn = (bn1 , . . . , bnM ). The notion of “limit problem” is intended in the sense of Γ-convergence, which, roughly speaking, prescribes the convergence of both the minimizers and the infimum values. Due to the type of growth involved in each problem (P)(bn) the (weak) limit of the functions (b2n1 , . . . , b 2 nM )—beside the limit (b1, . . . , bM ) of the (bn1 , . . . , bnM )—is crucial for the description of the limit problem. Of course, since the bn are L2 maps, the limit of the (b2n1 , . . . , b 2 nM ) may well be a (vector valued) measure μ = (μ1, . . . , μM ). It happens that when the problems (P)(bn) enjoy a certain commutativity property, then the pair (b, μ) is sufficient to characterize the limit problem. This is no longer true when the commutativity property is not in force. Indeed, we construct two sequences of problems (P)(bn) and (P)(b̃n) which are equal except for the coefficient bn(·) and b̃n(·), respectively. Moreover, both the sequences (bn, bn) and (b̃n, b̃n) converge to the same pair (b, μ). However, the infimum values of the problems (P)(bn) tend to a value which is different from the limit of the infimum values of the (P)(b̃n). This means that the mere information contained in the pair (b, μ) is not sufficient to characterize the limit problem. We overcome this drawback by embedding the problems in a more general setting where limit problems can be characterized by triples of functions (B0, B, γ) with B0 ≥ 0.