Lie algebraic obstructions to Gamma-convergence of optimal control problems

We investigate the possibility of describing the limit problem of a sequenc e of optimal control problems (P)((b n)), each of which is characterized by the presence of a time dependent vector valued coefficient b(n) = (b(n 1), ..., b(n M)). The notion of limit problem is intended in the sense of Gamma -convergence, which, roughly speaking, prescribes the convergence of both the minimizers and the in mum values. Due to the type of growth involved in each problem (P)((b n)) the ( weak) limit of the functions (b(n 1)(2),..., b(n M)(2))-beside the limit (b(1),...,b(M)) of the (b(n 1),...,b(n M)) is c rucial for the description of the limit problem. Of course, since the b(n) are L-2 maps, the limit of the (b(n 1)(2),...,b(n M)(2)) may well be a ( ve ctor valued) measure mu = (mu (1),...,mu (M)). It happens that when the pro blems (P) (b(n)) enjoy a certain commutativity property, then the pair (b,m u) is sufficient to characterize the limit problem. This is no longer true when the commutativity property is not in force. Ind eed, we construct two sequences of problems (P)((b n)) and (P)(((b) over ba r n)) which are equal except for the coefficient b(n) ((.)) and (b) over ti lde (n)((.)), respectively. Moreover, both the sequences (b(n), b(n)(2)) an d ((b) over tilde (n), (b) over tilde (2)(n)) converge to the same pair (b, mu). 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)(((b) ove r tilde n)). This means that the mere information contained in the pair (b, mu) is not sufficient to characterize the limit problem. We overcome this d rawback by embedding the problems in a more general setting where limit pro blems can be characterized by triples of functions (B0, B, y) with B0 great er than or equal to0.