Lie Algebraic Obstructions to $\Gamma$-Convergence of Optimal Control Problems

Abstract

We investigate the possibility of describing the of a sequence of optimal control problems $({\cal P})_{\piccolo (b_n)}$, each of which is characterized by the presence of a time dependent vector valued coefficient $b_n=(b_{\piccolo n_1},\ldots,b_{\piccolo n_M})$. The notion of is intended in the sense of $\Gamma$-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 $({\cal P})_{\piccolo (b_n)}$ the (weak) limit of the functions $(\bunq,\ldots,\bMnq)$---beside the limit (b1, . . . ,bM) of the $(\bun,\ldots,\bMn)$---is crucial for the description of the limit problem. Of course, since the bn are L2 maps, the limit of the $(b_{\piccolo n_1}^2,\ldots,b_{\piccolo n_M}^2)$ may well be a (vector valued) measure $\mu=(\mu_1,\ldots,\mu_M)$. It happens that when the problems $({\cal P})_{\piccolo (b_n)}$ enjoy a certain commutativity property, then the pair $(b,\mu)$ 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 $({\cal P})_{\piccolo (b_n)}$ and $({\cal P})_{\piccolo (\tilde{b}_n)}$ which are equal except for the coefficient $b_n(\cdot)$ and $\tilde{b}_n(\cdot)$, respectively. Moreover, both the sequences $(b_n,b^2_n)$ and $(\tilde{b}_n,\tilde{b}^2_n)$ converge to the same pair $(b,\mu)$. However, the infimum values of the problems $({\cal P})_{\piccolo (b_n)}$ tend to a value which is different from the limit of the infimum values of the $({\cal P})_{\piccolo (\tilde{b}_n)}$. This means that the mere information contained in the pair $(b,\mu)$ 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 $(B_0,B,\gamma)$ with $B_0 \geq 0$.