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Abstract: This survey paper presents some new advances in theoretical aspects of differential game theory. We particular focus on three topics: differential games with state constraints; backward stochastic differential equations approach to stochastic differential games; differential games with incomplete information. We also address some recent development in nonzero-sum differential games (analysis of systems of Hamilton–Jacobi equations by conservation laws methods; differential games with a large number of players, i.e., mean-field games) and long-time average of zero-sum differential games.

Twenty patients with hypogonadism (19 women with amenorrhoea and 1 man with impotence and infertility), galactorrhoea and hyper-prolactinaemia (range: 36 to 344 ng/ml) were studied. The radiological study of the sella turcica, including in all cases hypocycloidal tomograms, allowed classification of the patients into 3 groups: group I (n = 4) had a grossly enlarged sella turcica, group II (n = 12) had localized alterations indicating the probable existence of a prolactin-secreting microadenoma ("microdeformation") while group III patients presented no radiological abnormality.Before treatment, all the patients were submitted to a complete evaluation of the function of their anterior pituitary, including the LH and FSH responses to iv administration of Gn-RH. All the group I patients had low basal LH levels and a blunted response to Gn-RH. The basal LH and in response to Gn-RH were normal in most of the group II patients and in all of the group III patients. An exaggerated FSH response to Gn-RH was observed in 6/12 patients with microdeformation (group II) but not in groups I and III patients. A low LH and a blunted LH response to Gn-RH is highly suggestive of the existence of a pituitary prolactin-secreting adenoma in case of amenorrhoea and hyper-prolactinaemia patients; a normal response does not however rule out such a diagnosis. The reasons for an exaggerated FSH response to Gn-RH in patients with suspected prolactin-secreting microadenoma remain to be investigated though this pattern can also occur in other cases of amenorrhoea. Hence the Gn-RH test might contribute to the assessment of the hypothalamo-pituitary axis of patients with hyper-prolactinaemia.Six patients treated for 4 months with bromocriptine (CB-154) were submitted to re-evaluation of their pituitary gonadotrophins reserve. All the women experienced restoration of menses with 39 days of treatment and the male patient regained potency. It was observed that bromocriptine treatment and subsequent normalized prolactin levels in the 4 group II women tested were associated with normalization of their previously exaggerated FSH response to Gn-RH; LH responses were also diminished in these cases. These data are compatible with the hypothesis that hyper-prolactinaemia per secould interfere with the endogenous secretion of Gn-RH at the hypothalamic level. In one patient with grossly enlarged sella turcica and a previous lack of an LH and FSH response to Gn-RH, bromocriptine treatment restored a normal gonadotrophins response, confirming that, in this case, the alteration of this response was indeed due to a prolonged lack of endogenous Gn-RH secretion.

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of ptargets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided.