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We shall also provide explicit formulas for this problem. When we deal with Hamilton--Jacobi--Bellman equations, we obtain the existence and uniqueness of Frankowska contingent episolutions. We shall deduce these results from the fact that the graph of the solution is the viable-capture basin of the graph of the boundary conditions under an auxiliary system and then from their properties and their characterizations proved in [J.-P. Aubin, SIAM J. Control Optim., 40 (2001), pp. 853--881].<\/jats:p>","DOI":"10.1137\/s0363012900381510","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"425-456","source":"Crossref","is-referenced-by-count":12,"title":["Boundary-Value Problems for Systems of Hamilton--Jacobi--Bellman Inclusions with Constraints"],"prefix":"10.1137","volume":"41","author":[{"given":"Jean-Pierre","family":"Aubin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","unstructured":"Jean\u2010PierreAubin, Contingent derivatives of set\u2010valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Adv. in Math. Suppl. Stud., Vol. 7, Academic Press, New York, 1981, 159\u201322983m:58014"},{"key":"R2","volume-title":"Viability theory","author":"Aubin Jean\u2010Pierre","year":"1991"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-60756-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1576-9"},{"key":"R5","unstructured":"J.\u2010P. Aubin,\n                      Impulse Differential Inclusions and Hybrid Systems: A Viability Approach\n                      , Lecture notes, Universit\u00e9 Paris\u2010Dauphine, Paris, France, 2001."},{"key":"R6","unstructured":"J.\u2010P. Aubin,\n                      Optimal impulse control problems and quasi\u2010variational inequalities thirty years later: A viability approach\n                      , in Contr\u00f4le optimal et EDP: Innovations et Applications, IOS Press, Amsterdam, 2000, pp. 311\u2013324."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/PL00001423"},{"key":"R8","unstructured":"J.\u2010P. Aubin,\n                      Lyapunov functions for impulse and hybrid control systems\n                      , in Proceedings of the IEEE Conference on Decision and Control, Sydney, Australia, 2000."},{"key":"R9","unstructured":"J.\u2010P. Aubin,\n                      The caliber of impulse and hybrid control systems\n                      , in Proceedings of the CIB International Symposium on Evolution, Trento, 2000, Birkh\u00e4user, Berlin, 2002."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"J.\u2010P. Aubin,\n                      The substratum of impulse and hybrid control systems\n                      , in Hybrid Systems: Computation and Control, Proceedings of the HSCC 2001 Conference, Lecture Notes in Comput. Sci. 2034, Springer\u2010Verlag, New York, 2001, pp. 105\u2013118.","DOI":"10.1007\/3-540-45351-2_12"},{"key":"R11","unstructured":"J.\u2010P. 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