Geodesic
On singular controls of a maximum principle for the problem of the Goursat–Darboux system optimization
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 4, pp. 483-491 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper deals with the terminal optimization problem connected with the Goursat–Darboux control system. The right-hand side of the differential equation is a full nonlinear Caratheodory function. We consider the case in which solutions of the Goursat–Darboux system necessarily belong to a class of functions with $p$-integrable (for some $p>1$) mixed derivatives. In our case a choice of this class is defined by boundary functions. We study singular controls in the sense of the pointwise maximum principle that are controls for which this principle is strong degenerate, i.e., degenerate together with second-order optimality conditions. It is shown that for strong degeneration of the pointwise maximum principle it is sufficient that right-hand side with respect to state derivatives is affine and these derivatives and control are separated additively. Necessary optimality conditions of the singular controls are given for this case. These conditions generalize similar necessary optimality conditions which were obtained for more smooth right-hand sides in the case of solutions with bounded mixed derivatives.
Keywords: nonlinear Goursat–Darboux system, solutions having summable mixed derivatives, terminal optimization problem, maximum principle, singular controls. 
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I. V. Lisachenko; V. I. Sumin. On singular controls of a maximum principle for the problem of the Goursat–Darboux system optimization. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 4, pp. 483-491. http://geodesic.mathdoc.fr/item/VUU_2015_25_4_a4/

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