Archive of Issues
Russia Nizhni Novgorod
Section  Mathematics 
Title  On singular controls of a maximum principle for the problem of the GoursatDarboux system optimization 
Author(s)  Lisachenko I.V.^{ab}, Sumin V.I.^{b} 
Affiliations  Nizhni Novgorod State Technical University^{a}, Nizhni Novgorod State University^{b} 
Abstract  The paper deals with the terminal optimization problem connected with the GoursatDarboux control system. The righthand side of the differential equation is a full nonlinear Caratheodory function. We consider the case in which solutions of the GoursatDarboux 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 secondorder optimality conditions. It is shown that for strong degeneration of the pointwise maximum principle it is sufficient that righthand 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 righthand sides in the case of solutions with bounded mixed derivatives. 
Keywords  nonlinear GoursatDarboux system, solutions having summable mixed derivatives, terminal optimization problem, maximum principle, singular controls 
UDC  517.95 
MSC  49K20 
DOI  10.20537/vm150405 
Received  2 October 2015 
Language  Russian 
Citation  Lisachenko I.V., Sumin V.I. On singular controls of a maximum principle for the problem of the GoursatDarboux system optimization, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 4, pp. 483491. 
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