# American Institute of Mathematical Sciences

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December  2018, 11(6): 1233-1258. doi: 10.3934/dcdss.2018070

## Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems

 Escola de Matemática Aplicada, Fundação Getulio Vargas, Praia de Botafogo 190, 22250-900 Rio de Janeiro - RJ, Brazil

* Corresponding author: M. Soledad Aronna

Received  March 2017 Revised  June 2017 Published  June 2018

Fund Project: This work was supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO.

In this article we study optimal control problems for systems that are affine with respect to some of the control variables and nonlinear in relation to the others. We consider finitely many equality and inequality constraints on the initial and final values of the state. We investigate singular optimal solutions for this class of problems, for which we obtain second order necessary and sufficient conditions for weak optimality in integral form. We also derive Goh pointwise necessary optimality conditions. We show an example to illustrate the results.

Citation: M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
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