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

## TL;DR

This paper develops second order necessary and sufficient optimality conditions for singular solutions in partially-affine control problems, enhancing understanding of optimality in systems with mixed affine and nonlinear controls.

## Contribution

It introduces new second order conditions and Goh pointwise conditions for singular solutions in partially-affine control problems, expanding theoretical tools for optimal control analysis.

## Key findings

- Derived second order necessary and sufficient conditions for weak optimality.
- Established Goh pointwise necessary optimality conditions.
- Provided an illustrative example demonstrating the theoretical results.

## Abstract

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.