Singular solutions of optimization and control problems

In some optimization problems, extremal arcs ( H u = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315137667/ebc3ad23-d2ed-4b59-a8e1-8861837bea7d/content/eq1845.tif"/> ) occur on which the matrix H_uu is singular. Such arcs are called singular arcs; they satisfy the necessary condition on convexity [Equation (6.3.12)] but not the strengthened condition [Equation (6.3.1)]; that is, H_uu is only semidefinite. Additional tests are needed to determine if a singular arc is optimizing or not.