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Singular extremals in multi-input time-optimal problems: a sufficient condition

Chittaro F. C., Stefani G.

Control and Cybernetics

2010

Vol. 39, no 4

1029-1068

In this paper we study second order sufficient conditions for the strong-local optimality of singular Pontryagin extremals. In particular, we focus on the minimum-time problem for a control-affine system with vector inputs. We use Hamiltonian methods to prove that the coercivity of a suitably-defined second variation - plus an involutivity assumption on the distribution of the controlled fields - is a sufficient condition for the strong optimality of a candidate extremal.

Jacobi type conditions for singular extremals

Dmitruk A. V.

Control and Cybernetics

2008

Vol. 37, no 2

285-306

We consider the class of optimal control problems linear in the control, and study a singular extremal. If the Lagrange multipliers are unique, the quadratic order optimality conditions have the form of sign definiteness of a quadratic functional (the second variation of Lagrange function) with totally zero Legendre coefficient. Using the Goh transformation, we convert it to a functional possibly satisfying the strengthened Legendre condition, involving also an additional parameter, and by applying the Hestenes approach, determine its sign definiteness in terms of the conjugate point, i.e. give Jacobi type conditions.