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Curvatures of single-input control systems

Jakubczyk B.

Control and Cybernetics

2009

Vol. 38, no 4B

1375-1391

We introduce invariants of control-affine systems which we call curvatures. They are defined by the drift and the control distribution, given by the system. The curvatures allow us to analyse the variational equation along a given trajectory, as well as existence of conjugate points.

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.