"Jacobi type conditions for singular extremals"
We consider the class of optimal control problems linear in the control, and study a singular extremal.
Assuming for simplicity that the Lagrange multipliers are unique, we formulate the quadratic order conditions,
which have the form of sign definiteness of an integral quadratic functional (the second variation
of Lagrange function) with an additional parameter and zero Legendre coefficient.
So, this functional is nonclassical. Using the Goh transformation, we to convert it to a functional
of the classical type, and by applying the Hestenes approach, determine its sign definiteness in terms
of the conjugate point, i.e. give equivalent Jacobi type conditions.
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