In Chap. 20 we established that the sign of the quadratic form λ t Hess ũ F t is related to optimality of the extremal control ũ. Under natural assumptions, the second variation is negative on short segments. Now we wish to catch the instant of time where this quadratic form fails to be negative. We derive an ODE (Jacobi equation) that allows to find such instants (conjugate times). Moreover, we give necessary and sufficient optimality conditions in these terms.
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