We denote the first conjugate time along x by t c.
The first conjugate time is defined by the infimum of times t>0 such that Q t has a nontrivial kernel.
In case of bang-bang control, a conjugate time theory has been developed (see [79] for a brief survey of the approaches), but the computation of conjugate times remains difficult in practice (see, e.g., [60]).
Essentially this result states that computing a first conjugate time t c reduces to finding the zero of some determinant along the extremal.
Some classical techniques of optimal control have been recalled, including Pontryagin Maximum Principle, first and second-order optimality conditions, and conjugate time theory.
Murphy et al. (2014) studied two substorms on April 9, 2011 during the interval 4 7 UT, using ground optical, magnetic, and riometer data, and space data from near-conjugate Time History of Events and Macroscale Interactions During Substorms (THEMIS and Geostationary Operational Environmental Satellite GOESES) probes.
Then, we have the following result (see, e.g., [1, 15] for the proof and more precise results): The time t c is a conjugate time along xif and only if the mapping (exp _{x_{0}}(t_{c},cdot)) is not an immersion at p 0, i.e., the differential of the mapping (exp _{x_{0}}(t_{c},cdot)) is not injective.
The time t c is a conjugate time along xif and only if the mapping (exp _{x_{0}}(t_{c},cdot)) is not an immersion at p 0, i.e., the differential of the mapping (exp _{x_{0}}(t_{c},cdot)) is not injective.
In [27], the authors indicate that if (lambda =lambda (bar {s})) is a turning point of order one, then the corresponding final time t f is a conjugate time, and the corresponding point (E_{x_{0},t_{f},lambda } u(x_{0},p_{0},t_{f},lambda))) is the corresponding conjugate point 4. By assuming the absence of conjugate points over Ω for all λ∈[0,1], the possibility of turning points is discarded.
Therefore, K is a convolution matrix, whose rows are reversed, conjugated, and time-shifted versions of A 's sampling.
A new conjugation method was developed to decrease the conjugation time and increase conjugate yields [ 23, 24].
