We need to find a control (u(t)) and corresponding trajectory (x t)) that minimize the functional J=frac{1}{2} int_{0}^{T} bigl[x'(t)Q t)x t)+u'(t)C t)u(t) bigr],dt, (3) where (Q t)=Q'(t ge0), (C t)=C'(t)>0) are the matrices of dimensions (ntimes n) and (mtimes m), correspondingly.
We refer to such SOAPs as generalized SOAPs, and the corresponding trajectory as a generalized trajectory.
We observe that for short step protocols, the corresponding trajectory makes only a small excursion down toward the shifted critical manifold without reaching it.
One of the most important optimal control problems is how to drive the corresponding trajectory of the equation from an initial state to a given target set in the shortest time, through applying constrained controllers.
Even if a system is not persistent (i.e., if there exist an initial population densities such that the corresponding trajectory leads to extinction of one or more species), there still can be locally stable attractors [ 19- 22].
Figure 1(b) shows a raster graph of the ensemble activity during a time interval, and Figure 1(c) plots the corresponding trajectory of the position in 2D plane.
After a single spike, as seen in Fig. 12b, the corresponding trajectory remains to the right of the threshold manifold.
For each t ∈ R, we define the mapping T t : D → D as T t = M S ˜ t L. Theorem 4.2 Given ( z ¯, μ ¯ ) ∈ D, if we denote t → ( z, μ ) ( t ) = T t ( z ¯, μ ¯ ) the corresponding trajectory, then z = ( u, v ) is a weak solution of the two-component Camassa-Holm equations (2.1), which constructs a continuous semigroup.
This weight represents the probability of how well the corresponding trajectory characterizes the segment.
