Three kinds of effective error bounds of the quadrature formulas with multiple nodes that generalize the well-known Micchelli-Rivlin quadrature formula, when the integrand
is an analytic function in the regions containing the confocal ellipses, were considered recently in [13, 15].
Furthermore, using the Gaussian mixture model representation from (26) and (27) and assuming Gaussian measurement noise, the integrand
can be expressed as a sum of weighted Gaussian PDFs; namely,
Therefore, the integrand
in (37) contains a constant component and [mathematical expression not reproducible].
The existence of optimal control [u.sup.*] can be established since the integrand
in (19) and the right hand side of system (18), denoted by f = ([f.sub.1], [f.sub.2], [f.sub.3], [f.sub.4]), are continuously differentiable functions and concave in both (S, I, C, T) and u = ([omega], [sigma]).
When regarding the integrand
in (24) as the Lagrangian, we can define the "Hamiltonian" H as
Next, proceeding as in [17,24] makes it possible to deform the integration contour so that it is wrapped around the singularities of the integrand
located in the upper-half of the complex plane, as shown in Fig.
Proof: Noting that both [mathematical expression not reproducible] are totally independent of [r.sub.0], the partial derivative w.r.t [r.sub.0] of the integrand
in the definition of [??] is
(iv) The integrand
of the objective functional is convex.
By the definition of [u.sub.n+1] and the fact that the Ito integral coincides with the Skorohod integral if the integrand
is predictable, it follows that, for any (t, x) [member of] [0, T] x R,
The neural network completed the mapping relationship between the input variable X and the integrand