Finally, we combine (20) and (22) to characterize the

extremals of [W.sub.[PHI]](S) as the solutions of the following Euler-Lagrange equation:

One main goal of this paper is to show without any controllability assumption that any linear-quadratic problem on time scales of the form (L[Q.sup.[sigma]]) or (LQ) has the property that its "critical pairs" are exactly "

extremals" at which the problem is "weakly normal." In other words, feasible pairs ([bar.x], [bar.u]) at which the first variation of the problem vanishes are exactly those who admit at least one set of multipliers satisfying the weak maximum principle with [[lambda].sub.0] = 1.

Tarantello, On the symmetry of

extremals for the Caffarelli-Kohn-Nirenberg inequalities, Adv.

His topics include the first variation, base generalizations, the homogeneous problem, variable-endpoint conditions, broken

extremals, and sufficient conditions.

Therefore the positive

extremals are presented by annual sediment yield which make <3% of the supply, the major anomalies make 36%, the small anomalies make 6-15%; the provision of negative

extremals makes >97%, the corresponding provision of large

extremals makes 97-94% and the provision of small

extremals makes 94-85%.

This definition provides the common classical

extremals; see Remark 43.

(a) Following the microvariational principle, the state of the agent in the SF is realized along admissible

extremals of the square root from the energy functional, which is associated with the distribution of the SD.

More precisely, the solutions of the implicit PDE system of first order (2.2) are

extremals for the Lagrangian

Moreover, the

extremals are given by u(x) = [alpha] sin ([[??].sup.-1] [[integral].sup.x.sub.0] a dt + [delta]) for some [alpha] [not equal to] 0 and [delta] [member of] R, where [??] = [(2[pi]).sup.-1] [[integral].sup.2[pi].sub.0] a dt.

Now the set can be plotted as a cluster of points on a plane, and more

extremals are possible.

Section 3 deals with the structure of

extremals, that is, the integral curves of the Hamiltonian vector fields [Mathematical Expression Omitted] associated with H, based on the decomposition of the cotangent bundle T*G of G as the product G x L*.

In this particular case we have studied the relating problems in a various aspect and obtained interesting results such as the exact values of the best constant S = S(p,q,[alpha]) in certain cases, the existence and nonexistence of the

extremals and so on.