By Agarwal R.P. (ed.)

International medical sequence in acceptable research (WSSIAA) goals at reporting new advancements of excessive mathematical normal and present curiosity. every one quantity within the sequence can be dedicated to the mathematical research that has been utilized or in all likelihood acceptable to the recommendations of medical, engineering and social difficulties. For the prior 25 years, there was an explosion of curiosity within the research of nonlinear dynamical structures. Mathematical options constructed in this interval were utilized to big nonlinear difficulties starting from physics and chemistry to ecology and economics. these types of advancements have made dynamical platforms thought an incredible and engaging department of arithmetic to scientists in lots of disciplines. This wealthy mathematical topic has been in part represented during this choice of forty five papers via a number of the top researchers within the quarter. This quantity comprises forty five cutting-edge articles at the mathematical idea of dynamical platforms through major researchers. it really is was hoping that this assortment will lead new course during this box

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**Example text**

For any u £ Uad, by the convexity of Uad, ut = u0-\- e(u — u0) £ Uad for 0 < £ < 1. 1, the state equation 34 (5) has a unique strong solution x, — x(u€,fi0) corresponding to the control u e and parameter /JoUsing the second part of the inequality (8), we have j l(t,xe(t),uc(t))dtDefine yc = xc — x0/e. j l(t,x0{t),uo(t))dt>(i, V ueUad. (9) Note that yc satisfies the following equation ye + A(t)yc = [ 5 (<,x,,ti £) /j 0 ) = [

19) with constant A(t) = A and diagonal B(t) = b(t)I. A simple computation provides the relation 1 r™+™ M +1 — / \\Ka)\\da = N + -^— for all N, M € IN . 2Af J2N 4 (24) On the other hand, it is easy to see that Urn f sup \\X(£ + 1,0 - Y(( + 1,0||1 = 0 . (25) This shows that condition (20) is in fact truely weaker than condition (22). Corollary 2. (l) and (19) satisfy hypothesis H. Then for the respective upper Bohl exponents of (1) and (19) the relation X(i) = X(i9) (26) holds true, provided that lim f sup \\X(( + 1,0 - Y(( + 1,0||1 = 0 .

2 ( Necessary Condition) Suppose Assumptions (Al) - (A4) hold. 2), where B(t) = DxG{x0,u0,nQ){t), l°{t) = lx{t,x0(t),u0{t)); 3) //(C*V> + / ; , « - i i o ) y . y « f t > 0 Vu£Uad, where C{t) = duG(x0,u0,fi0)(t), and /Jj(i) = lu(t,x0(t),u0(t)); 4 ) Ji(4>,g{t,xo,u0,n))dt < Jt{ij>,g(t,xo,uo,iio))dt for all n £ M(E). Proof. Let (u 0 , /*o) be an optimal (saddle) solution of problem (P) and x0 = X(UQ, /io) , the corresponding optimal trajectory. For any u £ Uad, by the convexity of Uad, ut = u0-\- e(u — u0) £ Uad for 0 < £ < 1.