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By Sudhakar G. Pandit, Sadashiv G. Deo (auth.)

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Is suitably chosen (indeed, it suffices to take 0 < ~ < e/32). 3. Let A(t) = - a, where a is a positive real number. Then X(t) x-l(s) = e -a(t-s), t ! s ! O. Take F(t,y)= e -t sin (e ty2) - t2 e t, O ~ t, y < c o . 5. 5 are satisfied. 4. Let A(t) be as in Example 3-3 and F(t,y) = t3 e -2t (y+l) + t(t+l) -I e-t + (t+2) e -3t. Then Choose 48 IF(t,y) J & ~ (t,lyJ) , where ~(t,r) : (t3r + t3 + 2t + 2)e -t. ~. °° Consider the equations x ! 29) where u(t) 0 , O

L o e t e fixed LuJ P" co of __ -'I L~J e ~ uniform bounded oo ~. esetoa variation with fixed initial time t = O. oootros Consider the controls u(k)ct) defined on [0, I]. = I 0 if 0 ~ t < 1/k 1 if 1/k £ t Then t t x2(t) = x2(o) ÷ S - x2(s)ds ÷ S u(s)ds + u(t) - u(o) o o and ~k) x { 0 (t) i Hence if 0£ t < 1/k if t : i/k . = 1/k I x2(slds -- o -- xl(1/k) , o and the sequence u(k)(t) is a sequence infimum of the cost functional lim k -~co The only control C(u (k)) = u~(t) in is the multi-valued control control does not exist.

_~. Suppose that (i) t X(t,s) ~ exp ( S a(r)dr), t A s A to S where the function a (ii) is continuous on J! there exist non-negative, continuous functions J such that [F(t,y)[ for t > to and ~ p(t)[y[, IYl ~ r, p [ G ( t , y ) [ f~ q ( t ) [ y [ r > O ! and q on 54 (iii) u is of type ~ ! Ibklq(tk) < i forock = 1,2,... 30 ) possesses the null solution and if p, qvd e L ! (J) and Sa(s)ds = _oo J then the null solution is uniformly asymptotically Proof. stable. 35), t t z(t) K IXol ÷ S p(s)z(s)ds + S q(s)z(s)dvu(S).

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