Optimal Control Theory for the Damping of Vibrations of by Vadim Komkov (auth.)

By Vadim Komkov (auth.)

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By Vadim Komkov (auth.)

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Extra resources for Optimal Control Theory for the Damping of Vibrations of Simple Elastic Systems

Example text

I (x) - w ~x 2 = f (x,t) ' -6/2 <= x <= +6/2, t >_ 0.

Uml}. In this situation it is clear that the optimum control will be in general nonunique. e. I[ ul ]. = I. Take any nonzero vector ~ v(t) which is orthogonal to % (t), and such that [lul I® = ~ ~ II u + vl I~ = i, (which is possible in this case). u(t), ueU a that is if u(t) obeys the m a x i m u m principle, the control Now, if then so does (u + v). 6 a) . is unique again the unique It is not hard to see if t h e definition II I I is such that U forms a unit ball which (L~ if its boundary Conversely, norm contains no straight I I II m then there w i l l exist non-unique of such a case is analogous given above for the sup.

If Wl(X,t) A [w (x, t), 8w (x, t)/~t] is the displacement vector corresponding to ;l(X,t) and w 2 (x,t) corresponds l(w I + w 2) corresponds construction to ~2 (x,t) , then to ~(;i + ;2 ) by linearity. By of ~i and $2 and using the Lemma 5, we have (wl ($1 (x, t), t) ) = ~(w 2 ($2 (x, t), t) for all t E [0,T]. Hence, = = ~ <(W 1 + w2),(w I + w2)>. 57 the use of Cauchy-Schwartz inequality shows that we have in fact equality <~I'W2 >2 = ' , and the equality ~i = ~2 easily follows for all t ~ [0,T].

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