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Neittaanmaki P., Dan Tiba. Optimal Control of Nonlinear Parabolic Systems (M.Dekker 1994)(ISBN 0824790812)(T)(C)(411s)_MOc_.djvu |
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In each of the relations giving p\, рг, рз, Qe denotes the mappings defined in
Propositions 5.2-5.5 and associated to the control problems (Pilt,«), (Рг.г.б),
(Рз,г,«) respectively...
Multiplying
by j9tn(yn) and integrating over [0,t], we get that {j9c"(y")} is bounded in
L°°@,T;H)...
II, for variational inequalities give that {yc} is bounded in H^2'llP(Q), and
the Sobolev theorem again implies that yE —» y* uniformly on Q...
For instance, if we require
и G I7ad = {u G U : и > 0 a.e.}
then (8.2) becomes u* = (B*p*)~ (the negative part of B*p*)...
We also refer to Barbu [1985], Barbu and Barron [1987] and He and
Morosanu [1989], also related to this topic...
(8.26)
Thus if у = 0 in Г = {(x,t) : x = s(t)}, then it is the solution of the
one-phase Stefan problem with Г as the free boundary...
In the previous paragraph we have shown that, by means of an appropriate
sequence of controls, any given surface can be considered as a free boundary
in some approximating sense...
In the case of nonlinear governing state systems, the regularity properties of
the optimal control or the adjoint optimal state are too weak in order to apply
the error estimates established in Ch...
A.9)
Obviously, A.8) may be equivalently expressed as
w = G + д1кГ*(р) = Vk(j>), A.Ю)
where V^: I<2{Q) —» K...
In fact, we have computed the Gateaux gradient of the cost functional
A.15) at the point Щ and in the direction v/, — Ил, which is given by the
left-hand side in relation A.23)...
We suppose y(t) € V/, or, in the case this is
not true, we replace у by its piecewise linear interpolant
Furthermore, let M = ((Vi,Vi)z,2(n))I^=i denote the mass matrix and A =
((V<pi, V^,)i,i(n))-j=1 denote the stiffness matrix...
Numerical results with errors in measurements;
top: a = 0.001, middle: a - 0.01, bottom: a =
0.1...
Jo Jo
Since {un} is weakly convergent in L2@,T;U) this implies that Bun —* Bu
weakly in L2@,T;H) and the corresponding sequence {у'„} is bounded in
L2@,T;H)...
Finally, we pass B.31) to the limit for h —» 0 and we obtain using (H.10)
/ e(y,u)dt< f
o Jo
The above inequality in connection with the conclusion of Step 2 allows us
to infer that [у,п] is an optimal pair for Problem (P), thereby completing the
proof...
B.65)
Let us point out that ?j is approximated in practice by some combination of
the gradients of / in nearby points if / is not differentiable at x^kK That is,
the algorithm performs operations of the same type as a gradient algorithm in
each iteration...
C.12)
Let us point out that the nonlinear algebraic system C.10)+C.12) has a unique
solution, according to Elliott and Ockendon [1982]...
The next step is to show that for к —* 0
vk —* v strongly in Z,2@,T; (Я1 (Я))*),
where w* is the polygonal function on [0,T] induced by the vector («][.), and
Ук —» у weakly in L2@, T; Я1 (ft)),
where у к is the polygonal function corresponding to (ykn )...
The uniqueness of the weak solution yields
that the convergences are valid without taking subsequences...
From C.25) we infer that
, m—1
| E ciwiS - «i"}is+i«8ii»(«)) ^ M> C-26)
n=0
where M > 0 is independent of h and fc...
The solution of the state problem for t = 1.0 (left initial, right
optimized with obtained boundary control)...
For every и 6 U, denote by yu E W^^C^L^a.c)) П Cw@,T;Hl(a,c)) the
solution of the boundary value problem C.43)-C.47) (by C.47) w 6 Wl-2@, T)
and one can apply Remark 3.19, Ch...
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