Prova do Teorema 6.4

Introduzindo o espa¸co de Banach W , com

W := {(f, z, p1_{, p}2_{) : f ∈ C}1₂,1_{4(Q), z, p}1_{, p}2 _{∈ C}1+1₂,1+1_4(Q)}.

Definamos a fun¸c˜ao Λ : Z 7→ 2Z_{, com}

Λ(w) = {y ∈ Z : (f, y, p1, p2) ´e o controle-estado em W, (6.42) e (6.43) ´e satisfeito},

e fixando R0 > 0. Ent˜ao, se ky0k_C2+ 1_2(Ω) ´e suficientemente pequeno, a fun¸c˜ao multi-

valuada Λ satisfaz as hip´oteses do Teorema do Ponto Fixo de Kakutani.

• Λ est´a bem definida; tamb´em para cada w ∈ Z, Λ(w) ´e n˜ao vazio e convexo (´e consequˆencia da Proposi¸c˜ao 6.3).

• Existe  > 0 (dependendo somente de Ω, O, T, a0, a1e R0) tal que, se ||y0||_C2+ 1 2(Ω) ≤

, tem-se Λ(w) ⊂ BZ[0; R0] para todo w ∈ BZ[0; R0]. Isto ´e consequˆencia da

Proposi¸c˜ao 6.3.

• Existe um conjunto compacto K ⊂ BZ[0; R0] tal que, se ||y0||_C2+ 1₂

(Ω) ≤  e

w ∈ BZ[0; R0], implica que Λ(w) ⊂ K. Isto ´e uma consequˆencia do fato de que

C2+1₂,1+1_{4(Q) ,→ Z compacta.}

• Λ tem o gr´afico fechado em Z. Isto n˜ao ´e dif´ıcil de verificar. Obviamente, dado wn

com wn→ w forte em Z, e assumindo que zn ∈ Λ(wn) e zn → z forte em Z. Ent˜ao,

existem controles fn tais que (fn, yn, p1n, p2n) pertencem a W e satisfazem

(6.59)                  yn,t− ∇ · (a(zn)∇yn) = fn˜1O− 1 µ1 p1_nβ₁21O1 − 1 µ2 p2_nβ₂21O2 em Q, − pi n,t− a(zn)∆pin= αi(yn− yid)1Oid em Q, yn(x, t) = pin(x, t) = 0 sobre Σ, yn(x, 0) = z0(x), pin(x, T ) = 0 em Ω.

Consequentemente, fazendo-se limite em (6.59) tem-se                  yt− ∇ · (a(z)∇y) = f ˜1O− 1 µ1 p1β₁21O1 − 1 µ2 p2β₂21O2 em Q, − pi t− a(z)∆φi = αi(y − yid)1Oid em Q, y(x, t) = pi_{(x, t) = 0 sobre Σ,} y(x, 0) = z0(x), φi(x, T ) = 0 em Ω.

Assim, Λ possui pelo menos um ponto fixo y. Obviamente, y ´e o estado associado ao controle f tal que (6.42) e (6.43) s˜ao satisfeitos. Portanto a prova do Teorema 6.4 est´a conclu´ıda.

Quest˜oes Abertas

• A controlabilidade por trajet´orias do sistema            yt− a( Z Ω

y(x0, t) dx0)∆y + b(x, t)y = v1ω em Q,

y(x, t) = 0 sobre Σ,

y(x, 0) = y0(x) em Ω,

´e um problema aberto.

Se tentamos usar as mesmas t´ecnicas do cap´ıtulo 2 para este problema, a dificuldade ´e encontrada na necessidade de ter uma desigualdade de Observabilidade. Porem, ´e poss´ıvel conseguir resultados similares `a Proposi¸c˜ao 2.1 quando b = b(x) ou b = b(t). • A controlabilidade local por trajet´orias do sistema (3.1) ´e um problema aberto. A principal dificuldade ´e encontrar uma desigualdade de Carleman para o seguinte sistema:            −ϕt− (ν0+ ν1||∇y||2)∆ϕ + 2ν1 R Ω∆yϕdx 0_{+ (Dϕ)y + ∇π = g em Q,} ∇ · ϕ = 0 em Q, ϕ = 0 sobre Σ, ϕ(T ) = ϕT _{em Ω,}

os argumentos tradicionalmente usados n˜ao funcionam.

• Sera poss´ıvel a controlabilidade nula de (4.1) em dimens˜ao trˆes com um controle escalar ? No caso do sistema de Navier-Stokes foi resolvido por J. M. Coron e Pierre Lissy em [36].

• A controlabilidade seguindo a estrat´egia de Stackelberg-Nash para o sistema N - dimensional        yt− ∇ · (a(∇y)∇y) = f 1O+ v1β11O1+ v 2_β 21O2 em Q, y(x, t) = 0 sobre Σ, y(x, 0) = y0(x) em Ω.

A t´ecnica desenvolvida no cap´ıtulo 6 n˜ao ´e suficiente. De fato, para estabelecer a estimativa de Carleman de equa¸c˜oes parab´olicas lineares, exigimos que os coefi- cientes das partes principais perten¸cam ao espa¸co de Sobolev W1,∞_{(Q). Portanto,}

para resolver o problema quase linear, temos que procurar por um ponto fixo em um espa¸co contendo as fun¸c˜oes y tal que ∇yt perten¸ca ao espa¸co L∞(Q)N. Pela

abordagem desenvolvida no cap´ıtulo 6, precisamos escolher o espa¸co de controles para ser C12,

1

4(Q). Ent˜ao, pelas estimativas de Schauder para equa¸c˜oes parab´olicas

lineares de segunda ordem, a solu¸c˜ao y da equa¸c˜ao linearizada para (6.13) satisfaz ∇yt ∈ (L2(Q))N. Isso n˜ao parece suficiente para estabelecer a estimativa de Carle-

man desejada para equa¸c˜oes parab´olicas lineares.

• A controlabilidade seguindo a estrat´egia de Stackelberg-Nash para os seguintes sistemas (p1)        yt− ∇ · (a(y)∇y) = v1β11O1 + v 2_β 21O2 em Q, y = f 1S sobre Σ, y(x, 0) = y0_{(x) em Ω,} e (p2)        yt− ∇ · (a(y)∇y) = f 1O em Q, y = v1β11S1 + v 2_β 21S2 sobre Σ, y(x, 0) = y0(x) em Ω.

Conclus˜ao

Nesta pesquisa, conclui-se que problemas associados `as equa¸c˜oes do tipo Parab´olico, s˜ao extremadamente desafiadores. O desenvolvimento da Tese proporcionou uma completa compreens˜ao da “aplicabilidade” da Teoria de Controle e mostrou que algumas quest˜oes abertas, podem ser respondidas!

Como resultado de esta Tese tem-se conseguido publicar em revistas cientificas os seguintes trabalhos titulados:

– On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equations (ver [7])

– Local Null Controllability of the N-Dimensional Ladyzhenskaya-Smagorinsky with N-1 Scalar Controls (ver [5])

– Exact Controllability to the Trajectories for Parabolic PDEs with Nonlocal Nonlinearities (ver [10])

– On the approximate null controllability of the “true” Boussinesq system in dimension 3 (Em prepara¸c˜ao)

Al´em disso, espera-se que os resultados de controlabilidade obtidos no cap´ıtulo 4 e cap´ıtulo 6 junto com seu desenvolvimento seja sometido para sua respectiva publica¸c˜ao em revistas cient´ıficas.

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