一类耦合系统的最优控制解的存在性证明

Existence of Solution for the Optimal Control of Coupling Systems

主要讨论如下最优控制解的存在性问题,即对给定的正数T和已知函数uT(x)∈L2(Ω),寻找一个最优控制q(·)∈L∞(0,T)满足0≤q(t)≤1,使得J(q)=∫Ω|u(x,T)-uT(x)|2dx+δH∫T0|q(t)|2dt,达到最小,其中δ>0为一给定常数,(,u)为下列耦合方程组初边值问题的解:{t+?×[a(x,t)?×]=F(x,t)(x,t)∈QT(1.1)u-▽(k(x,u)▽u)=q(t)a(x,t)|▽×(x,t)QT(1,2)N×(x,t)=N×G(x,t),u(x,t)=g(x,t)x∈?Ω,0<t<T(1,3)(x,0)=H0(x),u(x,0)=u0(x)x∈Ω(1.4)其中QT=Ω×(0,T],Ω为有界区域,?=(?/?x1,?/?x2,?/?x3),H=(H1,H2,H3),G(x,t),g(x,t)为给定函数,0(x),u0(x)为给定初始函数,N为边界?Ω的法向导数。

The paper mainly discussese the following optimal control problem, namely, for a given positive T and known function uT(x)∈L2(Ω) , to find an optimal control q(·)∈L∞(0,T) meet 0≤q(t)≤1 ,make, J(q)=∫Ω|u(x,T)-uT(x)|2dx+δ∫0T|q(t)|2dt become mininum,Where δ>0 is a given constant, (H,u) for the following equations solution of the the initial boundary value problem:ìHt+?× [a(x, t)?×H ]=F (x, t) (x, t)∈QT (1.1) where QT=Ω×(0,T] ,Ω is a bounded í ? ? ? ? ut-?(k(x,u)?u)=q(t)a(x,t)|?×H|2 (x, t)∈QT (1.2) ?N×H(x,t)=N×G(x,t), u(x,t)=g(x,t) x∈?Ω,0<t<T (1.3)H(x,0)=H0(x), u(x,0)=u0(x) x∈Ω(1.4) domain, ?=?è? ??÷??x1,??x2,??x3 ,H=(H1,H2,H3) ,G(x,t),g(x,t) is a given function, H0(x),u0(x) for a given initial function, N is the boundary ?Ωof the noamal derivative.