科技通报
科技通報
과기통보
Bulletin of Science and Technology
2015年
11期
10-13
,共4页
耦合系统%最优控制%存在性%收敛
耦閤繫統%最優控製%存在性%收斂
우합계통%최우공제%존재성%수렴
coupling systems%optimal control%existence%convergence
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+δ∫0T|q(t)|2dt ，达到最小，其中δ>0为一给定常数，(H , u)为下列耦合方程组初边值问题的解：ìH t+?×[a(x, t)?×H ]=F (x, t)(x, t)∈QT (1.1)???? ut-?(k(x,u)?u)=q(t)a(x,t)|?×H|2(x, t)∈QT (1.2)íN ×H(x,t)=N ×G(x,t), u(x,t)=g(x,t) x∈?Ω,0<t<T其中 QT=Ω×(0,T]，Ω为有界区域，(1.3)?H(x,0)=H0(x), u(x,0)=u0(x) x∈Ω(1.4)?=?è???÷??x1,??x2,??x3，H=(H1,H2,H3)，G(x,t),g(x,t)为给定函数，H0(x),u0(x)为给定初始函数，N为边界?Ω的法向导数。
主要討論如下最優控製解的存在性問題，即對給定的正數T和已知函數uT(x)∈L2(Ω)，尋找一箇最優控製q(·)∈L∞(0,T)滿足0≤q(t)≤1，使得J(q)=∫Ω|u(x,T)-uT(x)|2dx+δ∫0T|q(t)|2dt ，達到最小，其中δ>0為一給定常數，(H , u)為下列耦閤方程組初邊值問題的解：ìH t+?×[a(x, t)?×H ]=F (x, t)(x, t)∈QT (1.1)???? ut-?(k(x,u)?u)=q(t)a(x,t)|?×H|2(x, t)∈QT (1.2)íN ×H(x,t)=N ×G(x,t), u(x,t)=g(x,t) x∈?Ω,0<t<T其中 QT=Ω×(0,T]，Ω為有界區域，(1.3)?H(x,0)=H0(x), u(x,0)=u0(x) x∈Ω(1.4)?=?è???÷??x1,??x2,??x3，H=(H1,H2,H3)，G(x,t),g(x,t)為給定函數，H0(x),u0(x)為給定初始函數，N為邊界?Ω的法嚮導數。
주요토론여하최우공제해적존재성문제，즉대급정적정수T화이지함수uT(x)∈L2(Ω)，심조일개최우공제q(·)∈L∞(0,T)만족0≤q(t)≤1，사득J(q)=∫Ω|u(x,T)-uT(x)|2dx+δ∫0T|q(t)|2dt ，체도최소，기중δ>0위일급정상수，(H , u)위하렬우합방정조초변치문제적해：ìH t+?×[a(x, t)?×H ]=F (x, t)(x, t)∈QT (1.1)???? ut-?(k(x,u)?u)=q(t)a(x,t)|?×H|2(x, t)∈QT (1.2)íN ×H(x,t)=N ×G(x,t), u(x,t)=g(x,t) x∈?Ω,0<t<T기중 QT=Ω×(0,T]，Ω위유계구역，(1.3)?H(x,0)=H0(x), u(x,0)=u0(x) x∈Ω(1.4)?=?è???÷??x1,??x2,??x3，H=(H1,H2,H3)，G(x,t),g(x,t)위급정함수，H0(x),u0(x)위급정초시함수，N위변계?Ω적법향도수。
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.
