С.В. Конягин - Вариационное исчисление и оптимальное управление, страница 4

Òîãäà èíòåãðàë âûøå ïåðåïèñûâàåòñÿ ñëåäóþùèìåññìîòðèì óðàâíåíèåèêêàòè, ïóñòüîáðàçîì:wZt1t0Íåñëîæíîâèäåòü,ẇ(t) =(C(t) − 2δ)÷òîïðè2A(t) − 2δ − ẇ(t)h(t) + ḣ(t) dtC(t) − 2δäîñòàòî÷íîìàëûõδ(C(t) − 2δ≥0)è,ñëåäîâàòåëüíî, âåñü èíòåãðàë íåîòðèöàòåëåí. Çàìåòèì, ÷òî ñóùåñòâîâàíèå ðåøåíèÿó ìîäèèöèðîâàííîãî óðàâíåíèÿ èêêàòè äëÿ ìàëûõδñëåäóåò èç ñóùåñòâîâàíèÿðåøåíèÿ ó ñàìîãî óðàâíåíèÿ èêêàòè (êîòîðîå ñîîòâåòñòâóåò çíà÷åíèþδ = 0)ïî òåîðåìå î íåïðåðûâíîé çàâèñèìîñòè ðåøåíèÿ äèåðåíöèàëüíîãî óðàâíåíèÿ îòïàðàìåòðà.Èòàê,J(x̂(·) + h(·)) − J(x̂(·)) ≥ 0,è, çíà÷èò,ìèíèìóì.

Òåîðåìà äîêàçàíà ïîëíîñòüþ.x̂äåéñòâèòåëüíî äîñòàâëÿåò ñëàáûéÈãîëü÷àòûå âàðèàöèè. ÓñëîâèåÂåéåðøòðàññà íåîáõîäèìîå óñëîâèåñèëüíîãî ýêñòðåìóìà.8ÏóñòüV = intV ⊂ Rn+1 , L ∈ C (V × Rn ); ∀t ∈ [t0 , t1 ] (t, x̂(t)) ∈ V , x̂(·) ∈ C ([t0 , t1 ], R)- ýêñòðåìàëü.19J(x(·)) =Zt1L(t, x(t), ẋ(t))dt → extr;t0x(t0 ) = x0 ; x(t1 ) = x1 ;Äàëåå ïîëîæèìn=1èx̂(·)- äîñòàâëÿåò ñèëüíûé ìèíèìóì.Òåïåðü ââåäåì ïîíÿòèå èãîëü÷àòîé âàðèàöèè: ðàññìîòðèì- íåêèé ìàëûé ïàðàìåòð. Ïóñòü˙ ).u = x̂(ττ ∈ (t0 , t1 ), v ∈ R, α > 0Ïîñòðîèì ñëåäóþùóþ óíêöèþ:t0 ≤ t ≤ τ − α;x̂(t),x(t, α) = x̂(t) + (v − u)(t − τ + α), τ − α ≤ t ≤ τ ;x̂(t) + (v−u)α(t1 − t),τ ≤ t ≤ t1 .t1 −τx̂(t)x(t, α)t0τ −ατèñ. 2: Ôóíêöèÿt1x(t, α)∀α - ýòî êóñî÷íî-íåïðåðûâíî äèåðåíöèðóåìàÿ óíêöèÿ.

(ïðè÷åìçíà÷åíèå ýòîé óíêöèè â òî÷êå τ ñîñòûêîâàíî, ò.å. x̂(t) + (v − u)(t − τ + α)|t=τ =x̂(t) + (v−u)α(t1 − t)|t=τ = x̂(τ ) + (v − u)α.) Ïî îïðåäåëåíèþ òàêîãî ðîäà âàðèàöèèt1 −τÇàìåòèì, ÷òîíàçûâàþòñÿ èãîëü÷àòûìè.àññìîòðèì ðàçíîñòü âèäà:τ −αZ˙L(t, x(t, α), ẋ(t, α)) − L(t, x̂(t), x̂(t))dt +J(x(·, α)) − J(x̂(·)) =t+Zτ τ −α||0{z}I0Zt1 ˙˙L(t, x(t, α), ẋ(t, α)) − L(t, x̂(t), x̂(t))dt +L(t, x(t, α), ẋ(t, α)) − L(t, x̂(t), x̂(t))dt{z}I1Òåïåðü îöåíèìI0 , I 1 , I2 .|τ{zI2˙I0 = 0 ò.ê. ïî îïðåäåëåíèþ óíêöèè x(t, α), L(t, x(t, α), ẋ(t, α)) − L(t, x̂(t), x̂(t))=˙˙L(t, x̂(t), x̂(t)) − L(t, x̂(t), x̂(t)) = 020}Äàëåå ïî îðìóëå âàðèàöèè èíòåãðàëüíîãî óíêöèîíàëà ñ ïîäâèæíûìè êîíöàìèèìååì: t=tdI2 dx 1= p(t) = −p(τ )(v − u) ⇒ I2 = −L̂ẋ (τ )(v − u)α + ō¯(α), α → 0dα α=0dα α=0 t=τÂîñïîëüçîâàâøèñü òåîðåìîé î ñðåäíåì, ïîëó÷àåì, ÷òî˙I1 = α L(t̃, x(t̃, α), ẋ(t̃, α)) − L(t̃, x̂(t̃), x̂(t̃)) , t̃ ∈ (τ − α, τ )Çàìå÷àåì, ÷òî˙ t̃)) → L(τ, x̂(τ ), u), α → 0.L(t̃, x̂(t̃), x̂(èL(t̃, x(t̃, α), ẋ(t̃, α)) → L(τ, x̂(τ ), v), α → 0.I1 = α (L(τ, x̂(τ ), v) − L(τ, x̂(τ ), u)) + ō¯(α), ïðè α → 0 ⇒J(x(·, α)) − J(x̂(·)) = α L(τ, x̂(τ ), v) − L(τ, x̂(τ ), u) − L̂ẋ (τ )(v − u) + ō¯(α), α → 0Ñëåäîâàòåëüíî,Îïðåäåëåíèå 8.1 Ôóíêöèÿ Âåéåðøòðàññà:E(t, x, u, v) = L(t, x, v) − L(t, x, u) − Lu (t, x, u)(v − u)J(x(·, α)) − J(x̂(·))˙ ))−(v−x̂(τ˙ ))L̂ẋ (τ, x̂(τ ), x̂(τ˙ )) == L(τ, x̂(τ ), v)−L(τ, x̂(τ ), x̂(τα→0+α0 ≤ lim˙ ), v)= E(τ, x̂(τ ), x̂(τÈòàê îêîí÷àòåëüíî ïîëó÷àåì, ÷òî˙ ), v)0 ≤ E(τ, x̂(τ ), x̂(τÏðîâåðèì, ÷òî ýòî óñëîâèå èìååò ìåñòî è íà êîíöàõ îòðåçêà[t0 , t1 ]:˙ 0 ), v) = lim E(τ, x̂(τ ), x̂(τ˙ ), v) ≥ 0E(t0 , x̂(t0 ), x̂(tτ →t0Àíàëîãè÷íî˙ 1 ), v) = lim E(τ, x̂(τ ), x̂(τ˙ ), v) ≥ 0E(t1 , x̂(t1 ), x̂(tτ →t1Òåì ñàìûì ìû äîêàçàëè ñëåäóþùóþ òåîðåìó:Òåîðåìà 8.1 Åñëè x̂(·) ∈ C [t , t ] - äîñòàâëÿåò ñèëüíûé ìèíèìóì (ìàêñèìóì), òî,101(,)˙˙∀t ∈ [t0 , t1 ] ∀v ∈ R E(t, x̂(t), x̂(t),v) ≥ 0 ∀t ∈ [t0 , t1 ] ∀v ∈ R E(t, x̂(t), x̂(t),v) ≤ 0Çàìå÷àíèå 8.1 Óòâåæäåíèå âåðíî è äëÿ x̂(·) ∈ P C .Çàìå÷àíèå 8.2 Óòâåðæäåíèå âåðíî è äëÿ n ≥ 1 (âåêòîðíûé ñëó÷àé).121àññìîòðèìÒàê êàêL(t, x, ẋ),ïóñòü∃Lẋẋ .Ìû èìååìE(t, x, u, u) = 0;∂= Lv (t, x, v)|v=u − Lu (t, x, u) = 0E(t, x, u, v)∂vv=u∂2E(t, x, u, v)∂v 2= Lvv (t, x, v)òî óñëîâèå˙E(t, x̂(t), x̂(t),v) ≥ 0 ∀vâëå÷åò˙Lẋẋ (t, x̂(t), x̂(t))≥0✻L(t, x, v)✲v˙x̂(t)èñ.

3: Ïðè èêñèðîâàííûõt, xãðàèêL(t, x, v)ëåæèò âûøå êàñàòåëüíîé.Ïîëó÷èëè óñëîâèå Ëåæàíäðà. Çàìåòèì, ÷òî óñëîâèå Âåéåðøòðàññà ãîâîðèò íàìî òîì, ÷òî ãðàèê óíêöèè ëåæèò âûøå êàñàòåëüíîé, ñì. (ðèñ.3). Ýòî ñâîåãî ðîäàãëîáàëüíîå óñëîâèå.Ýëåìåíòû òåîðèè ïîëÿ.9àññìîòðèì ïðîñòåéøóþ çàäà÷ó êëàññè÷åñêîãî âàðèàöèîííîãî èñ÷èñëåíèÿ:J(x(·)) =Zt1L(t, x(t), ẋ(t))dt → extrt0x(t0 ) = x0 ; x(t1 ) = x1 .(t, x̂(t)) ∈ VÏîëîæèìL ∈ C(V × Rn ), V = intV ⊂ Rn+1 , x̂(·)àññìîòðèì íåêîå ñåìåéñòâî ýêñòðåìàëåé:α − ïðîáåãàåòÏóñòüîêðåñòíîñòüx(t, 0) = x̂(t),α = 0)0;′′′- ýêñòðåìàëü,′{x(t, α)} t ∈ (t0 , t1 ) (t0 < t0 < t1 < t1 ),ò.å. ñåìåéñòâî ýêñòðåìàëåé ñîäåðæèò, â ÷àñòíîñòè, è íàøóýêñòðåìàëü (ïðèÍàëîæèì íà ýòî ñåìåéñòâî íåêîòîðûå óñëîâèÿ ãëàäêîñòè:x, ẋ,∂x ∂ ẋ,− íåïðåðûâíû.∂α ∂α22Îïðåäåëåíèå 9.1 x̂(·) - îêðóæåíà ïîëåì ýêñòðåìàëåé, åñëè îïðåäåëåíà óíêöèÿ α :ïðè ýòîì x = x(t, α) ⇔ α = α(t, x(t)), ïîñëåäíåå îçíà÷àåò,÷òî ÷åðåç êàæäóþ òî÷êó x îáëàñòè V ïðîõîäèò ðîâíî îäíà ýêñòðåìàëü.V → Rn (α = α(t, x(t)))Ïðèìåð 9.1Z1ẋ2 dt → extr0;.

Íàéòè ñîîòâåòñòâóþùåå ïîëå ýêñòðåìàëåé.x(0) = 0 x(1) = 0✻✻(a)(b)✲0✲1-101èñ. 4: x̂(t) = 0- äîïóñòèìàÿ ýêñòðåìàëü. Òåïåðü ðàññìîòðèì ñåìåéñòâî ýêñòðåìàëåéâèäà:x(t, α) = αt;Òîãäà íà îáëàñòè âûäåëåííîé íà (ðèñ. 8(a)) ñóùåñòâóåò òî÷êà, ÷åðåç êîòîðóþïðîõîäÿòâñåýêñòðåìàëèñåìåéñòâà,÷òîïðîòèâîðå÷èòîïðåäåëåíèþïîëÿ.àññìîòðèì äðóãîå ñåìåéñòâî:x(t, α) = (t + 1)α;Òîãäà ÷åðåç êàæäóþ òî÷êó îáëàñòè íà (ðèñ. 8(b)) ïðîõîäèò ðîâíî îäíà ýêñòðåìàëü.Ïîýòîìó ýòî ñåìåéñòâî ìîæíî ðàññìàòðèâàòü êàê ïîëå ýêñòðåìàëåé.Áóäåì äàëåå òðåáîâàòü, ÷òîáûα(·) ∈ C 1 .Îïðåäåëåíèå 9.2 Ïîëå ýêñòðåìàëåé íàçûâàåòñÿ öåíòðàëüíûì ïîëåìýêñòðåìàëåé (ö.ï.ý.), åñëè ∃(t∗, x∗) òàêàÿ, ÷òî ∀α x(t∗, α) = x∗. (ò.å. âñåýêñòðåìàëè ïðîõîäÿò ÷åðåç îäíó òî÷êó.)Íàïðèìåð, ïîëå ýêñòðåìàëåé èç ïðåäûäóùåé çàäà÷è ÿâëÿåòñÿ öåíòðàëüíûì ïîëåìýêñòðåìàëåé.

Î÷åâèäíî, ÷òît∗ ∈/ [t0 , t1 ].Ïîëîæèì äëÿ îïðåäåëåííîñòè, ÷òîÎïðåäåëåíèå 9.3 Ôóíêöèåé íàêëîíà ïîëÿ íàçûâàåòñÿ óíêöèÿ âèäà:u(t, x) = ẋ(t, α(t, x)), u(·) ∈ C(V )23t∗ < t0Îïðåäåëåíèå 9.4 S - óíêöèåé íàçûâàåòñÿ óíêöèÿ âèäà:S(τ, ξ) =ZτL(t, x(t, α(τ, ξ)), ẋ(t, α(τ, ξ)))dt;t∗ÄëÿíàõîæäåíèÿäèåðåíöèàëàäèåðåíöèðîâàíèèÔèêñèðóåìèíòåãðàëüíîãîτ, ξ, ∆τ, ∆ξ .S -óíêöèèâîñïîëüçóåìñÿóíêöèîíàëàñòåîðåìîéïîäâèæíûìèàññìîòðèì ñåìåéñòâî ýêñòðåìàëåéX(t, β),ãäåîêîíöàìè.βïðîáåãàåòîêðåñòíîñòü íóëÿ. ÏîëîæèìX(t, β) = x(t, α(τ + β∆τ, ξ + β∆ξ)),t0 (β) = τ∗ , t1 (β) = τ + β∆τ,x0 (β) = x∗ , x1 (β) = ξ + β∆ξ,J(β) =Zt1 (β)L(t, X(t, β), Ẋ(t, β))dtt0 (β)Ïî òåîðåìå î äèåðåíöèðîâàíèè èíòåãðàëüíîãî óíêöèîíàëà ñ ïîäâèæíûìèêîíöàìèJ ′ (0) = Lẋ (τ, ξ, u(τ, ξ))∆ξ − (Lẋ (τ, ξ, u(τ, ξ))u(t, ξ) − L(τ, ξ, u(τ, ξ))) ∆τÇíà÷èò,dS = Lẋ (τ, ξ, u(τ, ξ))dξ − (Lẋ (τ, ξ, u(τ, ξ))u(t, ξ) − L(τ, ξ, u(τ, ξ))) dτ− ∂S= H(τ, ξ, ∂S),∂τ∂ξÄàííîåãäåH(τ, ξ, p) = pu(τ, ξ) − L(τ, ξ, u(τ, ξ))äèåðåíöèàëüíîåóðàâíåíèå,êîòîðîìóóäîâëåòâîðÿåòS-óíêöèÿ,íàçûâàåòñÿ óðàâíåíèåì àìèëüòîíà-ßêîáè.Ïóñòü åñòü êàêàÿ-òî äîïóñòèìàÿ óíêöèÿ÷òî∀t ∈ [t0 , t1 ] (t, x(t)) ∈ V .x(·).ÍàéäåìJ(x(·)) − J(x̂(·)),ñ÷èòàåì,Èìååì:ZJ(x̂(·)) =t1L(t, x(t, 0), ẋ(t, 0))dt =t0= {âñå=Zýêñòðåìàëè íàøåãî ñåìåéñòâà ïðîõîäÿò ÷åðåçt1t∗L(t, x(t, 0), ẋ(t, 0))dt −=ZZt0t∗L(t, x(t, 0), ẋ(t, 0))dt = S(t1 , x1 ) − S(t0 , x0 ) =t1t0t∗ } =dS(t, x(t)) = {ξ = x(t) ⇒ dξ = ẋ(t)dt} =24Zt1− (Lẋ (t, x(t), u(t, x(t))u(t, x(t)) − L(t, x(t), u(t, x(t)))) dt+t0+Zt1Lẋ (t, x(t), u(t, x(t)))ẋ(t)dt =t0=Zt1t0(L(t, x(t), u(t, x(t))) + Lẋ (t, x(t), u(t, x(t)))(ẋ(t) − u(t, x(t)))) dtÄàëåå èìååìJ(x(·)) − J(x̂(·)) =Zt1t0(L(t, x(t), ẋ(t)) − L(t, x(t), u(t, x(t)))−−Lẋ (t, x(t), u(t, x(t)))(ẋ(t) − u(t, x(t))))dt == {ïîäèíòåãðàëüíîå=âûðàæåíèå - ýòî óíêöèÿ Âåéåðøòðàññà}Z=t1t0E(t, x(t), u(t, x(t)), ẋ(t))dtÏîëó÷åííàÿ îðìóëà íàçûâàåòñÿ îñíîâíîé îðìóëîé Âåéåðøòðàññà.Òåîðåìà 9.1 Ïóñòü x̂(·) - äîïóñòèìàÿ ýêñòðåìàëü, îêðóæåííàÿ öåíòðàëüíûìïîëåì ýêñòðåìàëåé (ö.ï.ý.) è ïóñòü ∃δ > 0 : ∀t ∈ [t0, t1], ∀x : |x − x̂(t)| < δ,˙∀u : |u − x̂(t)|< δ , ∀v ∈ R E(t, x, u, v) ≥ 0 (≤ 0), ò.å.

âûïîëíåíî óñèëåííîåóñëîâèå Âåéåðøòðàññà. Òîãäà x̂(·) - ñèëüíûé ìèíèìóì (ñîîòâåòñòâåííî ñèëüíûéìàêñèìóì).Ïðîâåðèì ïî îïðåäåëåíèþ. Ïóñòüx(·)- äîïóñòèìàÿ óíêöèÿ.˙kx(·) − x̂(·)kC[t0 ,t1 ] < ε ⇒ ∀t |x(t) − x̂(t)| < ε ≤ δ, |u(t, x(t)) − x̂(t)|<δJ(x(·)) − J(x̂(·)) ≥ 0. Ò.î. äëÿ x(·) èç C 1 [t0 , t1 ]x(·) èç P C 1 [t0 , t1 ] ïîëó÷àåì, âîñïîëüçîâàâøèñüÏî îñíîâíîé îðìóëå Âåéåðøòðàññàòåîðåìà äîêàçàíà.

Åå âåðíîñòü äëÿëåììîé î ñêðóãëåíèè óãëîâ.Ñëåäñòâèå 9.1 Ïóñòü x̂(·) îêðóæåíà ö.ï.ý., L(t, x, ẋ) - âûïóêëàÿ ïî ẋ. Òîãäà x̂(·) -ñèëüíûé ìèíèìóì.Äåéñòâèòåëüíî, èç âûïóêëîñòè ñëåäóåò, ÷òîE(t, x, u, v) ≥ 0. Ñëåäóþùóþ òåîðåìó ïðèâåäåì áåç äîêàçàòåëüñòâà:Òåîðåìà 9.2 Ïóñòü L, âûïîëíåíû óñèëåííûå óñëîâèÿ Ëåæàíäðà,ßêîáè, x̂(·) - äîïóñòèìàÿ ýêñòðåìàëü. Òîãäà x̂(·) ìîæíî îêðóæèòü ö.ï.ý.∈ C 3 (V × R)25Ñëåäñòâèå 9.2 Ïðè âûïîëíåíèè óñëîâèé òåîðåìû 9.2 è óñèëåííîãî óñëîâèÿÂåéåðøòðàññà x̂(·) - ñèëüíûé ýêñòðåìóì.Çàìåòèì â çàêëþ÷åíèå, ÷òî âñå ñêàçàííîå â äàííîì ïàðàãðàå ïðàêòè÷åñêèáåçèçìåíåíèéïåðåíîñèòñÿíàâåêòîðíûéñëó÷àé,èñîðìóëèðóåìíåñêîëüêîîáÿçàòåëüíûõ çàäà÷.Çàäà÷à 3 J(x(·)) = Rt1t0,,äîïóñòèìàÿ ýêñòðåìàëü,ìèíèìóì (â êëàññå äîïóñòèìûõ óíêöèé).Çàäà÷à 4 J(x(·)) = R; ïóñòüA11 ẋ2 (t))dtìèíèìóì.t1(A(t)t0x̂(·);.

Òîãäà- ãëîáàëüíûéL(t, ẋ(t))dt → extr x(t0 ) = x0 , x(t1 ) = x1 x̂(·) ∈ C 1 [t0 , t1 ]˙∀t ∈ [t0 , t1 ] ∀v E(t, x̂(t), x̂(t),v) ≤ 0x̂(·)+ A0 (t)x(t) + A1 (t)ẋ(t) + A00 (t)x2 (t) + 2A01 (t)x(t)ẋ(t) +x̂(·)- ñëàáûé ìèíèìóì. Òîãäàäîñòàâëÿåò ãëîáàëüíûéÇàäà÷à 5 Ïðèâåñòè ïðèìåð ïðîñòåéøåé çàäà÷è ê.â.è., òàêîé ÷òî íåêîòîðàÿýêñòðåìàëüóäîâëåòâîðÿåò óñèëåííîìó óñëîâèþ Ëåæàíäðà, óñëîâèþ ßêîáè,, íî x̂(·) íå ÿâëÿåòñÿ ñèëüíûì ìèíèìóìîì.x̂(·)˙E(t, x̂(t), x̂(t), v) ≤ 0 ∀t, vÒàêîé çàäà÷åé ÿâëÿåòñÿ, íàïðèìåð, ñëåäóþùàÿ:Z01(ẋ2 − xẋ3 )dt → extr; x(0) = x(1) = 0; x̂(t) = 026Çàäà÷à î áðàõèñòîõðîíå.10 1696 ãîäó Èîãàíí Áåðíóëëè â ïåðâîì â èñòîðèè ìàòåìàòè÷åñêîì æóðíàëå Ataeruditorum(îñíîâàííîìâ1682ã.)îïóáëèêîâàëad ujus solutionem Mathematii invitanturçàìåòêóProblema novum, Íîâàÿ çàäà÷à, ðåøèòü êîòîðóþïðèãëàøàþòñÿ ìàòåìàòèêè, â êîòîðîì ïðåäëàãàë âíèìàíèþ ìàòåìàòèêîâ çàäà÷óî ëèíèè áûñòðåéøåãî ñêàòà áðàõèñòîõðîíå.

 ýòîé çàäà÷å òðåáóåòñÿ îïðåäåëèòüëèíèþ, ñîåäèíÿþùóþ äâå çàäàííûå òî÷êèA è B , íå ëåæàùèå íà îäíîé âåðòèêàëüíîéïðÿìîé, è îáëàäàþùóþ òåì ñâîéñòâîì, ÷òî ìàòåðèàëüíàÿ òî÷êà ïîä äåéñòâèåì ñèëûòÿæåñòè ñêàòèòñÿ ïî ýòîé ëèíèè èç òî÷êèAâ òî÷êóBâ êðàò÷àéøåå âðåìÿ.x✲x✲(x0 , y0 )y = y(x)y(x1 , y1 )y❄(x + dx, y + dy)❄èñ. 5: Áðàõèñòîõðîí â êîîðäèíàòàõ(x, y)Ââåäåì äåêàðòîâó ñèñòåìó êîîðäèíàò íà ïëîñêîñòè (ñì.

ðèñ 6); ïóñòü îñüOxOy íàïðàâëåíàA(x0 , y0 ) è B(x1 , y1 ) èìåþò ïîëîæèòåëüíûå îðäèíàòû,ds=à x0 < x1 . Ñîëàñíî çàêîíàì ìåõàíèêè ñêîðîñòü äâèæåíèÿ ìàòåðèàëüíîé òî÷êèdt√2gy , îòêóäà íàõîäèì âðåìÿ, çàòðà÷èâàåìîå íà ïåðåìåùåíèå òî÷êè èç ïîëîæåíèÿ Aâ ïîëîæåíèå B :Z x1 p1 + ẏ 21dx; y(x0 ) = y0 , y(x1 ) = y1 .T [y(x)] = √√y2g x0ñîîòâåòñòâóåò íóëåâîé ñêîðîñòè è íàïðàâëåíà ãîðèçîíòàëüíî, îñüâåðòèêàëüíî âíèç. Ïóñòü òî÷êèÁóäåì ïðåäïîëàãàòüL(y, ẏ) =√y(·)íåïðåðûâíî äèåðåíöèðóåìîé óíêöèåé.