Kapustyan, V.Y.

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

OPTIMAL CONTROL IN PARABOLIC SINGULARPERTURBATED PROBLEM WITH OBSTACLE.c© Vladimir Y. Kapustyan1. OPTIMALITY CONDITIONS.Consider such optimal control problem with an obstacle: to find u(t) ∈ U = {v :v(t) ∈ L2(0, T ), | v(t) |≤ ξ for a.e. t ∈ [0, T ]} such thatI(v) =12T∫0(∫Ω(y(x, t)− z(x))2dx+ νv2(t))dt → min, (1)where y(x, t) is the solution of variational inequality of parabolic type in [1-2](yt(x, t)− ²24y(x, t)− g(x)v(t))(y(x, t)− ψ(x)) = 0a.e.inQyt(x, t)− ²24y(x, t)− g(x)v(t) ≥ 0,y(x, t) ≥ ψ(x) a.e. in Q, (2)y(x, 0) = y0(x), a.e. in Ω, y(x, t) = 0, a.e. in Σ;here Q = Ω × (0, T ), Σ = ∂Ω × (0, T ), Ω ∈ Rn–has compact closure and smooth(from C∞) (n−1)-dimensional boundary ∂Ω, z(x) ∈ L2(Ω), g(x) ∈ Lq(Ω), y0(x) ∈W2−2/q,q0 (Ω), ψ(x) ∈ H2(Ω), ψ(x) ≤ 0 a.e. on ∂Ω, y0 ≥ ψ(x) a.e. in Ω, q >max(n,2),0 < ²¿ 1, ν =const> 0, 4 is the Laplace operator.The problem (1)-(2) has at least one solution u. Let (y, u) be an pair from the problem(1)-(2). Then ([1]) there exists a function p ∈ L2(0, T ;H1(Ω))⋂BV ([0, T ];Y ∗), Y =Hs(Ω)⋂H1(Ω), s > n/2 which satisfies the following equations:−pt − ²24p = y(x, t)− z(x)a.e.in{(x, t) : y(x, t) > ψ(x)},p(x, t) = 0, a.e. inΣ;p(x, t)(g(x)u(t) + ²24y) = 0a.e.in{y = ψ}, (3)p(x, T ) = 0a.e.inΩ,u(t) =−ξ, (g, p(·, t))− νξ > 0,−ν−1(g, p(·, t)),ξ, (g, p(·, t)) + νξ < 0,(4)where (g, p(·, t)) = ∫Ωg(x)p(x, t)dx.2. FORMAL ASYMPTOTICS.We shall find the outer [6] decomposition of the solution of (2)-(4) in the formy¯(x, t) =∞∑i=0²iy¯i(x, t); p¯(x, t) =∞∑i=0²ip¯i(x, t). (5)ASSUMPTION 1. Suppose z(x), y0(x), ψ(x), 0 ≤ g(x) ∈ C∞(Ω)• Zero componentsof the decomposition (5) are defined like the solution of the problem(y¯0t(x, t)− g(x)u¯0(t))(y¯0(x, t)− ψ(x)) = 0 in Q,y¯0t(x, t)− g(x)u¯0(t) ≥ 0, y¯0(x, t) ≥ ψ(x) in Q,y¯0(x, 0) = y0(x) in Ω; (6)−p¯0t = y¯0(x, t)− z(x) in {(x, t) : y¯0(x, t) > ψ(x)},p¯0(x, t)g(x)u¯0(t) = 0 a.e. in {y¯0 = ψ},p¯0(x, T ) = 0 in Ω,u¯0(t) =−ξ, (g, p¯0(·, t))− νξ > 0,−ν−1(g, p¯0(·, t)),ξ, (g, p¯0(·, t)) + νξ < 0.(7)Introduce setsQ0 = {(x, t) : y(x, t) = ψ(x)a.e.}, Q+ = {(x, t) : y(x, t) > ψ(x)a.e.}, (8)Q = Q0⋃Q+, Q0⋂Q+ = ∅,where y(x, t) is the solution of (2)-(4).Then Q¯0, Q¯+ are their zeroth-order approximations.ASSUMPTION 2. Suppose that p(x, t) = 0 in Q0 and let Q0 be a cylinder inRn+1 and its base be two-connected domain (Ω\Ω0). Suppose that the outer boundaryis equivalent to ∂Ω and the inner boundary (∂Ω) possesses properties of the outerboundary •Then the correlations are fulfilled in Q¯+(x ∈ Ω0, t ∈ T 01 ) :{˙¯y0 = −g(x)ξ,− ˙¯p0 = y¯0 − z, (g, p¯0)0 − νξ > 0;(9)(x ∈ Ω0, t ∈ T 02 ) :{˙¯y0 = g(x)ξ,− ˙¯p0 = y¯0 − z, (g, p¯0)0 + νξ < 0;(10)(x ∈ Ω0, t ∈ T 03 ) :{˙¯y0 = −ν−1g(x)(g, p¯0)0,− ˙¯p0 = y¯0 − z,(11)T =3⋃i=1T 0i , T0i⋂T 0j = ∅, i 6= j,where (·, ·) denotes the scalar products by Ω¯0.Let’s go over to the problems for (g, y˜0)0, (g, p¯0)0 for the definition of zero componentsof the control switching momentst ∈ T 01 :{(g, ˙˜y0)0 = − ‖ g ‖20 ξ,−(g, ˙¯p0)0 = (g, y˜0)0, (g, p¯0)0 − νξ > 0;(12)t ∈ T 02 :{(g, ˙˜y0)0 =‖ g ‖20 ξ,−(g, ˙¯p0)0 = (g, y˜0)0, (g, p¯0)0 + νξ < 0;(13)t ∈ T 03 :{(g, ˙˜y0)0 = −ν−1 ‖ g ‖20 (g, p¯0)0,−(g, ˙¯p0)0 = (g, y˜0)0, y˜0 = y¯0 − z.(14)The problems (12)-(14) may be solved by dint of phase picture [3] which allows to definethe control structure. Then two cases are possible: 1) phase point ((g, y˜0)0, (g, p¯0))doesn’t go on bounds, i.e. it belongs to set P1 = {0 < (g, p¯0)0 ≤ νξ, ν−1/2 ‖ g ‖0(g, p¯0)0 < (g, y˜0)0 < ∞}⋃{−νξ ≤ (g, p¯0)0 < 0, −∞ < (g, y˜0)0 < ν−1/2× × ‖ g ‖0(g, p¯0)0}; 2) phase point goes on bounds, i.e. it belongs to set P2 = {(g, p¯0)0 ≥ νξ,ν−1/2 ‖ g ‖0 (g, p¯0)0 ≤ (g, y˜0)0 < ∞}⋃{−νξ > (g, p¯0)0, −∞ < (g, y˜0)0 ≤ ν−1/2‖ g ‖0 (g, p¯0)0}. In the case 1) the solution has an appearance:(g, y˜0)0 = (y0 − z, g)0ch((ν−1/2 ‖ g ‖0 ××(T − t))(ch(ν−1/2 ‖ g ‖0 T ))−1,(g, p¯0)0 = ν1/2 ‖ g ‖−10 (y0 − z, g)0××sh(ν−1/2 ‖ g ‖0 (T − t))××(ch(ν−1/2 ‖ g ‖0 T ))−1(15)on condition that initial data satisfy the inclusion{(y0 − z, g)0, ν1/2 ‖ g ‖−10 (y0 − z, g)0th(ν−1/2 ‖ g ‖0 T )} ∈ P1. (16)In the case 2) systems (12)-(13) have the solution in 0 ≤ t ≤ τ0 (τ0 is the moments ofdescent of control from limitation){(g, y˜0) = ∓ξ ‖ g ‖20 t+ (g, y0 − z)0,(g, p¯0) = ±ξ(ν − 1/2 ‖ g ‖20 [τ20 − t2]) + (g, y0 − z0)0(τ0 − t).(17)On the segment t ∈ (τ0, T ] system (14) has the solution(g, y˜0)0 = ±ξν1/2 ‖ g ‖0 ch(ν−1/2 ‖ g ‖0 (T − t))××(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1,(g, p¯0)0 = ±ξνsh(ν−1/2 ‖ g ‖0 ××(T − t))(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1.(18)Let’s regard futher for definition that the condition{(y0 − z, g)0, ξ(ν − 1/2 ‖ g ‖20 τ20 ) + (g, y0 − z)0τ0} ∈ P2⋂{(g, p¯0)0 > νξ, ν−1/2 ‖ g ‖0 (g, p¯0)0 ≤ (g, y˜0)0 <∞},(19)which guarantees uniqueness of the solution of equation−ξ ‖ g ‖0 (ν1/2cth(ν−1/2 ‖ g ‖0 (T − τ0))+ ‖ g ‖0 τ0) = (g, y0 − z)0, (20)is fulfilled. Thus in case 1) the couple (y¯0(x, t), p¯0(x, t)) is fined from (11). In particular,y¯0(x, t) = y0(x)− g(x) ‖ g ‖−20 (y0 − z, g)0(ch(ν−1/2 ‖ g ‖0 T )−−ch(ν−1/2 ‖ g ‖0 (T − t)))(ch(ν−1/2 ‖ g ‖0 T ))−1.(21)The question about choice of domain Ω¯0 is solved this way. Let Ω˜0 be a set fromΩ, which satisfies the condition y0(x) > ψ(x) and suppose that for any x ∈ Ω˜0 theinequalityy¯0(x, T ) > ψ(x) (22)is fulfilled, at that the function ch(ν−1/2 ‖ g ‖0 T )− ch(ν−1/2 ‖ g ‖0 (T − t)) increasesmonotonically. Systems (9)-(11) are the conditions of optimality in the optimal controlproblem: find u¯0(t) ∈ U such thatI0(v) =12T∫0(∫Ω¯0(y¯0(x, t)− z(x))2dx++νv2(t))dt→ minby bounds˙¯y0(x, t) = g(x)v(t), y¯0(x, 0) = y0(x).Let ˜˜Ω0 be a system of expanded sets which belong to Ω and contain Ω˜0 (boundaries ofthe indicated sets have the properties of the boundary ∂Ω). Then Ω¯0 is the solution ofthe optimization problem12T∫0(∫˜˜Ω0(y¯0(x, t)− z(x))2dx+∫Ω/ ˜˜Ω0(ψ(x)− z(x))2dx++ν(∫˜˜Ω0g(x)p¯0(x, t))2)dt→ min(23)by bound (16),(22), y¯0(x, t) is given by the representation (21) and scalar products (·, ·)0are calculated by ˜˜Ω0 in all terms.In case 2) the solution of (9),(11), continuous for t ∈ [0, T ] and smooth for x ∈ Ω¯0, isgiven by the couple (y¯0(x, t), p¯0(x, t)). In particular, for t ∈ [0, τ0]y¯0(x, t) = −ξg(x)t+ y0(x),for t ∈ (τ0, T ]y¯0(x, t) = −ξg(x)τ0 + y0(x) + ξν1/2 ‖ g ‖−10 ××g(x)(sh(ν−1/2 ‖ g ‖0 (T − τ0)))−1(ch(ν−1/2×× ‖ g ‖0 (T − t))− ch(ν−1/2 ‖ g ‖0 (T − τ0))).(24)The question about choice of domain Ω¯0 is solved by analogy with preceding case withnext changes: the function (23) is minimized by bounds (19),(20),(22) and y¯0(x, t) isgiven by representation (24). Let’s supplement the solutions (y¯0(x, t), p¯0(x, t)) on ∂Ω¯0by following boundary layer functions y˜0(t¯, s, t), p˜0(t¯, s, t) [5,7].Thereby the solution of (6)-(7) is constructed completely, i.e. zeroth components ofdecomposition (5) are fined.ASSUMPTION 3. Suppose the problem’s data such that the moment of the controlswitching τ0 ∈ (0, T ) exists and ∂Ω¯0 = ∂Ω •Let τ be a moment of the control descent from the bound of the initial problem.Let’s to find it in the form of an asymptotic seriesτ =∞∑j=0²jτj .The algorithm of the specification of the control switching moment is constructed in [4].THEOREM. Let’s suppose that the assumptions 1-3 are true and (19) takes place.Then the next inequalities hold||grad(y − y(N))||L2(Q) + ||grad(p − p(N))||L2(Q) ≤ C²N ,||y − y(N)||L2(Q) + ||p − p(N)||L2(Q) ≤ C²N+1,||u − u(N)||L2(0,T ) ≤ C²N+1, |I(u) − I(u(N))| ≤ C²2(N+1),whereτN =N∑i=0²iτi, y(N)(x, t) =N∑j=0(y¯j(x, t) + y˜j(t¯, s, t))²j ,p(N)(x, t) =N∑j=0(p¯j(x, t) + p˜j(t¯, s, t))²j ,u(N)(t) ={ −ξ, 0 ≤ t ≤ τN ,−ν−1 (g, p(N)(., t)), τN ≤ t ≤ T.References1. Barbu V., Optimal control 0f variation inequalities, Pitman, London, 1984.2. Barbu V., Analysis and control of nonlinear infinite dimensional systems, Academic Presspubl,Ins, 1993.3. Boltyansky V.G., Mathematical methods of optimal control, Nauka, Moscow.4. Kapustyan V. Y., Asymptotics of locally boundedcontrol in optimal parabolic problems., Ukr. math.J. 48,N1, (1996), 50-56..5. Kapustyan V. Y., Asymptotics of control in optimal singular pertyrbated parabolic problems. Globalbounds on control., Docl. AN (Russia), 333, N4, (1993), 428-431..6. Nazarov S. A., Asymptotic solution of variation inequalittes for linear operator with small param-eter by senior derivatives., Izv. AN USSR. Series of math., 54, N4, (1990), 754-773..7. Vasil’eva A. B. and Butuzov V. F., Asymptotic methods in singular perturbations theory., VysshayaShkola, Moscow., 1990.8 - a / 43, Pisargevsky str.,320005, Dnepropetrovsk, Ukraine.

Optimal control in parabolic singular perturbated problem with obstacle

http://dspace.nbuv.gov.ua/xmlui/bitstream/123456789/169288/1/26-Kapustyan.pdf

Optimal control in parabolic singular perturbated problem with obstacle

Abstract

Similar works

Full text

Available Versions

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)