Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION H.F. Gulyev 1, Z.R. Safarova 2 1 Baku State Unversty, Baku, Azerbajan 2 Nakhchvan State Unversty, Nakhchvan, Azerbajan Abstract. In ths paper, the problem of determnng the ntal functons for the second order lnear hyperbolc equaton s consdered. The problem s reduced to the optmal control problem. The necessary suffcent optmalty condton s derved for the obtaned new problem. Keywords: hyperbolc equaton, ntal functons, optmal control problem, necessary suffcent condton. AMS Subject Classfcaton: 31A25, 31B20, 49J15. Correspondng author: Z.R. Safarova, Nakhchvan State Unversty, Unversty campus, AZ7012, Nakhchvan, Azerbajan, e-mal: seferovazumrud@ymal.com Receved: 24 October 2018; Revsed: 23 November 2018; Accepted: 06 December 2018; Publshed: 28 December 2018. 1 Introducton In drect problems of mathematcal physcs, t s requred to determne the functons descrbng varous physcal phenomena. In these cases, the coeffcents of the equatons of the studed processes, rght-h sde, ntal state of the process, boundary condtons are assumed to be known. However, these parameters are often unknown. Then nverse problems arse, n whch, accordng to nformaton about the solutons of the drect problem, t s requred to determne certan characterstcs of the process. These problems are usually ll-posed. Snce these parameters are mportant characterstcs of the process under consderaton, the study of nverse problems for the equatons of mathematcal physcs s one of the mportant problems of the modern appled mathematcs (Kabankhn, 2009). In ths paper, the problem of determnng the ntal functons from the observable values of the boundary functons for the lnear second-order hyperbolc equaton s consdered. Ths problem s reduced to the optmal control problem s nvestgated by the methods of optmal control theory. We note that the smlar problem for a parabolc equaton was studed by Lons (1972). 2 Formulaton of the problem Let the state of the system u(x, t) be descrbed by the followng second order hyperbolc equaton where Au 2 u Au = f(x, t), (x, t), (1) 2 (a j (x, t) u ) a 0 (x, t)u, x x j 215
ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, 2018 = (0, T ) s a cylnder n R n1, R n s a bounded doman wth smooth enough boundary; T > 0 s a gven number, f(x, t) L 2 (), a j (x, t) C 1 ( ), a 0 (x, t) C( ) are gven functons a j (x, t) = a j (x, t),, j = 1, n, ν ξ 2 a j (x, t)ξ ξ j ; ν = const > 0, (x, t), ξ = (ξ 1,..., ξ n ) R n. Consder the problem of determnng the ntal state (u(x, 0), u(x,0) ) at the observed values of the state of the system at the boundary u S = g 0, u S = g 1, (2) where S = Γ (0, T ) s a lateral surface of the cylnder, u n a j (x, t) u x j cos(ν, x ) s a conormal dervatve; ν s outward normal to the boundary Γ of the doman, g 0 W2 1(S), g 1 L 2 (S) are gven functons. It s supposed that there exsts the functon u(x, t) from W2 1 (), satsfyng relatons (1),(2). If T > 0 s large enough then such functon s unque. Indeed f ũ(x, t) s the second such functon, then the dfference ū = u ũ satsfes the relaton 2 ū 2 Aū = 0 n, ū S = 0, u S = 0, for enough T ū 0 (T depends on the geometry of the doman, see. Lattes et.al. (1970), p.196). Thus, there exsts some defned ntal state (u(x, 0), u(x,0) ). Consequently, one can set the problem on determnaton of the value (u(x, 0), u(x,0) ) of the soluton to Cauchy problem (1),(2). 3 Transformaton of the problem We consder u(x, 0) = v0 1 = v0 2(x), v 0(x) = (v0 1(x), v2 0 (x)) as a control that should be defned by the optmal way n the sense gven below. Let v(x) = (v 1 (x), v 2 (x)) be an arbtrary control, v 1 (x) W2 1 (), v2 (x) L 2 (). Denote H W2 1() L 2(). Introduce two systems the states of whch u 1 = u 1 (x, t; v) u 2 = u 2 (x, t; v) are defned as solutons of the correspondng boundary value problems (x), u(x,0) 2 u 1 2 Au1 = f n, (3) u S = g 0, u 1 (x, 0; v) = v 1 (x), u1 (x, 0; v) = v 2 (x) n (4) 2 u 2 2 Au2 = f n, (5) u S = g 1, u 2 (x, 0; v) = v 1 (x), u2 (x, 0; v) = v 2 (x) n. (6) Here the compatblty condton g 0 t=0 = v 1 Γ = 0 216
H.F. GULIYEV, Z.R. SAFAROVA: ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM... should be met. Note that boundary value problem (3),(4) (also (5),(6)) for each control v(x) from H has the only soluton from W2 1 () (Ladyzhenskaya, 1973, pp.209-215; Lons et al., 1971, pp.296-302) the followng estmatons are vald u 1 W 1 2 () c( f L 2 () v 1 W 1 2 () v2 L2 () g 0 W 1 2 (S) ) (7) u 2 W 1 2 () c( f L 2 () v 1 W 1 2 () v2 L2 () g 1 L2 (S)). (8) Here further on by c we denote dfferent constants not dependng on the admssble controls quanttes under estmaton. As a generalzed soluton of boundary value problem (3),(4) at gven v H we assume the functon u 2 W2 1(), whch s equal to v1 (x) at t = 0 satsfes to the ntegral equalty u1 φ 1 a j (x, t) u1 φ 1 a 0 uφ dxdt v 2 (x)φ 1 (x, 0)dx = fφ 1 dxdt, for all φ 1 = φ 1 (x, t) W2,0 1 (), φ 1(x, t) = 0. Smlarly, the defnton of the generalzed soluton to problem (5),(6) may be gven. It s obvous that f v(x) = v 0 (x) = (v0 1(x), v2 0 (x)), then u 1 (v 0 ) = u 2 (v 0 ). Introduce the functonal J(v) = 1 u 1 (v) u 2 (v)] 2 dxdt. (9) 2 Then one can state that there exsts a control v 0 H for whch nf J(v) = J(v 0) = 0. (10) v H We search the value (u(x, 0) = v0 1 = v0 2 (x)) consderng relaton (10). Snce relaton (10) s equvalent to the ntal problem, t contans the nstablty property of ths problem. We reduce t to the stable problem. The functonal J(v) gven by formula (9) s unconducve n the space H, therefore we have to regularze ths functonal. For ε > 0 consder the functonal u(x,0) (x), J ε (v) = J(v) ε 2 v1 2 v 2 2 W2 1() L 2 ()], (11) that wll be mnmzed n H. Snce problems (3),(4) (5),(6) are lnear wth respect to v = (v 1, v 2 ), the functonal J(v) s convex, due to the second term J ε (v) s strong convex on H. Therefore, by vrtue of the well-known theorem (Lons, 1972, p.13) n new problem (3)-(6),(11) there s a unque element v ε = (vε, 1 vε) 2 H mnmzng J ε (v) by vrtue of the theorem (Lons, 1972, p.48) v ε v 0 n H at ε 0. Thus, we have reduced the problem under consderaton to the fndng the element v ε. Ths new problem s already stable. Indeed, relaton (10) s vald only for some partcular values g 0 g 1, satsfyng the compatblty condtons. Therefore, the element v 0 (x) n relaton (10) does not depend contnuously on the functons g 0 g 1. At the same tme, the element v ε depends contnuously on these functons. 4 Search of v ε optmalty condton Consder more general problem: t needs to fnd nf J ε (v) at v U H, where U s a convex closed subset of H. 217
ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, 2018 We nvestgate the Frechet dfferentablty of functonal (11). Introduce the adjont state ψ = (ψ 1, ψ 2 ) settng ψ 1 = ψ 1 (x, t; v ε ), ψ 2 = ψ 2 (x, t; v ε ), as solutons of the problems 2 ψ 1 2 Aψ1 = u 1 (v ε ) u 2 (v ε ) n, (12) ψ 1 S = 0, ψ 1 (x, T ; v ε ) = 0, ψ1 (x, T ; v ε ) = 0 n (13) 2 ψ 2 2 Aψ2 = u 1 (v ε ) u 2 (v ε ) n, (14) ψ 2 S = 0, ψ 2 (x, T ; v ε ) = 0, ψ2 (x, T ; v ε ) = 0 n. (15) As a generalzed solutons from W2 1 () of boundary value problems (12),(13) (14),(15) for the gven v ε = (vε, 1 vε) 2 U we assume the vector functon ψ = (ψ 1, ψ 2 ) = (ψ 1 (x, t; v ε ), ψ 2 = ψ 2 (x, t; v ε )) equal to zero at t = T, components of whch satsfy the ntegral equaltes below ψ1 for all η 1 = η 1 W2,0 1 (), ψ2 η 1 a j (x, t) ψ1 η 1 a 0 ψ 1 η 1 dxdt ψ 1 (x, 0; v ε ) η 1 (x, 0)dx = η 2 u 1 (v ε ) u 2 (v ε )]η 1 dxdt, (16) a j (x, t) ψ2 η 2 a 0 ψ 2 η 2 dxdt ψ 2 (x, 0; v ε ) η 2 (x, 0)dx = u 1 (v ε ) u 2 (v ε )]η 2 dxdt (17) for all η 2 = η 2 W2 1(). As follows from Ladyzhenskaya (1973), p.209-215; Lons et.al. (1971), p.296-302 under the above assumptons for each gven v ε U each of problems (12),(13) (14),(15) has the only soluton from W2 1 () the estmatons ψ 1 W 1 2 () c u1 (v ε ) u 2 (v ε ) L2 (), ψ 2 W 1 2 () c u1 (v ε ) u 2 (v ε ) L2 () are vald. Then from those estmaton also from (7),(8) follows that ψ 1 W 1 2 () c( f L 2 () v 1 ε W 1 2 () v2 ε L2 () g 0 W 1 2 (S) g 1 L2 (S)) (18) ψ 2 W 1 2 () c( f L 2 () v 1 ε W 1 2 () v2 ε L2 () g 0 W 1 2 (S) g 1 L2 (S)). (19) Theorem 1. Let the condtons of problems (3)-(6),(11) be satsfed. Then functonal (11) s contnuously Frechet dfferentable on U ts dfferental n the pont v ε U at the ncrement δv = (δv 1, δv 2 ) H, v ε δv U s defned by the expresson < J ε(v ε ), δv >= ψ 1 (x, 0; v ε ) ψ 2 (x, 0; v ε )]δv 2 (x)dx 218
H.F. GULIYEV, Z.R. SAFAROVA: ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM... ψ1 (x, 0; v ε ) ε vεδv 1 1 ψ2 (x, 0; v ε ) ]δv 1 (x)dx vε 1 δv 1 v x x εδv 2 2 ]dx. (20) Proof. Let v ε, v ε δv U be arbtrary controls δu = (δu 1, δu 2 ), δu 1 = δu 1 (x, t) = u 1 (x, t; v ε δv) u 1 (x, t; v ε ), δu 2 = δu 2 (x, t) = u 2 (x, t; v ε δv) u 2 (x, t; v ε ). From (3)-(6) follows that δu 1, δu 2 are solutons from W2 1 () to the followng boundary value problems 2 δu 1 2 Aδu 1 = 0 n, (21) δu 1 S =, δu 1 (x, 0) = δv 1 (x), δu1 (x, 0) = δv 2 (x) n (22) 2 δu 2 2 Aδu 2 = 0 n, (23) δu S = 0, δu 2 (x, 0) = δv 1 (x), δu2 (x, 0) = δv 2 (x) n. (24) From the knows results (Ladyzheskaya, 1973, pp.209-215; Lons et al., 1971, pp.296-302) we obtan that for the soluton of problems (21),(22) (23),(24) the estmatons δu 1 W 1 2 () c( δv1 W 1 2 () δv 2 L2 ()), (25) δu 2 W 1 2 () c( δv1 W 1 2 () δv 2 L2 ()). (26) are vald. Increment J ε (v ε ) = J ε (v ε δv) J ε (v ε ) of functonal (11) n the pont v ε U has a form J ε (v ε ) = 1 { u 1 (v ε δv) u 2 (v ε δv)] 2 u 1 (v ε ) u 2 (v ε )] 2} dxdt 2 ε 2 (v 1 ε δv 1 ) 2 (v 1 ε) 2 ( (v 1 ε δv 1 ) 2 ( ) ) v 1 2 ] ε x x (vε 2 δv 2 ) 2 (vε) 2 2] dx = u 1 (v ε ) u 2 (v ε )](δu 1 δu 2 )dxdt 1 2 ε v 1 εδv 1 (δu 1 δu 2 ) 2 dxdt ε 2 vε 1 δv 1 v x x εδv 2 2 ] (δv 1 ) 2 ( δv1 x ) 2 (δv 2 ) 2 ]dx. (27) It s obvous that the functons δu 1 (x, t) δu 2 (x, t) satsfy followng ntegral denttes δu1 φ 1 a j (x, t) δu1 φ 1 a 0 δu 1 φ 1 dxdt 219
for all φ 1 W 1 2,0 (), φ 1(x, T ) = 0, ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, 2018 δu2 φ 2 φ 1 (x, 0)δv 2 dx = 0 (28) a j (x, t) δu2 φ 2 a 0 δu 2 φ 2 dxdt φ 2 (x, 0)δv 2 dx = 0, (29) for all φ 2 W 1 2 (), φ 2(x, T ) = 0 condtons δu 1 (x, 0) = δv 1 (x), δu 2 (x, 0) = δv 1 (x) at t = 0. Takng η 1 = δu 1, η 2 = δu 2 n denttes (16), (17) φ 1 = ψ 1, φ 2 = ψ 2 n (28), (29) then subtractng (28) from (16) (29) from (17) we get ψ 1 (x, 0) δv 1 (x)dx ψ 2 (x, 0) δv 1 (x)dx From formulas (30) (31) follows that u 1 (v ε ) u 2 (v ε )]δu 1 δu 2 ]dxdt = = ψ 2 (x, 0) ψ 1 (x, 0)δv 2 (x)dx = u 1 (v ε ) u 2 (v ε )]δu 1 dxdt, (30) ψ 2 (x, 0)δv 2 (x)dx = u 1 (v ε ) u 2 (v ε )]δu 2 dxdt. (31) ] ψ1 (x, 0) δv 1 (x)dx ψ 1 (x, 0) ψ 2 (x, 0) ] δv 2 (x)dx. (32) If to consder formula (32) n the expresson for the ncrement of functonal (27), then we obtan ψ 2 ] (x, 0) J ε (v ε ) = ψ1 (x, 0) δv 1 (x)dx where R = 1 2 ψ 1 (x, 0) ψ 2 (x, 0) ] δv 2 (x)dx ε (δu 1 δu 2 ) 2 dxdt ε 2 v 1 ε δv 1 ] vε 1 δv 1 v 2 x x εδv 2 dx R, (33) (δv 1 ) 2 ( δv1 x ) 2 (δv 2 ) 2 ]dx (34) s a remander term. From estmatons (25), (26) expresson for R t s easy to get the estmaton R c( δv 1 W 1 2 () δv 2 L2 ()). (35) Thus from formula (33) estmaton (35) follows that the functonal J ε (v) s Frechet dfferentable on U ts dfferental s determned by expresson (20). Now we show that the mappng v ε J ε(v ε ), defned by expresson (20) acts contnuously from U nto H, where H s adjont to the space H H W 1 2 () L 2 (). Let δψ = 220
H.F. GULIYEV, Z.R. SAFAROVA: ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM... (δψ 1, δψ 2 ) = (ψ 1 (x, t; v ε δv ε ) ψ 1 (x, t; v ε ), ψ 2 (x, t; v ε δv ε ) ψ 2 (x, t; v ε )). From (12)-(15) follows that δψ 1 δψ 2 are solutons from the class W2 1 () to the boundary value problems 2 δψ 1 2 Aδψ 1 = δu 1 (v ε ) δu 2 (v ε ) n, δψ S = 0, δψ 1 (x, T ; v ε ) = 0, δψ1 (x, T ; v ε ) = 0 n 2 δψ 2 2 Aδψ 2 = δu 1 (v ε ) δu 2 (v ε ) n, δψ S = 0, δψ 2 (x, T ; v ε ) = 0, δψ2 (x, T ; v ε ) = 0 n. As n (18), (19), for the solutons to these problems the estmatons δψ 1 W 1 2 () c( δu1 L2 () δu 2 L2 ()) c( δv 1 ε W 1 2 () δv2 ε L2 ()), δψ 2 W 1 2 () c( δu1 L2 () δu 2 L2 ()) c( δv 1 ε W 1 2 () δv2 ε L2 ()). (36) are vald. Moreover (20) mples valdty of the nequalty J ε(v ε δv ε ) J ε(v) H c δψ 1 (x, 0; v ε ) δψ 2 (x, 0; v ε ) ] δψ1 (x, 0; v ε ) δψ2 (x, 0; v ε ) dx c δvε 1 δv1 ε δv x ε ]dx. 2 Then due to estmaton (36) the rght h sde of ths nequalty tends to zero at δv ε H 0. It leads us to the fact that v ε J ε(v ε ) s a contnuous mappng from U nto H. Theorem 1 s proved. The next theorem s on the necessary suffcent optmalty condtons n problem (3)- (6),(11) usng the dfferental of functonal (11). Theorem 2. Let the condtons of Theorem 1 be satsfed. Then for the optmalty of the controlv ε (x) = (vε(x), 1 vε(x)) 2 U n problem (3)-(6),(11) t s necessary suffcent fulflment of the nequalty ψ 1 (x, 0; v ε ) ψ 2 (x, 0; v ε )](v 2 (x) vε(x))dx 2 ε v 1 ε(v 1 (x) v 1 ε(x)) ψ1 (x, 0; v ε ) for arbtrary v = v(x) = (v 1 (x), v 2 (x)) U. ψ2 (x, 0; v ε ) ](v 1 (x) v 1 ε(x))dx vε 1 ( v 1 ) (x) v1 ε(x) v x x x ε(v 2 2 (x) vε(x))] 2 0, (37) Proof. The set U s convex n H the functonal J ε (v) s strongly convex on U accordng to Theorem 1 functonal (11) s contnuously Frechet dfferentable on U. Then due to Theorem 5 (Vaslyev, 1981, p.28)on the element v ε U t s necessary suffcent fulflment of the nequalty < J ε(v ε ), v v ε > for all v U. From ths (20) follows the valdty of nequalty (37). Theorem 2 s proved. 221
ADVANCED MATH. MODELS & APPLICATIONS, V.3, N.3, 2018 Now f to suppose that U = H n condton (37), then we get ψ 1 (x, 0; v ε ) ψ 2 (x, 0; v ε ) εv 2 ε(x) = 0 n, ψ 1 (x, 0; v ε ) ψ2 (x, 0; v ε ) ε(v 1 ε(x) vε(x)) 1 = 0 n n the sense of dstrbutons, where s the Laplace operator. As a result we get the followng Theorem 3. Let the condtons n the formulaton of problem (3)-(6),(11) hold. Then to fnd the optmal control v ε (x) = (vε(x), 1 vε(x)) 2 H t s necessary solvng the problem: { u 1 (x, 0) = u 2 (x, 0), u1 (x,0) 2 u 1 2 Au 1 = f, 2 u 2 2 Au 2 = f, 2 ψ 1 2 Aψ 1 = u 1 u 2, 2 ψ 2 2 Aψ 2 = u 1 u 2 n ; (38) u 1 = g 0, u2 = g 1, ψ 1 = 0, ψ2 = 0 on S; (39) = u2 (x,0), ψ 1 (x, T ) = 0, ψ1 (x,t ) = 0, ψ 2 (x, T ) = 0, ψ2 (x,t ) ψ 1 (x, 0) ψ 2 (x, 0) ε u1 (x,0) n the sense of dstrbutons. = 0 n, ψ1 (x,0) = 0, ψ2 (x,0) ε(u 1 (x, 0) u 1 (x, 0)) = 0 n, (40) It follows from ths fact that, to fnd the optmal control v ε (x) one frst have to solve system (38)-(40), then set vε(x) 1 = u 1 ε(x, 0), vε(x) 2 = u1 ε(x, 0). References Kabankhn S.I. (2009). Inverse Ill-Posed Problems. Sb. Scentfc Publshng House. Novosbrsk, 457 p. Lons J.-L. (1972). Optmal Control of the System Descrbed by Partal Dfferental Equatons. Moscow, Mr, 416 p. Lattes R., Lons J.L. (1970). The uas-inverson Method ts Applcatons. Moscow, Mr, 336 p. Ladyzhenskaya O.A. (1973). Boundary Value Problems of Mathematcal Physcs. Moscow, Nauka, 408 p. Lons J.-L., Magenes E. (1971). Nonhomogeneous Boundary Value Problems Ther Applcatons. Moscow, Mr, 372 p. Vaslev F.P. (1981). Methods for Solvng Extremal Problems. Moscow, Nauka, 400 p. 222