Key words. Optimal control, eikonal equation, Hamilton-Jacobi equation, approximate controllability

Transcription

1 OPTIMAL CONTROL OF THE PROPAGATION OF A GRAPH IN INHOMOGENEOUS MEDIA KLAUS DECKELNICK, CHARLES M. ELLIOTT AND VANESSA STYLES Abstract. We study an optimal control problem for viscosity solutions of a Hamilton-Jacobi equation describing te propagation of a one dimensional grap wit te control being te speed function. Te existence of an optimal control is proved togeter wit an approximate controllability result in te H norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference sceme and describe some numerical results. Key words. Optimal control, eikonal equation, Hamilton-Jacobi equation, approximate controllability AMS subect classifications. 49J, 49L5, 49M5. Introduction. We are concerned wit te formulation and numerical approximation of an optimal control problem for first order quasi-linear equations of Hamilton-Jacobi type describing te motion of a front in an inomogeneous medium. Tus we are concerned wit te initial value problem,.).) V Γ = a Γ) = Γ were V Γ denotes te normal velocity of an evolving surface Γt) R n from an initial surface Γ. Te strictly positive velocity function a is taken to be spatially dependent. It is known for suc an initial value problem tat wen formulated as finding Φx, t) satisfying Φ t = a Φ, Φx, ) = Φ x) and Γt) is te zero level set of Φ, t) tat tere is a unique viscosity solution. Suc a problem arises in etcing out a surface Γt) wit a prescribed etc rate wic may depend on te medium, [, 3, ]. It is natural to control te final surface at time T say using te etc rate. Suppose tat te surface can be written as a grap in te plane, i.e. Γt) = {x, yx, t)) x R} R. Te issue is te control of te location of te grap at a given time evolving from a planar surface by coice of te prescribed velocity. We suppose tat te speed function is positive and depends INSTITUT FÜR ANALYSIS UND NUMERIK, OTTO-VON-GUERICKE-UNIVERSITÄT MAGDEBURG, UNIVERSITÄTSPLATZ, 396 MAGDEBURG, GERMANY. MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK, COVENTRY CV4 7AL, UK. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF SUSSEX, BRIGHTON BN 9RF, UK.

2 only on te orizontal direction so tat a = ax) >. Ten.) takes te form.3) y t = ax) y x, x R, t, T ). Taking te speed to depend on x implies tat we are considering a vertically striated medium in wic etcing takes place at differing rates. An example would be a vertically layered medium in wic te etc speed is piecewise constant. We wis to fix te speed function in order tat a particular etced sape is acieved at a fixed time T. Actually by scaling te time scale in te equation we can coose tis time to be T =. Te setting is tat of a control problem for a first order quasi-linear partial differential equation of Hamilton-Jacobi type. In general it is known tat optimal control for nonlinear yperbolic equations is a difficult topic. Solutions of our state equation are considered in te context of viscosity solutions. In Section we formulate tis as a control problem wit a quadratic obective function and te grap eikonal equation as te state equation. Te idea is tat we seek te speed function a to minimize te quadratic energy functional J a) were J a) =.4) y a x, T ) y T x) dx a xdx I under certain constraints on a were y T is te grap of te target etc sape, y a is te solution of te state equation and. Te first term of te energy is a fidelity term wereas te second is a regularization for te speed in order to ensure tat te state equation is well posed. If we allow te speed to depend only on a finite number of parameters tis term can be omitted. Let us note tat a related problem involving a stationary eikonal equation for te first arrival time of a front was studied numerically in []. We also refer to [3, 4] for results concerning te optimal control of first order conservation equations. We begin by proving te existence of optimal controls for differing assumptions concerning te speed function. Since te solutions of te state equation are non-smoot and non-unique in general we need to consider viscosity solutions of te state equation. It is classical tat tere exists a unique viscosity solution wen te speed function is continuous. Wen considering optimal control in tis case one can apply W, constraints on te set of admissable functions wic ten would need to be applied wen applying iterative descent metods for te optimization. Tis can be avoided by regularising te functional.4) by adding a quadratic term in te gradient, tat is taking positive. On te oter and it is also interesting to consider discontinuous piece-wise constant a and tis is te main focus of te paper. In tis case we use te well-posedness teory of [7, 8] in order to prove existence of an optimal control. An interesting question is weter te target is acievable. Our next contribution is to sow approximate controllability by sowing convergence in te H norm as te number of sub-intervals increases. In order to realise te optimal control we turn to numerical discretization. Te idea is to discretize te state equation and te functional in suc a way tat te discrete optimization problem as a minimizer. Tis leaves te interesting and difficult question as to weter te discrete minimizers I

3 approximate a minimizer of te control problem. Our next result is to sow convergence as te mes size converges to zero. Tis relies on te use of monotone finite difference scemes for te Hamilton Jacobi equation. We formulate tis in te case of te discontinuous piece-wise constant speed function. We coose to implement monotone finite difference scemes wic are differentiable wit respect to te solution so tat te discrete minimization problem can be solved iteratively by a descent metod based on a discrete adoint equation. We found tat tis metod worked very well in practice. Our paper is organized as follows: Section discusses existence and uniqueness of te state equation togeter wit an existence result for te optimal control. An approximate controllability result in te H norm is proved in Section 3. Te numerical solution is considered in Section 4 wit a convergence result being proved for a monotone discretization of te state equation. We conclude wit some numerical results.. Formulation and existence... Matematical setting. Let I = [, ] and < α < β <. We sall assume tat te control function a belongs to one of te following sets: K) K : = {a : I R ax) = a i, x ˆx i, ˆx i ), aˆx i ) = a i a i, i =,..., L, a ) = a) = a a L, α ax) β, x I} were = ˆx < ˆx <... < ˆx L < ˆx L = is a partition of I. K) K := {a : I R ax) = L i= a iφ i x), α ax) β, x I}, were {φ i } L i= satisfy φ i W, I), φ i ) = φ i ), φ i x), i =,..., L and L i= φ ix) =, x I. K3) K := {a W, I) α ax) β, x I, a ) = a)}. In wat follows we tink of te function a as being extended to a -periodic function on R. Given a K we ten consider te eikonal equation.).).3) y t = ax) yx in R, T ] y, ) = in R yx, t) = yx, t) x R, t T. According to Teorem.3 in Section. te initial value problem.)-.3) as a unique solution y W, R, T )) wic we sall denote by y a in order to indicate its dependence on a K. Let us now consider te following control problems: P) min J a) = y a x, T ) y T x) dx I subect to a K, were K is given eiter by K) or K). 3

4 P ) min J a) = y a x, T ) y T x) dx a xdx I I subect to a K, were K is te set given in K3). Here, y T L I) is a given function and >. Remark.. In te definition of K) te coice of te values of a at te end points of te intervals can be replaced by aˆx i ) = a i were a i is any value in te interval [mina i, a i ), maxa i, a i )]... Existence and uniqueness for te state equation. Let us begin by recalling te notion of viscosity solution originally introduced by Crandall & Lions [5]) see also [6]). Since te function a is possibly discontinuous we use a generalized definition due to Isii [9]). Applied to our situation tis results in te following Definition.. A function y C R, T ]) is called a viscosity subsolution of.) if for eac ζ C R, )): if y ζ as a local maximum at a point x, t ) R, T ], ten ζ t x, t ) a x ) ζ x x, t ). A function y C R, T ]) is called a viscosity supersolution of.) if for eac ζ C R, )): if y ζ as a local mimimum at a point x, t ) R, T ], ten ζ t x, t ) a x ) ζ x x, t ). A viscosity solution of.),.) ten is a function y C R [, T ]) wic is bot a viscosity sub-and supersolution and wic satisfies yx, ) = for all x R. In te above, a x) := lim r sup{ay) x y < r}, a x) := lim r inf{ay) x y < r} denote te upper and lower semicontinous envelope of a respectively. Teorem.3. Let a K, were K is one of te sets given in K)-K3). Ten tere exists a unique viscosity solution y W, R [, T ]) of.)-.3) and y satisfies.4) y W, R [,T ]) C, were C depends on T, α and β. Proof. We will give te main ideas of a proof wic covers te cases K)-K3). Existence: Consider te following regularised problem.5).6).7) y ɛ xx yt ɛ ɛ yx) ɛ = aɛ yx) ɛ in R, T ] y ɛ, ) = in R y ɛ x, t) = y ɛ x, t) x R, < t T. 4

5 Here, ɛ > and a ɛ is a suitable mollification of a wit.8) α a ɛ β in R. Using te Leray-Scauder principle it is possible to prove tat.5)-.7) as a unique smoot solution, []. Te corresponding argument relies on te derivation of a-priori estimates on y ɛ t and y ɛ x for a solution of.5)-.7). As tese bounds also motivate wy.4) olds we sall briefly sketc teir proof for te convenience of te reader. Differentiating.5) wit respect to time gives y ɛ tt ɛ y ɛ t,xx y ɛ x) ɛ y ɛ xxy ɛ x y ɛ x ) ) yɛ t,x a ɛ y ɛ x y ɛ x ) yɛ t,x = in R, T ). Te maximum principle togeter wit.6) and.8) implies.9) max R [,T ] yɛ t = max yt, ɛ ) = max a ɛ β. R R In order to derive an estimate on y ɛ x we introduce zx, t) := y ɛ xx, t)). We can assume tat µ := max I [,T ] z >. Tere exists a point x, t ) I [, T ] suc tat zx, t ) = µ. Ten z x x, t ) = y ɛ xx, t ) y ɛ xxx, t ) =. Since µ >, y ɛ xx, t ) cannot vanis and we infer y ɛ xxx, t ) =. Substituting tis information into.5) gives and terefore by.9) and.8).) y ɛ tx, t ) = a ɛ x ) zx, t ) max I [,T ] yɛ x = zx, t ) y ɛ tx, t ) a ɛ x ) β ). α Since te estimates.9) and.) are uniform in ɛ tere exists a subsequence ɛ function y W, R [, T ]), wic is -periodic in space suc tat and a y ɛ y uniformly in R [, T ]. It can be sown tat y is a viscosity solution of.)-.3) see [6] if a is continuous and e.g. [7, 8] for te case of discontinuous a). Uniqueness: Tis follows from te results in [6] and [8] respectively. Wen using te latter result, note tat a satisfies assumption A) in [8], p. 64 wile A) is not needed for te uniqueness proof. Corollary.4. Suppose tat a, ã K wit a ã a.e. in I. Ten y a yã in R [, T ]. 5

6 Proof. Let us denote by y ɛ and ỹ ɛ te solutions of.5)-.7) corresponding to a ɛ, ã ɛ respectively. Here, a ɛ, ã ɛ are mollifications of a, ã so tat a ɛ ã ɛ in R. Recalling te proof of Teorem.3 we may assume tat for some sequence ɛ, y ɛ y a, ỹ ɛ yã uniformly in R [, T ]. Te functions v := y ɛ ỹ ɛ are -periodic in space and satisfy a differential inquality of te form v t b v xx c v x in R [, T ] for suitable functions c and b >. Since v, ) = te maximum principle implies tat v in R [, T ] and te result follows after sending. Corollary.5. Let K be defined by K) and suppose tat a K satisfies a k = β for some k {,..., L}. Ten y a x, t) = βt, x [ˆx k, ˆx k ], t >. Proof. Define ã K by ã i = β, i =,..., L. Clearly, yã βt so tat Corollary.4 implies tat y a x, t) βt, x R, t >. Let a ɛ and y ɛ be as in te proof of Teorem.3, so tat a ɛn a a.e. in I, y ɛn y a uniformly in R [, T ] for some suitable sequence ɛ n N t, T ] wit ɛ n, n. Recalling.6) and.5) we obtain for ˆxk ˆx k βt y a, t) = = lim n t ˆxk = lim n ɛ ˆxk a ɛ n yt ɛn ˆx k t ˆxk ˆx k βt y a, t)) = lim n ˆx k wic implies te desired result. ) limn t ˆxk ˆxk y ɛn xx y ɛn x ) ) = limn ɛ a ɛn t y ɛn, t)) ˆx k a ɛ n y ɛ n ) x ) yt ɛn ˆx k t arctany ɛn x ) ˆx k ˆx k ), n,.3. Existence for te optimal control problem. Teorem.6. Te control problems P) and P ) ave at least one solution a K. Proof. Let us first consider P) and suppose tat a m ) m N K is a minimizing sequence so tat J a m ) inf J a), as m. a K Te corresponding solutions y m = y a m of te state equation satisfy in view of.4) y m W, R [,T ]) C = CT, α, β). 6

7 Combining tis estimate wit te observation tat for K), K) te set K is contained in a finitedimensional space we obtain a subsequence m k ) k N and functions a K, y W, R [, T ]) suc tat a m k a uniformly in I, y m k y uniformly in R [, T ]. We claim tat y = y a. We sow first tat y is a viscosity subsolution in te sense of Definition.. Let ζ C R, )) and suppose tat y ζ as a local maximum at some point x, t ) R, T ]. Using standard arguments we can assume tat x, t ) is a strict maximum from wic we deduce te existence of a sequence x k, t k ) wit te properties tat lim x k, t k ) = x, t ) and y m k ζ as a local maximum at x k, t k ). k Since y m k is a viscosity solution of.) we infer tat ζ t x k, t k ) a m k ) x k ) ζ x x k, t k )). Observing tat we obtain after passing to te limit k, lim sup k a m k ) x k ) a x ) ζ t x, t ) a x ) ζ x x, t )). In a similar way one proves tat y is a viscosity supersolution, so tat y = y a. Finally we infer from te uniform convergence of y m k to y tat J a) = inf J ã). ã K In te case of P ) were K is given as in K3) we observe tat a minimizing sequence a m is bounded uniformly in H I) so tat tere is a subsequence m k ) k N and a K suc tat a m k a uniformly in I, a m k x a x in L I). Te remainder of te proof follows in a fasion similar to te previous case. 3. Approximate controllability in te H ) norm. Let us consider problem P) for a given target y T C I). We assume tat te set K is given by K), so tat te admissible control functions are constant on I i = ˆx i, ˆx i ), were = ˆx < ˆx <... < ˆx L < ˆx L = is a partition of I. It is natural to expect tat y T can be reaced in a better way as we decrease te fineness of te partition. It turns out tat tis can be made matematically precise if one works in a weak norm and allows te speeds to become small, but still positive. In wat follows we fix te upper bound β on te speeds in suc a way tat 3.) β > T max x I y T x). 7

8 Let us introduce te set K β := {a : I R ax) = a i, x ˆx i, ˆx i ), aˆx i ) = a i a i, i =,..., L, a ) = a) = a a L, < a i β, i =,..., L}. Te following lemma says tat if te speed is sufficiently small on some subinterval I i, ten te mean value of te corresponding solution over I i will lie below min x I y T x). Lemma 3.. Let y T C I) be positive. Ten tere exists < α < T min y T x) suc tat for x I every a K β : if a i = α for some i {,..., L} and α α ten y a, T ) < min I i y T x). I x I i Proof. Let a K β and suppose tat a i = α α for some i {,..., L}, were α will be determined later. Corollary.4 implies tat y a x, t) yâx, t), x, t) R [, ), were â K is given by â i = α and â k = β for k i. It is terefore sufficient to prove te claim for â. One cecks tat yâ is given on I [, T ] by te formula βt, ˆx k x ˆx k, k i, βt γx ˆx i ), ˆx i x ˆx i ct, yâx, t) = αt, ˆx i ct x ˆx i ct, βt γx ˆx i ), ˆx i ct x ˆx i, were β γ = α and c = β α. γ Here, α is cosen a-priori so small tat ct inf i ˆx i ˆx i ). A straigtforward calculation sows tat I i yâ, T ) = β α)ct γc T αˆx i ˆx i )T = γc T α I i T = αβ α) 3 T α I β α) i T αβt α I i T. Hence we ave tat y a, T ) < min I i y T x) provided tat I x I i αβt I i wic can be acieved for α > sufficiently small. α < T min x I y T x), 8

9 We can now formulate te main result of tis section. Teorem 3.. Let y T C I) be positive and = ˆx < ˆx <... < ˆx L < ˆx L =. Ten inf y a, T ) y T H ) a K ) βt y T L I) max ˆx i ˆx i ). β i=,...,l Proof. Coose < α < T min y T x) according to Lemma 3. and set Ω := α, β) L R L. We define x I F : Ω R L by F a) i := y a, T ), i =,..., L, T I i I i were a = a,..., a L ) t and a : I R is te corresponding piecewise constant function. Using similar arguments as in Section one can sow tat F is continuous. Let b R L be given by b i = y T, i =,..., L. T I i I i Recalling 3.) we ave 3.) α < T min x I y T x) b i T max x I y T x) < β, i =,..., L, so tat b Ω. We now consider te following omotopy H : Ω [, ] R L : Denoting by deg te Brouwer degree we ave Ha, σ) := σf a) σ)a b. degh, ), Ω, ) = degid b, Ω, ) = degid, Ω, b) =, since b Ω. In order to verify tat H is an admissible omotopy we ave to sow tat / H Ω, σ) for all σ, ]. Assume tat tere exist a Ω and σ, ] suc tat Ha, σ) =, i.e. b i = σ 3.3) y a, T ) σ)a i, i =,..., L. T I i I i We distinguis two cases: Case : a i = β for some i {,..., L}. Corollary.5 implies tat y a x, T ) = βt, x I i so tat 3.3) yields b i = β, a contradiction to 3.). Case : a i = α for some i {,..., L}. In tis case we conclude wit te elp of 3.3) and Lemma 3. b i = σ y a, T ) σ)α < σ T I i I i T min y T x) σ) x I T min y T x) b i, x I 9

10 wic is again a contradiction. Tus, H is an admissible omotopy so tat degh, ), Ω, ) = and by te properties of te Brouwer degree tere exists a Ω suc tat Ha, ) =, or equivalently 3.4) y a, T ) = I i y T, I i i =,..., L. Let ϕ H I). For te state corresponding to a we deduce I y a, T ) y T )ϕ = L i= I i y a, T ) y T )ϕ = by 3.4). Using Poincaré s inequality we deduce tat I L i= I i y a, T ) y T ) ϕ I i y a, T ) y T )ϕ L ) ya, T ) L I i) y T L I i) ˆxi ˆx i ) ϕ L I i) i= Since y a x, T ) βt, x I we obtain and te teorem is proved. max ˆx i ˆx i ) ) y a, T ) L i=,...,l I) y T L I) ϕ L I) max i=,...,l ˆx i ˆx i ) y a, T ) L I) y T L I) y a, T ) y T H ) βt y T L I)) max i=,...,l ˆx i ˆx i ) I i ϕ ) ) ϕ L I). let 4. Numerical approximation. Let us coose space and time steps and τ respectively and x =, Z, t n = nτ, n N, were = /J and τ = T/N. Furtermore we set G,τ := {x, t R [, T ] Z, n N}. For a grid function Y : G,τ R we denote by Y n its value at te point x, t. It is convenient to define te difference operators 4.) Y n := Y n Y n, Y n := Y n Y n, t Y n := Y n Y n. τ In wat follows we aim to find approximate solutions of P) were K is given by K). We begin by discussing te discretisation of te state equation. 4.. Discretisation of te state equation. Te solution of te state equation.)-.3) is approximated by te grid function Y : G,τ R wic for a given a K satisfies t Y n = ax )F ɛ Y n, Y 4.), =,..., J, n =,..., N 4.3) Y =, Z 4.4) Y n J = Y n, Z, n =,..., N,

11 were we ave cosen te discrete Hamiltonian to be F ɛ p, q) = p ) ɛ ɛ) q ) ɛ ɛ). Here, ɛ > is a regularization parameter and p = maxp, ), q = minq, ). Te purpose of introducing a regularization is to ensure tat F ɛ is differentiable. Te relation 4.) can be written in te form 4.5) Y n = GY, n N, were, 4.6) GV ) := V τax )F ɛ V, V ), =,..., J; GV )J = GV ), Z for a grid function V : R/Z R. Following [] we need to sow tat 4.)-4.4) gives rise to a consistent, monotone and stable approximation of te state equation. Te first point follows from te estimate F ɛ p, p) p = p ɛ ɛ) p 4.7) p ɛ ɛ p ɛ, p R. Let us next address te monotonicity of te sceme. Lemma 4.. Suppose tat τ. Ten G is monotone, i.e. V W implies tat GV ) GW ). β Proof. Let V W. A calculation sows tat for ɛ >, p, q R, F ɛ p, q), p F ɛ p, q). q As a consequence we ave for =,..., J GW ) GV ) = W V τax ) F ɛ W, W ) Fɛ V, V ) ) = W V ) τ ξax ) τ ηax ) ) τ ax ) ξw V ) ηw V ) ) for some ξ [, ], η [, ]. Recalling tat ax ) β, τ β GW ) GV ). as well as W V we deduce tat Finally, we ave te following stability bounds wic reflect corresponding estimates for te continuous problem. Corollary 4.. Let a K and suppose tat τ. Ten te solution Y of 4.)-4.4) satisfies: β a) Y n τ Y n β, Z, n =,..., N ;

12 Y b) n Y n β ɛ, Z, n =,..., N. α Proof. Te lower bound in a) follows immediately from te definition of te sceme and te fact tat a and F ɛ are nonnegative. We prove te upper bound by induction on n. Te assertion is clear for n =. Let us suppose tat it olds for some n {,..., N }. Ten we ave Y n yields Y n βτ, Z so tat Lemma 4. togeter wit te relation GV c) = GV ) c c R) Y n = GY GY βτ = Y n βτ. Since F ɛ p, q) maxp, q ) ɛ we deduce from a) for Z β Y n τ Y n = ax )F ɛ Y n, Y α max Y, Y ) αɛ. Observing tat Y n = Y n, b) follows. We extend te solution Y of 4.)-4.4) to a function Y,τ W, R, T )) via Y,τ x, t) := t n t τ x x Y n x x ) Y n t t n x x Y n τ x x ) Y n if x, t) [x, x ] [t n, t n ]. Note tat Y,τ x, t) = Y,τ x, t). Furtermore, Corollary 4. implies tat 4.8) Y,τ W, CT, α, β) for τ β, < ɛ. We are now in position to prove a convergence result for te approximation of te state equation. Teorem 4.3. Let, τ, ɛ) denote a sequence suc tat τ β,, ɛ and let Y,τ be te corresponding solutions of 4.)-4.4). Ten Y,τ y a uniformly in R [, T ]. Proof. In view of 4.8) and Arzela s teorem tere exists a subsequence k, τ k, ɛ k ) k N wit lim k k = lim k ɛ k =, τ k k β and a function y C R [, T ]), wic is -periodic in space suc tat 4.9) Y k) := Y k,τ k y uniformly in R [, T ]. We claim tat y is te viscosity solution of.)-.3), i.e. y = y a. To see tis, suppose tat ζ C R, )) and tat y ζ as a local maximum at a point x, t ) R, T ]. Using standard arguments from te teory of viscosity solutions we may assume tat tis maximum is strict and global. In view of 4.9) tere exists a sequence of gridpoints x k, t nk ) G k,τ k suc tat lim k x k, t nk ) = x, t ) and ξ k := Y n k k), k W n k k = max Y k) W ). G k,τ k

13 Here, W n = ζx, t. In particular, Y n k k) W n k ξ k so tat Lemma 4. implies Y n k k), k = GY n k k) )) k GW nk )) k ξ k = W n k k τ k ax k )F ɛk and ence, recalling te definitions of W n and ξ k, ζx k, t nk ) ζx k, t nk τ k ) τ k ax k )F ɛk W n k k W n k k, W k k., W k k, Sending k we obtain ζ t x, t ) a x ) ζ xx, t ), were we ave used te smootness of ζ, te fact tat lim k x k, t nk ) = x, t ) and 4.7). Hence, y is a viscosity subsolution of.) and in a similar way one proves tat y is also a viscosity supersolution. Since te solution is unique, te wole sequence Y,τ ) converges to y a. Te relations.) and.3) follow immediately from 4.9) and te definition of te sceme. 4.. Discrete optimal control problem. In wat follows we sall identify te function a K wit its values a, =,..., L and abbreviate a = a,..., a L ) t. We denote by Y a te solution of 4.)-4.4), were we always assume tat τ β. In addition we suppose tat te target y T satisfies y T ) = y T ) and denote by Y T te piecewise linear function on I wit values 4.) Y T := y T x ), =,..., J. A discrete obective function is defined by J a) := J Y N a, Y T ). = We consider te following discrete control problem: P,τ ) minimize J a), subect to a [α, β] L. Since te solution of te state equation depends continuously on a tere exists a minimizer of te finite dimensional optimization problem P,τ ), wic we denote by a. Teorem 4.4. Suppose tat y T H I), y T ) = y T ). Tere exists a sequence, ɛ suc tat a a and te associate function a K is a minimum of J. Proof. Combining te bounds α a, β wit 4.8) we obtain a sequence, ɛ and a [α, β] L, y C R [, T ]), wic is -periodic in space suc tat 4.) a a, Y a y uniformly in R [, T ]. Similarly as in te proof of Teorem 4.3 it can be sown tat y = y a. Our aim is to prove tat J a ) J a) for all a K. Since J a ) J a) for all a K te claim follows provided tat 3

14 we can sow tat 4.) J a) J a), J a ) J a ). To begin, note tat for a piecewise linear function η : I R wit η ) = η) we ave η J ηx ) ηx ) ) C I η L I). Hence, = J a) J a) C Ya,x N Yx T L I) C y T H I), as, ɛ, Y N ) C Y N a a Y T L I) y a, T ) y T L I) ) y a, T ) L I) y T H I) were we ave used 4.8) and Teorem 4.3. Recalling 4.), te second convergence in 4.) can be sown in a similar way so tat te result follows Adoint equation. In order to compute te derivative of J wit respect to a we formulate te following discrete adoint equation. For fixed a K let Y a be as above. We ten denote by P : G,τ R te solution of te following backward problem: 4.3) 4.4) 4.5) P N t P n = Ya, N Y T, Z, = ax ) F ɛ p ax ) F ɛ q P n J = P n, Z. Y a,, n Y a, P n ax ) F ɛ Y a,, n Y a, P n ax ) F ɛ q p =,..., J, n = N,...,. Y a,, n Y a, P n Y n a,, Y n a, Lemma 4.5. Let a K and P : G,τ R be te solution of 4.3), 4.4). Ten N J a) = τ a i J n= = χ i F ɛ Y a,, n Y a, P n, i =,..., L ) P n were χ i = if x ˆx i, ˆx i ) if x = ˆx i or x = ˆx i oterwise. 4

15 Proof. Fix i {,..., L}. We define Z : G,τ R by Z n = Y a, n a). In wat follows we simply a i write Y = Y a. Recalling 4.3) and observing tat Z = we ave 4.6) J a i a) = J Y N = N = J Y T )Z N P n Z n n= = N = J P n n= = = J = P N P n Z n ) Z N N P Z n J n= = P n Z n Z n ). Differentiating 4.) wit respect to a i yields 4.7) t Z n = χ i F ɛ Y n, Y ax ) F ɛ p Y n, Y n Hence from 4.6), 4.7) and 4.4) we obtain N J a) = τ a i J n= = Z n ax ) F ɛ q N J τ τ τ = τ n= = N J n= = N n= = N J J n= = ) Zn ax ) F ɛ q Y n, Y Zn. { ax ) F ɛ p Y, n Y P n ax ) F ɛ p Y n, Y P n Y n, Y P n ax ) F } ɛ q Y, n Y P n χ i F ɛ Y n, Y P n P n ax ) F ɛ p Y n, Y Z n Z P n ax ) F ɛ q Y n, Y Z n Z ) n χ i F ɛ Y n, Y P n by sifting indices and using te periodicity of Z and P Optimisation metod. We set P S a) i = maxα, mina i, β)) for i =,,..., L and now define a proected gradient algoritm to solve te problem P,τ ): Step Coose a [α, β] L, γ, ) and tol. 5

16 Step For k =,,,..., do Steps 3-6. Step 3 Set s k = J a k J ) = Step 4 Coose te minimum σ k {,, 4 ) a k ). a L,...} for wic a a k ),, J J P S a k σ k s k )) J a k ) γ σ k P S a k σ k s k ) a k. Step 5 Set a k = P S a k σ k s k ). Step 6 If a k a k < tol ten STOP. Here, denotes te euclidian norm in R L Numerical experiments. We consider six numerical experiments, using te following values of y T x): 4.8) y T x) =.7 cosπx )) 4.9) y T x) =.3 x ) x) 4.) y T x) =.5, x.5 3 x, x.5 3 x, x.5, x 4.) y T x) =., x. 3 x, x. 3 x, x., x 4.) y T x) =.5, x <., x.5, < x 6

17 4.3) y T x) =., x <.5, x., < x. In all our computations we take T =, α =., β = 4, =.5, τ = β, tol = 9, γ =. and ɛ =. Ten for eac of te six experiments we take L = 5, and 5. Te computational results for y T given by 4.8) are displayed in Figure 4.. We see six subplots, te upper tree plots sow Y T dased line) and Ya N bold line), wile te lower tree plots sow a. In te left and plots we took L = 5, in te centre plots we took L = and in te rigt and plots we took L = 5. Figures take te same form as Figure 4. except tat 4.9)-4.3) were used in place of 4.8). In Figures it is not always easy to distinguis Ya N from Y T ; we terefore display plots of te error Ya N Y T for te above six coices of y T in Figures Te left and plot of Figure 4.7 sows te difference Ya N Y T for L = 5 dased line), L = dotted line) and L solid line), for y T being given by 4.8). In te rigt and plot of Figure 4.7 and Figures 4.8 and 4.9 we sow corresponding results for te coices 4.9)-4.3) of y T. We conclude wit Figures in wic we look at te convergence of J a k ) as k increases; in Figure 4. we took y T given by 4.8) and compared te value of J a k ), wit k large, for tree values of L; L = 5, and 5. Eac plot sows J a k ) plotted against te iteration number k, te left and plot displays results for L = 5, te centre plot L = and te rigt and plot L = 5. Figures 4. and 4. take te same form as Figure 4. except tat 4.) and 4.3) were used in place of 4.8). From Figures we see tat for eac value of L, J a k ) converges, say to J a ), and tat as L increases J a ) decreases. 5. Conclusion. We ave formulated an optimal control problem in wic te final sape of a one dimensional grap evolving wit a prescribed inomogeneous speed is controlled by varying te speed function. Te problem is one of controlling a first order Hamilton Jacobi equation. Te matematical formulation is one of a quadratic obective functional wit a state equation wic is posed in te sense of viscosity solutions. Existence is sown for several variants for constraints on te speed function. An approximate controllability result in te H norm is proved wit respect to te number of intervals defining a piece-wise constant speed function. A discrete version of te control problem was formulated and sown to be convergent as te mes size goes to zero. Our numerical metod was based on a sceme for te state equation wic was differentiable wit respect to te discrete state. Tis allowed te derivation of an adoint equation. Finally we displayed some numerical results. Tis is an example of an optimal control problem in wic te nonlinear state equation is not formulated in a classical manner but in te sense of viscosity solutions and in wic te solution of te state equation is not differentiable wit respect to te control variable. Te approac of tis article could be extended to consideration of iger space dimensional problems. 7

18 .5.5.5!!!.5.5.5!!! Fig. 4.. Results for 4.8), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a !!!.5.5.5!!! Fig. 4.. Results for 4.9), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a. REFERENCES [] G. Barles, P.E. Souganidis, Convergence of approximation scemes for fully nonlinear second order equations, Asymptotic Anal. 4, ). [] J. M. Berg, A. Yezzi, A. R. Tannenbaum, Pase transitions, curve evolution and te control of semiconductor manufacturing processes, in: Proceedings of te IEEE Conference on Decision and Control, Kobe), pp ). 8

19 !!! !!! Fig Results for 4.), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a !!!.5.5!.5.5!.5.5! Fig Results for 4.), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a. [3] J. M. Berg and N. Zou, Sape-based optimal estimation and design of curve evolution processes wit application to plasma etcing, IEEE Trans. Auto. Control 46, ). [4] C. Castro, F. Palacios, E. Zuazua An alternating descent metod for te optimal control of te inviscid Burger s equation in te presence of socks Newton Institute Preprint INI-7-4 [5] M.G. Crandall, P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Mat. Soc. 77, 4 983). 9

20 .5.5!.5.5!.5.5!.5.5!.5.5!.5.5! Fig Results for 4.), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a !!!.5.5!.5.5!.5.5! Fig Results for 4.3), upper plots sow Y T dased line) and Y N a solid line), lower plots sow a. [6] M.G. Crandall, L.C. Evans, P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Mat. Soc. 8, ). [7] K. Deckelnick, C.M. Elliott, Uniqueness and error analysis for Hamilton-Jacobi equations wit discontinuities, Interfaces Free Bound. 6, No.3, ). [8] K. Deckelnick, C.M. Elliott: Propagation of graps in two-dimensional inomogeneous media, Applied Numerical Matematics 56, ).

21 ...!.!!!.!! Fig Plots of Y T Ya N for L = 5 dased line), L = dotted line) and L = 5 solid line) wit y T given by 4.8) in te left and plot and y T given by 4.9) rigt and plots...!.!.!.!!!.4!! Fig Plots of Y T Ya N for L = 5 dased line), L = dotted line) and L = 5 solid line) wit y T given by 4.) in te left and plot and y T given by 4.) rigt and plots.!!!!!!!! Fig Plots of Y T Ya N for L = 5 dased line), L = dotted line) and L = 5 solid line) wit y T given by 4.) in te left and plot and y T given by 4.3) rigt and plots.

22 x!3 x! Fig. 4.. Plots of J a k ) for L = 5 left), L = centre) and L = 5 rigt) wit y T given by 4.8). $ x!3 8 x!4 4 $% 4 4 ' 34 Fig. 4.. Plots of J a k ) for L = 5 left), L = centre) and L = 5 rigt) wit y T given by 4.) Fig. 4.. Plots of J a k ) for L = 5 left), L = centre) and L = 5 rigt) wit y T given by 4.3). [9] H. Isii, Hamilton-Jacobi equations wit discontinuous Hamiltonians an arbitrary open sets, Bull. Fac. Sci. Engrg. Cuo. Univ. 8, ). [] O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and quasilinear equations of parabolic type, Translations of Matematical Monograps Vol. 4, American Matematical Society, Providence 968. [] S. Leung, J. Qian, An adoint state metod for tree-dimensional transmission traveltime tomograpy using first-arrivals, Commun. Mat. Sci. 4, no., ). [] J.A. Setian, Level Set Metods, Cambridge University Press 996. [3] S. Ulbric, A sensitivity analysis and adoint calculus for discontinuous solutions of yperbolic conservation laws wit source terms, SIAM J. Control and Optimization 4, ).