Higher-order numerical scheme for linear quadratic problems with bang bang controls

Transcription

1 Compu Opm Appl DOI 1.17/s z Hgher-order numercal scheme for lnear quadrac problems wh bang bang conrols T. Scarnc 1 V. M. Velov Receved: 3 June 17 The Auhors 17. Ths arcle s an open access publcaon Absrac Ths paper consders a lnear-quadrac opmal conrol problem where he conrol funcon appears lnearly and akes values n a hypercube. Is assumed ha he opmal conrols are of purely bang bang ype and ha he swchng funcon, assocaed wh he problem, exhbs a suable growh around s zeros. The auhors nroduce a scheme for he dscrezaon of he problem ha doubles he rae of convergence of he Euler s scheme. The proof of he accuracy esmae employs some recenly obaned resuls concernng he sably of he opmal soluons wh respec o dsurbances. Keywords Opmal conrol Numercal mehods Bang bang conrol Lnear-quadrac opmal conrol problems Tme-dscrezaon mehods Mahemacs Subjec Classfcaon 49M5 65L99 49J3 49N1 49J15 T. Scarnc and V. M. Velov were suppored by he Ausran Scence Foundaon FWF under Gran No. P664-N5. T. Scarnc was also suppored by he Docoral Programme Venna Graduae School on Compuaonal Opmzaon, funded by Ausran Scence Foundaon under Projec No. W16-N35. B T. Scarnc eresa.scarnc@gmal.com V. M. Velov vladmr.velov@uwen.ac.a 1 Deparmen of Sasc and Operaon Research, Unversy of Venna, Venna, Ausra Insue of Sascs and Mahemacal Mehods n Economcs, Venna Unversy of Technology, Venna, Ausra

2 T. Scarnc, V. M. Velov 1 Inroducon Dscrezaon schemes for opmal conrol problems have been largely nvesgaed n he las 6 years see, e.g., [6 9,17,1], and he more recen paper [5] and he references heren. In he aforemenoned papers and n mos of he leraure, he opmal conrols are ypcally assumed o be suffcenly smooh a leas Lpschz connuous and resuls are usually based on second-order opmaly condons. On he oher hand, whenever he conrol appears lnearly n he sysem, he lack of coercvy ypcally leads o dsconnues of he opmal conrols. Recenly, new second-order opmaly condons for sysems ha are lnear wh respec o he conrol have been developed. We refer o [3,] for analyss of secondorder necessary condons for bang bang and sngular-bang conrols, respecvely. Resuls on he sably of soluons wh respec o dsurbances were also recenly obaned, see [4,1,14,5] and he bblography heren. Based on hese resuls, error esmaes for he accuracy of he Euler dscrezaon scheme appled o varous classes of affne opmal conrol problems were obaned n [1,,13,18,6,7]. The error esmaes are a mos of frs order wh respec o he dscrezaon sep, whch s naural n vew of he dsconnuy of he opmal conrol. For he same reason, usng hgher order Runge Kua dscrezaon schemes on a fxed grd does no help o mprove he order of accuracy. Seemngly, he frs paper ha addresses he ssue of accuracy of dscree approxmaons for affne problems s [3], where a hgher order Runge Kua scheme s appled o a lnear sysem, bu he error esmae s of frs order or less. A new ype of dscrezaon scheme was recenly presened n [3] for Mayer s problems for lnear sysems. The dea behnd hs scheme goes back o [15,,8] and s based on a runcaed Volerra Fless-ype expanson of he sae and adjon equaons. The analyss of he convergence and of he error esmae makes use of he srong Hölder merc sub-regulary of he map assocaed wh he Ponryagn maxmum prncple, proved n [5]. The goal of he presen paper s o exend hs dscrezaon scheme and he peranng error analyss o affne lnear-quadrac problems. Ths exenson s no a roune ask, due o he appearance of he sae funcon n he assocaed wh he problem swchng funcon, and of boh he sae and he conrol, n he adjon equaon. More precsely, we consder he problem subjec o Jx, u := 1 xt QxT + q xt T 1 + x W x + x Su d mn 1.1 ẋ = Ax + Bu, x = x, for a.e. [, T ], 1. u U := [ 1, 1] m for a.e. [, T ]. 1.3

3 Hgher-order numercal scheme for lnear quadrac Here [, T ] s a fxed me horzon, he sae x s n-dmensonal, he nal sae x s gven, he conrol u s m-dmensonal, Q R n n, q R n, and he marx funcons A, W : [, T ] R n n and S, B : [, T ] R n m are gven daa; he superscrp denoes ransposon. Admssble conrols are all measurable funcons wh values n he se U for a.e. [, T ]. Lnear erms are noncluded n he negrand n 1.1, snce hey can be shfed n a sandard way no he dfferenal equaon 1.. The opmal conrols n he problem are ypcally concaenaons of bang bang and sngular pahs. In hs paper, we assume ha he opmal conrols are of srcly bang bang ype wh a fne number of swches, and he componens of he swchng funcon have a ceran growh rae a her zeros, characerzed by a number κ 1. Ths number appears n he error esmae obaned n hs paper for he proposed dscrezaon scheme. Genercally, κ = 1, and n hs case he error esmae s of second order. The paper s organzed as follows. In Sec. we recall some noaons and formulae he assumpons. In Sec. 3 we nroduce our dscrezaon scheme and presen he man resul he error esmae. Secon 4 conans he proof. Secon 5 presens an error esmae n case of nexac soluon of he dscrezed problem. A numercal expermen confrmng he heorecal fndngs s gven n Sec. 6. Concludng remarks and furher perspecves are dscussed n Sec. 7. Prelmnares Frs of all we pose some assumpons, whch are sandard n he conex of problem Assumpon A1 The marx funcons A, B, W and S, [, T ], have Lpschz connuous frs dervaves, Q and W are symmerc. The se of all admssble conrols wll be denoed by U L. A par x, u formed by an admssble conrol u and he correspondng soluon x of 1. s referred o as an admssble process, and he se of all admssble processes s denoed by F.These F wll be consdered as a subse of he space Wx 1,1 L 1, where Wx 1,1 = Wx 1,1, T s he affne space of all absoluely connuous funcons x :[, T ] R n wh x = x, and L 1 = L 1, T has he usual meanng. 1 Due o he compacness and he convexy of U,heseF s compac wh respec o he L -weak opology for u and he unform norm for x. Thus a mnmzer ˆx, û does exsn he space Wx 1,1 L 1 nfac,alsonwx 1, L. The followng assumpon requres a sor of dreconal convexy of he objecve funconal J a ˆx, û. 1 To avod a confuson, we menon ha he admssble conrols, hence he dervave of he sae funcon, ẋ, as well as he dervave of he adjon funcon, ṗ, appearng below, belong o L. However, we use he L 1 -norms of hese dervaves n mos of he consderaons.

4 T. Scarnc, V. M. Velov Assumpon A 1 T 1 zt QzT + z W z + z Sv d for every z,v F ˆx, û. Le ˆx, û be an opmal process n problem Accordng o he Ponryagn mnmum prncple, here exss ˆp W 1, such ha ˆx, û, ˆp sasfes he followng sysem of generalzed equaons: for a.e. [, T ], =ẋ Ax Bu, x = x,.1 = ṗ + A p + W x + Su,. B p + S x + N U u,.3 = pt QxT q,.4 where N U u s he normal cone o U a u: { f u / U, N U u = {l R m : l,v u v U} f u U. Sysem.1.4 can be shorly rewren as where F s he se-valued map defned by Fx, p, u,.5 ẋ Ax Bu Fx, p, u := ṗ + A p + Wx + Su B p + S x + N U u. pt QxT q The mappng F s consdered as acng from he space X o he space Y, where X := W 1,1 x W 1,1 L 1, Y := L 1 L 1 L R n. The norms n hese spaces are defned as usual: for x, p, u X and ξ,π,ρ,ν Y, x 1,1 + p 1,1 + u 1 and ξ,π,ρ,ν := ξ 1 + π 1 + ρ + ν,

5 Hgher-order numercal scheme for lnear quadrac where x 1,1 abbrevaes x W 1,1 and s s he norm n L s, s {1, }. Noce ha he normal cone N U u o he se U L 1 a u U has he pon-wse represenaon {ξ L : ξ N U u a.e. on [, T ]}. We also defne he dsance d # u 1, u = meas { [, T ]:u 1 = u } n he space L. We menon ha U L s a complee merc space wh respec o hs merc [11, Lemma 7.]. Observe ha he ncluson.3 s equvalen o u Argmn σ w, where σ :[, T ] R m s he so-called swchng funcon, defned for all [, T ] as Thus, for j = 1,...,m, σ = B ˆp + S ˆx. w U û j = { 1 fσ j >, 1 fσ j <,.6 where σ j and û j say for he j-componen of σ and û, respecvely. Assumpon B Src bang bang propery There exs posve real numbers κ 1, m and δ such ha for every j = 1,...,m, and for every zero τ [, T ] of σ j,he nequaly σ j m τ κ holds for all [τ δ, τ + δ] [, T ]. Remark.1 Clearly, Assumpon B mples ha each componen σ j has a fne number of zeros n [, T ] and hen each componen of û s pecewse consan wh values 1 and 1. The followng heorem plays a crucal role n he error analyss of he dscrezaon scheme presened below. Is a modfcaon under weaker condons and a dfferen space seng of [4, Theorem 8], and s proved n [4]. Theorem. Le Assumpon A1 and A be fulflled. Le ˆx, ˆp, û be a soluon of generalzed equaon.5 and le Assumpon B be fulflled wh some real number κ 1. Then for any b > here exss c > such ha for every y := ξ,π,ρ,ν Y wh y b, here exss a rple x, p, u X solvng y Fx, p, u and every such rple sasfes he nequaly x ˆx 1,1 + p ˆp 1,1 + u û 1 c y 1/κ..7 We menon ha he above propery of he mappng F and he reference pon ˆx, û, ˆp X and Y s a sronger verson non-local wh respec o x, p, uof he so-called merc sub-regulary [1].

6 T. Scarnc, V. M. Velov 3 Dscrezaon scheme In hs secon we propose a dscrezaon scheme for problem whch has a hgher accuracy han he Euler scheme whou a subsanal ncrease of he numercal complexy of he dscrezed problem. We recall ha he Euler mehod has already been profoundly nvesgaed n he case where bang bang conrols appear e.g. [1,,13,18,6]. As menoned n he nroducon, n dong hs we use an dea ha orgnaes from [15,8] and was mplemened n [3] n he case of Mayer s problems. The approach uses second order runcaed Volerra Fless seres, as descrbed n he nex subsecon. 3.1 Truncaed Volerra Fless seres Gven a naural number N, denoe h = T/N, and = h for =,...,N. Leu be an admssble conrol on [, +1 ]. The soluon x of 1. on he nerval [, +1 ] can be represened as see [3, Secon 3] x = [I + A + ] A + A x + B + AB us ds + AB + B s us ds + Oh 3, 3.1 where for shorness we skp he fxed argumen n he appearng funcons, has, A := A, B := B, ec. As usual, we denoe by Oε, ε>, any funcon ha sasfes Oε /ε c, where c s a generc consan, has, dependng only on he daa of he problem hus, ndependen of and, alhough Oh 3 may depend on and n he above conex. The second-order expanson of he soluon of he adjon equaon. dffers from han [3, Secon 3] due o he presence of an negral erm n he objecve funconal 1.1, herefore we derve below. For all [, +1 ], p = As ps + W sxs + Ssus ds + p Applyng he frs order Taylor expanson for A, W, S a, he represenaons 3.1 of xs and 3.ofps, and skppng all hrd order erms we oban ha p = + s A + s A p+1 A p+1 + Wx + Suζ dζ ds

7 Hgher-order numercal scheme for lnear quadrac + + W + s W x + s Ax + B s Sus + s S us ds + p+1 + Oh 3. uζ dζ ds Hence, we oban he followng runcaed Volerra Fless expanson for he adjon funcon: p = I + +1 A + h A p A A p+1 + Wx + A S s uζ dζ ds + +1 Wx + h W + WAx s + WB uζ dζ ds + S us ds + S s us ds + Oh Now we shall derve a second order approxmaon o he negral erm of he objecve funconal Jx, u on [, +1 ] n erm of he frs and second order momenum of he conrols. Concernng he quadrac erm n he x-varable we make use of he frs order par of represenaon 3.1 and of he Taylor expanson a for W. Rememberng ha W s a symmerc-marx-valued funcon, we oban ha x + A x + B us ds x W x d = W + W x + A x +B uτ dτ d + Oh 3 = hx W + hw A + h W x +x W B us ds d + Oh 3. Noe ha an easy calculaon mples ha uτ dτ d = h u d u d. 3.4

8 T. Scarnc, V. M. Velov Now we consder he mxed erm n he negral n 1.1. A calculaon of he same fashon as he prevous one yelds x Su d = Le us focus on he las erm: = x + A x + B us ds S + S u d + Oh 3 = x S u d + S u d + A x S u d + B us ds S u d + Oh B B us ds S u d Inegrang by pars we oban he relaon = us ds S d us ds. u B S + S B us dsb S us ds. us ds d Followng [8], n order o oban a second-order expanson expressed n erm of he frs and second-order momenum of u, we assume he followng. Assumpon I The marx B S s symmerc for all [, T ]. Indeed, usng Assumpon I we oban from he las exposed equaly he expresson B us ds S u d = 1 us dsb S us ds, whch can be subsued n 3.5. Noce ha Assumpon I s always fulflled f m = 1. The above obaned second order approxmaons wll be used n he nex subsecon o defne an approprae dscree-me approxmaon of problem

9 Hgher-order numercal scheme for lnear quadrac 3. The numercal scheme Frs of all, observe ha he represenaon 3.1 ofx for [, +1 ] depends on he conrol u only hrough he negrals +1 u d and +1 u d.thesame apples o he approxmaons of he negral erms of he objecve funconal obaned n he las subsecon. By changng he varable = + hs, hs par of negrals can be represened n he form hz 1 and h z, respecvely, where z 1 = 1 ϕs ds, z = 1 sϕs ds, and ϕs = u + hs s a measurable funcon wh values n [ 1, 1]. By varyng u, hence ϕ, n he se of all admssble conrols on [, T ], he couple z 1, z R m generaes a srcly-convex and compac se Z m R m. Noe ha Z m can be expressed as he Caresan produc m 1 Z, where Z s he Aumann negral 1 1 Z := [ 1, 1]ds. 3.6 s As poned oun [3], s a maer of sandard calculaon o represen he se Z n he more convenen way as Z = {α, β : α [ 1, 1], β [φ 1 α, φ α]}, where φ 1 α := α + α and φ α := α α. Followng he hn provded by he represenaon 3.1, we nroduce he noaons A := A + h A + A, B := B + ha B, C := A B + B, and replace he dfferenal equaon 1. wh he dscree-me conrolled dynamcs x +1 = x + ha x + B u + hc v, =,...,N 1, x gven, 3.7 u,v Z m =,...,N Takng no accoun he approxmaons of he objecve funconal n he prevous subsecon, we nroduce s dscree-me counerpar: for x = x,...,x N, u = u,...,u N 1, v = v,...,v N 1, J h x, u,v := 1 x N Qx N + q + h N 1 = x W x + ha x + h x W x

10 + h N 1 = hx W B u v + x T. Scarnc, V. M. Velov S u + hs v + ha x S v + h N 1 B S u, u. 3.9 We denoe by P h he dscree problem of mnmzng 3.9 subjec o The Karush Kuhn Tucker heorem gves he followng necessary condons for opmaly of x,...,x N, w,...,w N 1, wh w := u,v Z m : here s an adjon sequence p,...,p N such ha = x +1 + x + ha x + B u + hc v, 3.1 = p + I + ha p +1 + h S u + hs v + ha S v + h W + h W A + h A W + h W + h W B u v, 3.11 N Z m w B + p +1 + S x + hb W x + hb S u h C p +1 B W x + S A + S, 3.1 x = p N + Q x N + q = In order o oban 3.1 we use agan Assumpon I. x 3.3 Consrucon of connuous-me conrols and order of convergence Le {x, u,v, p } be any soluon of sysem Based on he sequence {u,v } = N 1 we shall consruc a connuous-me admssble conrol u such ha usds = hu, s usds = h v, =,...,N The consrucon s by dea smlar o han [3] wh he essenal dfference ha now u akes values only n he se { 1, 1} and he consrucon s smpler. For α, β Z, wh α = 1 has, α 1, 1] and β [φ 1 α, φ α] defne τα,β := 1 + β 1 + α 1 + α, θα,β := 1 + β α α. 4 For α = 1weseτ = θ =. Gven ha β [φ 1 α, φ α], hs s, n fac, an exenson by connuy. Clearly, τ θ, whle τ s mpled by β φ 1 α and θ 1smpledbyβ φ α. Then defne he admssble conrol u componen-wse

11 Hgher-order numercal scheme for lnear quadrac as follows: for j = 1,...,mand =,...,N 1seτ j and 1 u j := 1 f [, + hτ j, 1 f [ + hτ j, + hθ j f + hθ j, +1]. = τu j,vj, θ j = θu j,vj, ], 3.15 The funcons τ and θ are defned n such a way ha he relaons 3.14 are fulflled. To show hs, s enough o subsue he above defned u n 3.14 and calculae he negrals. We skp hs rval bu cumbersome calculaon. We menon han our framework he pars u j,vj, =,...,N 1, j = 1,...,M, ypcally belong o he boundary of he se Z. In such a case every componen of he conrol u defned n 3.15 has a mos one swchng pon per mesh nerval [, +1 ] and we can dsngush he followng possbles: f u j = 1oru j -a f u j 1, 1 and v j = 1, hen u j = 1, respecvely u j = 1, n [, +1 ]; = φ 1 u j hen τ j =, θ j = 1 + u j /, hus { u j := 1 f [, + hθ j ], 1 f + hθ j, +1]; b f u j 1, 1 and v j = φ u j hen τ j = 1 u j /, θ j = 1, hus { u j 1 := f [, + hτ j ], 1 f + hτ j, +1]. A hrd possbly s ha u j,vj happens o belong o he neror of Z: f u j 1, 1 and v j φ 1 u j, φ u j hen formula 3.15 has o be used o defne u. In fac, formula 3.15 gves a unfed descrpon of all he above cases, where some of he hree subnervals nervals n 3.15 degenerae n he ypcal cases and. Theorem 3.1 Le Assumpon A1, A and I be fulflled. Le ˆx, û be a soluon of problem for whch Assumpon B s fulflled wh some κ 1. Le ˆp be he correspondng adjon funcon so ha ˆx, ˆp, û sasfes he Ponryagn sysem.1.4. Then for every naural number N sysem has a soluon {x, u,v, p }. Moreover, for he connuous embeddng of u,v defned n 3.15, holds ha max xk ˆx k + p k ˆp k + d # u, û ch /κ k=,...,n

12 T. Scarnc, V. M. Velov We menon ha, for me-nvaran problems whou sngular arcs, Assumpon B s ypcally fulflled wh a number κ {1,, 3,...}, correspondng o he mulplcy of he zeros of he swchng funcon. As argued n [3], he case κ = 1 s n a ceran sense generc and he error esmae 3.17 s of second order n hs case. Also n he case κ>1 he order of accuracy s doubled n comparson wh ha proved n [6] for he Euler scheme. Ulzaon of hgher order Runge Kua schemes on a fxed mesh could nomprove he accuracy of he Euler scheme due o he dsconnuy of he opmal conrol. A soluon {x, u,v, p } of sysem can be obaned by any mehod for solvng he dscree problem The adjon varables p do no need o be drecly nvolved. For example, we use for numercal compuaons a verson of he seepes descen mehod, where he adjon equaon s only ndrecly nvolved for calculaon of he dervave of he objecve funcon 3.9 wh respec o he varables u,v. In any case, he adjon funcons ˆp and {p } are well defned and he error esmae 3.17 s vald. As we argue n Sec. 5, he soluon {x, u,v, p } can be nexac, whch leads o a modfcaon of he error esmae as saed here. 4 Proof of Theorem Prelmnares. Le {x, u,v, p } be a soluon of he dscree sysem and le u be he connuous embeddng of {u,v } defned n We embed he sequences {x } and {p } no he spaces W 1,1 x and W 1,1 usng he hn provded by he expansons developed n Sec Namely, for [, +1,we defne x := I + A + A + A x + B + A B us ds + C s us ds 4.1 and p := [I + +1 A + +1 A + h ] A + +1 W x + h W + W A x s + W B uτ dτ ds + +1 A W x + A S + S s uζ dζ ds + S us ds p +1 s us ds. 4.

13 Hgher-order numercal scheme for lnear quadrac We show below ha p W 1,1.From3.4 and 3.14 follows ha ] lm [I p = + ha + h + A + h A p +1 + hw x + h W x + W A x + h W B u v + h A W x + hs u + h A S + S v. 4.3 Snce ] [I + ha + h A + h A p +1 = I + ha p +1, he rgh-hand sde of 4.3 s equal o he expresson of p gven by Thus, p s connuous a, and hence p W 1,1. The proof ha x Wx 1,1 s analogous and can be found n [3, Secon 5]. By Theorem., for every b > here exs a number c such haf y b hen x ˆx 1,1 + p ˆp 1,1 + u û 1 c y 1/κ, 4.4 where y = ξ,π,ν,ρs he resdual ha x, p, u gves n.1.4, has, y Fx, p, u. Thus we have o esmae he norm of hs resdual. The esmae ξ 1 Oh of he frs resdual s obaned n [3, Secon 4], where he prmal dfferenal equaon s he same as n he presen paper. We shall analyze below he resdual n he remanng Eqs Resdual n. and.4. Frs, we dfferenae he expresson n 4. for [, +1 : [ ṗ = A + +1 A + A ] p +1 + W x + W + W A x + W B us ds + +1 A W x + A S uζ dζ + S u + S u = A I + +1 A p +1 + W x + W A x + W B us ds + +1 A W x + A S uζ dζ + Su + O; h.

14 T. Scarnc, V. M. Velov Here and below O; h Ch for some consan C > whch s ndependen of [, T ].Usng4.1 and hen 4. we oban ha ṗ = A I + +1 A p A W x + A S uζ dζ + W x + Su + O; h = A p + W x + Su + O; h. Hence, we deduce ha π Oh. Noce ha he Eq. 3.1 gves ha he resdual n.4 s zero, has, ν =. 3. Resdual n.3. Frs of all, we derve a second order expanson of he erm B p + S x appearng n.3. By 4.1, 4. and he Taylor expanson for B and r we have B p + S x = B + B p A p W x + S us ds + S + S I + A x +B us ds + O; h. Then, by usng he defnon of B and C we oban ha B p + S x = B + C p +1 + B +1 W x +S us ds + S B us ds + S I + A x Snce Assumpon I means B S = S B, + S x + O; h. B p + S x = B + C p +1 + B +1 W x + S + S I + A x us ds + S x + O; h. 4.5 Our goal s now o esmae he norm of he resdual n.3. Snce N U u = j=1,...,m N [ 1,1]u j, we analyze a sngle componen j of.3. Moreover,.3 s

15 Hgher-order numercal scheme for lnear quadrac a pon-wse relaon, herefore we consder on an arbrarly fxed nerval [, +1 ]. We also menon ha he se Z s he area surrounded by he wo parabolas β = φ 1 α and β = φ α, where φ 1 α φ α. Thus he normal cone o Z s easy o calculae and he followng expresson s provded n [3, Secon 4]: f α, β / Z {αλ, μ λ N Z α, β = : μ,λ } f α { 1, 1} {μζ + α, ζ : μ } f α 1, 1 β {φ 1 α, φ α} {} f α 1, 1 β φ 1 α, φ α, 4.6 where ζ = sgnα β. We consder separaely each of he cases, and for consrucon of connuous me conrol ha appear n Sec Case u j { 1, 1}. To be specfc, le us assume ha u j = 1, hence v j = φ 1 1 = φ 1 = 1/. The case u j = 1 s smlar. The normal cone o Z a he pon 1, 1/ s [see he second lne n 4.6] N Z 1, 1/ = { } μ, 1 + λ 1, 1 : μ, λ. Then due o 3.1 for every j = 1,...,m here exs μ and λ such ha j B p +1 + S x + hb W x + hb S u = λ, 4.7 h C p +1 B W x + S A + S j x = μ λ. 4.8 Observe ha, for [, +1 ], λ + μ λ h = μ h + λ 1, 4.9 h hus he quany above s non-negave for all [, + h. Thus, addng up 4.7 and 4.8, he laer mulpled by /h, we oban ha [ B + C p +1 + B +1 W x + S u + S I + A x + S x ] j. By 3.14 he quany above s dencal o he j-h componen of he rgh-hand sde of 4.5, modulo O; h. By he fac ha u j = 1 n case, we hus oban Case u j he case v j B p + S x j + Oh N [ 1,1] u j , 1, v j {φ 1 u j, φ u j j }. We consder he case v = φ 1 u j ; = φ u j can be reaed smlarly. The connuous-me conrol u j s

16 defned by 3.16, where he jump pon θ j u j,φu j s see he hrd lne n 4.6 T. Scarnc, V. M. Velov = 1 u j /. The normal cone o Z a { } N Z u j,φu j = μ 1 + u j, : μ. By 3.1, here exss μ such ha B p +1 + S x + hb W x + hb S u j = μ1 + u j, 4.11 j h C p +1 B W x + S A + S x = μ. 4.1 Observe ha, from he defnon of θ j, follows ha he quany μ1 + u j + μ h 4.13 s non-posve whenever [, + hθ j, and non-negave whenever [ + hθ j, +1]. Thus, addng up 4.11 and 4.1, he laer mulpled by /h, and usng he defnon of u j we oban ha B p +1 + S x + hb W x + hb S u j + C p +1 B W x + j +S x N [ 1,1] u j. S A By 3.14 and 4.5, he lef-hand sde erm of he relaon above s equal o B p+ S x + r j + Oh. Ths proves 4.1 n he case. Case By 3.1 and he fac ha u j k,vj k In Z, wehave Then, = B p +1 + S x + hb W x + hb S u j =, h C p +1 B W x + S A + S x j =. B + p +1 + S x + hb W x + hb S u C p +1 B W x + S A + S x j.

17 Hgher-order numercal scheme for lnear quadrac Ths and 4.5 yeld 4.1 n he case. We can fnally conclude ha ρ = Oh. Summarzng, we have obaned ha y c 1 h, where c 1 s ndependen of N. Snce c 1 h c 1 T =: b, Theorem. mples exsence of c such ha for every naural N x ˆx 1,1 + p ˆp 1, + u û 1 ch /κ. We know ha x = x and p = p, hence max x ˆx + p ˆp + u û 1 c h /κ. =,...,N Now we focus on he las erm n he lef-hand sde. Snce û and u ake only values ±1, as already poned ou for nsance n [5, Secon 4], we have ha d # u, û c 3 u û 1 for some c 3, and hs concludes he proof. 5 Error esmae n case of nexac soluon of he dscree problem The esmaon 3.17 n Theorem 3.1 s vald on he assumpon ha he dscree-me problem s exacly solved. In he presen secon we ncorporae n he error esmaon possble naccuracy n solvng he dscree-me problem. The basc argumen for has dencal wh he one for Mayer s problems, presened n [3, Secon 5], herefore we only skech. We assume ha as a resul of a numercal procedure for solvng he mahemacal programmng problem we have obaned an approxmae soluon { x }, { p }, { w } of he frs order opmaly Karush Kuhn Tucker sysem Ths means ha he relaons are sasfed by he sequences { x }, { p }, { w } wh some resdual ξ,π,ρ,ν= he sze or he resdual by he number ε := ξ l1 + π l + ρ l + ν =h + max =,...,N 1 ρ + ν. {ξ } N 1, {π } N 1, {ρ } N 1,ν N 1 = ξ + max =,...,N 1 π. We measure Usng he approxmae soluon { w }, one can defne an approxmaon, ũ,ofhe opmal conrol û n he same way as descrbed n Sec Then he esmaon 3.17 n Theorem 3.1 akes he form max x ˆx + p ˆp + d # ũ û c ε + h 1/κ. 5.1 =,...,N The proof of hs saemens no sraghforward, bu he argumens dencal wh han [3, Secon 5], herefore we do no repea here.

18 T. Scarnc, V. M. Velov Clearly, n order o make he error arsng n he proposed dscrezaon scheme and he error n solvng he dscrezed problem conssen, one has o solve he laer wh accuracy ε proporonal o h. 6 A numercal expermen Example 6.1 Le us consder he followng opmal conrol problem on he plane: subjec o { 1 } 1 mn by1 + x d ẋ = y, x = a, ẏ = u, y =, wh conrol consran u [ 1, 1] and for a > 1/, b >. Here for approprae values of a and b here s a unque opmal soluon wh a swch from u = 1ou = 1 a me τ whch s a soluon of he equaon 5τ 4 + 4τ 3 1a + 36τ + 4a + τ + 4b 1a 3 =. Moreover, τ s a smple zero of he swchng funcon, hus κ = 1. Takng, for example, a = 1 and b =.1, he equaon above becomes 5τ 4 + 4τ 3 48τ + 44τ 1.6 = and he sngle real soluon of hs equaon n [, 1] s τ = wh all dgs beng correc. We solved sysem for varous values of he dscrezaon sep h = T/N N s a naural number by usng a verson of he graden projecon mehod. The compuaon of he approxmae soluons { x }, { p }, { w } s done wh accuracy measured by he resdual ε see Sec. 5 hgher han h, so ha he heorecal error esmae 5.1sch. As before, ˆx, ˆp, û s he exac soluon of he consdered problem, and ũ s he connuous me conrol consruced as descrbed n Sec. 3.3 for he compued { w }={ũ, ṽ } noe ha ũ depends on sze of he mesh N, herefore furher we use he noaon ũ N nsead of ũ. Table 1 Here e N = d # û ũ N s he error of he numercally obaned conrol ũ N for varous values of N N e N e N /h The las lne gves he values e N /h

19 Hgher-order numercal scheme for lnear quadrac In Table 1 we repor numercal resuls, focusng on he mos crcal error e N = d # û ũ N. We also calculae he value e N /h, whch accordng o 5.1 should be bounded. Ths s confrmed by he resuls. 7 Concludng remarks In hs paper we exend he analyss of he dscrezaon scheme nroduced n [3]for Mayer s problems o embrace convex lnear-quadrac problems, affne wh respec o he conrol. The opmal conrols are assumed o be purely bang bang, wh an addonal assumpon nvolvng he assocaed swchng funcon. Our dscrezaon approach ops for solvng a dscree-me opmzaon problem nvolvng an addonal conrol varable. Ths yelds an order of convergence whch doubles ha of he Euler s mehod. The prce for has ha he dscree problem nvolves quadrac conrol consrans nsead of he box-ype consrans n he orgnal problem. Is worh nong ha he componens of he opmal conrols could be, n general, concaenaons of bang bang and sngular arcs. Ths challengng case wll be a subjec of furher nvesgaon. I requres, among oher hngs, a deeper analyss of he merc sub-regulary of he sysem of frs order opmaly condons under perurbaons a sep n hs drecon s made n [13]. Anoher challengng ssue s o avod Assumpon I. Assumpon of hs knd s presen also n [9], as well as n [16], n a nonlnear conex, where requres he Le brackes of he nvolved conrolled vecor felds o vansh. An dea n hs drecon s presened n [19], bu does no seem o be effcen for numercal mplemenaon. Acknowledgemens Open access fundng provded by Ausran Scence Fund FWF. Open Access Ths arcle s dsrbued under he erms of he Creave Commons Arbuon 4. Inernaonal Lcense hp://creavecommons.org/lcenses/by/4./, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded you gve approprae cred o he orgnal auhors and he source, provde a lnk o he Creave Commons lcense, and ndcae f changes were made. References 1. Al, W., Baer, R., Gerds, M., Lempo, F.: Error bounds for Euler approxmaons of lnear-quadrac conrol problems wh bang bang soluons. Numer. Algebra Conrol Opm. 3, Al, W., Baer, R., Lempo, F., Gerds, M.: Approxmaons of lnear conrol problems wh bang bang soluons. Opmzaon 61, Aronna, M.S., Bonnans, J.F., Dmruk, A.V., Loo, P.A.: Quadrac order condons for bang-sngular exremals. Numer. Algebra Conrol Opm. 3, Al, W., Schneder, C., Seydenschwanz, M.: Regularzaon and mplc Euler dscrezaon of lnearquadrac opmal conrol problems wh bang bang soluons. Appl. Mah. Compu. 87/88, Bonnans, J.F., Fesa, A.: Error esmaes for he Euler dscrezaon of an opmal conrol problem wh frs-order sae consrans. SIAM J. Numer. Anal. 55, Donchev, A.L.: An a pror esmae for dscree approxmaons n nonlnear opmal conrol. SIAM J. Conrol Opm. 34, Donchev, A.L., Hager, W.W.: Lpschzan sably n nonlnear conrol and opmzaon. SIAM J. Conrol Opm. 31,

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