Optimal Bounds for POD Approximations of Infinite Horizon Control Problems Based on Time Derivatives

Jou nal o Scien i ic Compu ing (2025) 103:19 h ps://doi.o g/10.1007/s10915-025-02833-0 Op imal Bounds o POD App oxima ions o In ini e Ho izon Con ol P oblems Based on Time De i a i es Ja ie de F u os1 ·Bosco Ga cía-A chilla2 ·Julia No o3 Recei ed: 30 Oc obe 2023 / Re ised: 30 Janua y 2025 / Accep ed: 17 Feb ua y 2025 / Published online: 1 Ma ch 2025 © The Au ho (s) 2025 Abs ac In his pape we conside he nume ical app oxima ion o in ini e ho izon p oblems ia he dynamic p og amming app oach. The alue unc ion o he p oblem sol es a Hamil on– Jacobi–Bellman equa ion ha is app oxima ed by a ully disc e e me hod. I is known ha he nume ical p oblem is di icul o handle by he so called cu se o dimensionali y. To mi iga e his issue we apply a educ ion o he o de by means o a new p ope o hogonal decomposi ion (POD) me hod based on ime de i a i es. We ca y ou he e o analysis o he me hod using ecen ly p o ed op imal bounds o he ully disc e e app oxima ions. Mo eo e , he use o snapsho s based on ime de i a i es allows us o bound some e ms o he e o ha could no be bounded in a s anda d POD app oach. Some nume ical expe imen s show he good pe o mance o he me hod in p ac ice. Keywo ds Dynamic p og amming ·Hamil on–Jacobi–Bellman equa ion ·Op imal con ol · P ope o hogonal decomposi ion ·Snapsho s based on ime de i a i es ·E o analysis 1 In oduc ion In his pape we conside he nume ical app oxima ion o op imal con ol p oblems. The subjec is o impo ance o many applica ions such as ae ospace enginee ing, chemical p ocessing and esou ce economics, among o he s. The alue unc ion o an op imal con ol p oblem is ob ained in e ms o a i s -o de nonlinea Hamil on–Jacobi–Bellman (HJB) pa ial di e en ial equa ion. A bo leneck in he compu a ion o he alue unc ion comes om he need o app oach a nonlinea pa ial di e en ial equa ion in dimension n, which is a challenging p oblem in high dimensions. BJulia No o julia.no [email p o ec ed] Ja ie de F u os [email p o ec ed]a.es Bosco Ga cía-A chilla [email p o ec ed] 1Ins i u o de In es igación en Ma emá icas (IMUVA), Uni e sidad de Valladolid, Valladolid, Spain 2Depa amen o de Ma emá ica Aplicada II, Uni e sidad de Se illa, Se ille, Spain 3Depa amen o de Ma emá icas, Uni e sidad Au ónoma de Mad id, Mad id, Spain 123 19 Page 2 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Se e al me hods ha e been s udied in he li e a u e ying o mi iga e he so called cu se o dimensionali y al hough i is s ill a di icul ask. As s a ed in [10], he ele ance o e i- cien nume ical me hods can be seen by he ac ha me hods sol ing he HJB equa ion a e a ely used in p ac ice due o he necessa y compu a ional e o . We men ion some ela ed e e ences ha a e no in ended o be a comple e lis . In [14], a domain decomposi ion ech- nique is conside ed. In [26] semi-Lag angian me hods a e s udied. The au ho s in [22] apply da a-based app oxima e policy i e a ion me hods. A p ocedu e o he nume ical app oxima- ion o high-dimensional HJB equa ions associa ed o op imal eedback con ol p oblems o semilinea pa abolic equa ions is p oposed in [19]. In [9] a enso decomposi ion app oach is p esen ed. In [10] an app oach based on low- ank enso ain decomposi ions is applied. Me hods using spa se g ids o HJB equa ions a e p esen ed in [6]. The solu ion o HJB equa- ions on a ee s uc u e was p esen ed in [2]. The au ho o [23,24] discusses an app oach o ce ain nonlinea HJB PDEs which is no subjec o he cu se o dimensionali y. The app oach u ilizes he max-plus algeb a. In [8] a da a-d i en app oach based on he knowledge o he alue unc ion and i s g adien on sample poin s is de eloped. The au ho s o [3]p esen anew app oach whe e he alue unc ion is compu ed using adial basis unc ions. Expanded li e - a u e on he con ol o pa ial di e en ial equa ions using dynamic p og amming app oach can be ound in he las wo e e ences. In he p esen pape we concen a e on educed o de models based on p ope o hogonal decomposi ion (POD) me hods. Ou wo k is ela ed o [1]. In his e e ence he au ho s p opose wo di e en ways o apply POD me hods in he nume ical app oxima ion o he ully-disc e e alue unc ion. In he i s app oach, he au ho s choose a se o nodes in he o iginal domain ⊂Rnand p ojec hen on o a educed space  ⊂R wi h <n o ge a new se o nodes. The p oblem in his p ocedu e is ha i p oduces a nonuni o m g id in which he mesh diame e canno be p edic ed a p io i. Consequen ly, he me hod is no sui able o implemen in p ac ice. Fu he mo e, al hough his is no e lec ed in he e o bounds in [1], he e o also depends on he in e pola ion p ope ies o he a p io i unknown educed mesh in  ⊂R . In he second app oach, he au ho s use a uni o m mesh o e he educed space  . This second me hod can be implemen ed in p ac ice ( he nume ical expe imen s in [1] a e ca ied ou wi h his me hod). Howe e , as he au ho s s a e, he compu a ion o an uppe bound o he e o in his case is much mo e in ol ed and he e o bound p o ed in [1]has some d awbacks see [1, Rema k 4.7, Rema k 4.9]. Recen ly in [7], a new e o analysis is in oduced in which a bound o size O(h+k)is ob ained o he ully disc e e app oxima ions o in ini e ho izon p oblems ia he dynamic p og amming app oach. In his e o bound, his he ime s ep while kis he spa ial mesh diame e . This e o bound imp o es exis ing esul s in he li e a u e, whe e only O(k/h) e o bounds a e p o ed, see [12,13]. To bound he e o in he i s me hod in [1] he au ho s ollow he echnique in [12, Co olla y 2.4], [13, Theo em 1.3] ob aining a bound o he e o o size O(k/h).Fo he second me hod in [1], he ac o 1/halso mul iplies all he e ms on he igh -hand side o he a p io i e o bound. In his pape we p esen a new app oach, simila o he second me hod in [1], bu wi h snapsho s based on he alue a di e en imes o he ime de i a i e o he s a e o he con olled nonlinea dynamical sys em, ins ead o alues o he s a e a di e en imes. This new app oach is inspi ed in he ecen esul s in [15] whe e he au ho s p o e ha he use o snapsho s based on ime de i a i es has he ad an age o p o iding poin wise es ima es o he e o be ween a unc ion and i s p ojec ion on o he POD space.The idea o using snapsho s app oaching he ime de i a i es is no new, al hough mos o he e e ences in he li e a u e employ i s o de di e ence quo ien s (DQs) (i.e. i s o de di ided ini e 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 3 o 28 19 di e ences) ins ead o Gale kin ime de i a i es, as in [15] and he p esen pape . In [21] he se o snapsho s (a di e en imes) is inc eased wi h DQs o ca y ou he e o analysis o he case in which p ojec ions espec o he H1 0no m a e conside ed. In a mo e ecen pape , [20], he au ho s show ha he use o DQs has he added p ope y o allowing o p o e poin wise es ima es in ime. In a la e pape , [11], he au ho s p o e ha one does no need o double he se o snapsho s wi h alues a di e en imes plus DQs since only DQs plus a single ini ial alue a e enough o ge poin wise es ima es. This is a e y in e es ing esul because one can wo k wi h he same numbe o snapsho s as in he s anda d case ( he one wi h only alues o he s a es a di e en imes). In [15], he au ho s p o e ha his is also he case wi h ime de i a i es. A se o snapsho s based on ime de i a i es plus he snapsho a he ini ial ime (o he mean alue o he s a es) is able o p o ide poin wise in ime e o es ima es. This is he idea we apply in he p esen pape . Mo eo e , we ca y ou a di e en e o analysis based on he ecen esul s ob ained in [7] ha allow us o ge sha pe e o bounds ee o 1/h ac o s. This is in ag eemen wi h he nume ical in es iga ions in he li e a u e whe e he 1/hbeha iou in he e o bounds o ully disc e e me hods has ne e been obse ed. Also, he use o snapsho s based on ime de i a i es allows us o gi e a bound o some e ms ha could no be bounded wi h he s anda d app oach. Bo h ac s, he new echnique used o bound he e o ha ollows ideas in [7] oge he wi h he use o snapsho s based on ime de i a i es, a e he key ing edien s o ge e o bounds o he new me hod ha a e op imal in e ms o he ime s ep hand he mesh diame e o he educed space k .As usual, ou e o bounds o he POD me hod depend also on he size o he ail o eigen alues in he singula alue decomposi ion. The ou line o he pape is as ollows. In Sec .2we s a e he model p oblem and some p elimina y esul s. In Sec .3we in oduce he POD app oxima ion and ca y ou he e o analysis o he me hod. Finally, in Sec .4we show some nume ical expe imen s in which we implemen he me hod we p opose in he pape . In he expe imen s o Sec .4we choose he same nume ical es s as in [1] o compa e ou esul s wi h hose in his ela ed e e ence. The me hod in oduced in he p esen pape seems o p oduce be e esul s han hose shown in [1]. We inish he pape wi h some conclusions. 2 Model P oblem and S anda d Nume ical App oxima ion In he sequel, ·deno es any no m associa ed o an inne p oduc and ·∞deno es he maximum no m o ec o s in Rn,n≥1. We will also deno e by ·2 he s anda d euclidean no m. In pa icula , in he nume ical expe imen s, we use a weigh ed no m ·sligh ly di e en om he s anda d euclidean no m ·2. Fo a nonlinea mapping :Rn×Rm→Rn, and a gi en ini ial condi ion y0∈Rnle us conside he con olled nonlinea dynamical sys em ˙y( )= (y( ), u( )) ∈Rn, >0,y(0)=y0∈Rn,(1) oge he wi h he in ini e ho izon cos unc ional J(y,u)=∞ 0 g(y( ), u( ))e−λ d .(2) 123 19 Page 4 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 In (2)λ>0 is a gi en weigh ing pa ame e and g:Rn×Rm→R. The se o admissible con ols is Uad ={u∈U|u( )∈Uad o almos all ≥0}, whe e U=L2(0,∞;Rm)and Uad ⊂Rmis a compac con ex subse . As in [1, Assump ion 2.1] we assume he ollowing hypo heses: •The igh -hand side in (1) is con inuous and globally Lipschi z-con inuous in bo h he i s and second a gumen s; i.e., he e exis s a cons an L >0 sa is ying  (y,u)− (˜y,u)≤L y−˜y,∀y,˜y∈Rn,u∈Uad,(3)  (y,u)− (y,˜u)≤L u−˜u,∀u,˜u∈Uad,y∈Rn.(4) •The igh -hand side in (1) sa is ies ha he e exis s a cons an M >0 such ha he ollowing bound holds  (y,u)∞≤M ,∀y∈⊂Rn,u∈Uad,(5) whe e is a bounded polyhed on such ha o su icien ly small h>0 he ollowing inwa d poin ing condi ion on he dynamics holds y+h (y,u)∈, ∀y∈, u∈Uad.(6) •The unning cos gis con inuous and globally Lipschi z-con inuous in bo h he i s and second a gumen s; i.e., he e exis s a cons an Lg>0 sa is ying |g(y,u)−g(˜y,u)|≤Lgy−˜y,∀y,˜y∈Rn,u∈Uad,(7) |g(y,u)−g(y,˜u)|≤Lgu−˜u,∀u,˜u∈Uad,y∈Rn.(8) •Mo eo e , he e exis s a cons an Mg>0 such ha |g(y,u)|≤Mg,∀(y,u)∈×Uad.(9) F om he assump ions made on he e exis s a unique solu ion o (1)y=y(y0,u)de ined on [0,∞) o e e y admissible con ol u∈Uad and o e e y ini ial condi ion y0∈Rn,see [4, Chap e 3]. We de ine he educed cos unc ional as ollows: ˆ J(y0,u)=J(y(y0,u), u), ∀u∈Uad,y0∈Rn,(10) whe e y(y0,u)sol es (1). Then, he op imal con ol can be o mula ed as ollows: o gi en y0∈Rnwe conside min u∈Uad ˆ J(y0,u). The alue unc ion o he p oblem is de ined as :Rn→Ras ollows: (y)=in ˆ J(y,u)|u∈Uad,y∈Rn.(11) This unc ion gi es he bes alue o e e y ini ial condi ion, gi en he se o admissible con ols Uad. I is cha ac e ized as he iscosi y solu ion o he HJB equa ion co esponding o he in ini e ho izon op imal con ol p oblem: λ (y)+sup u∈Uad {− (y,u)·∇ (y)−g(y,u)}=0,y∈Rn.(12) 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 5 o 28 19 The solu ion o (12) is unique o su icien ly la ge λ,λ>max(Lg,L ),[4]. Le us conside i s a ime disc e iza ion whe e his a s ic ly posi i e s ep size. We conside he ollowing semidisc e e scheme o (12): h(y)=min u∈Uad {(1−λh) h(y+h (y,u)) +hg(y,u)},y∈Rn.(13) As i is well-known equa ion (13) ep esen s a nume ical app oxima ion ela ed o he HJB equa ion (12) (see Rema k 7). The ollowing con e gence esul o he semidisc e e app ox- ima ion [12, Theo em 2.3] equi es ha o (y,˜y,u)∈Rn×Rn×Uad  (y+˜y,u)−2 (y,u)+ (y−˜y,u)≤C ˜y2,(14) g(y+˜y,u)−2g(y,u)+g(y−˜y,u)≤Cg˜y2.(15) Theo em 1 Le assump ions (3),(5),(6),(7),(9),(14)and (15)hold and le λ> max(2Lg,L ).Le and hbe he solu ions o (12)and (13), espec i ely. Then, he e exis s a cons an C ≥0, ha can be bounded explici ly, such ha he ollowing bound holds sup y∈Rn| (y)− h(y)|≤Ch,h∈[0,1/λ). (16) As in [1] le us suppose ha he e exis s a bounded polyhed on ⊂Rnsuch ha o h>0 small enough (6) holds. We conside a ully-disc e e app oxima ion o (12). Le Sjms j=1be a amily o simplices which de ines a egula iangula ion o  = ms  j=1 Sj,k=max 1≤j≤ms (diam Sj). We assume we ha e ns e ices/nodes ˆy1,..., ˆynsin he iangula ion. Le Vkbe he space o piecewise a ine unc ions om  o Rwhich a e con inuous in ha ing cons an g adien s in he in e io o any simplex Sjo he iangula ion. Then, a ully disc e e scheme o he HJB equa ions is gi en by h,k(ˆyi)=min u∈Uad (1−λh) h,k(ˆyi+h (ˆyi,u)) +hg(ˆyi,u),(17) o any e ex ˆyi∈. The e exis s a unique solu ion o (17) in he space Vk,see[4, Theo em 1.1, Appendix A]. Fo he ully disc e e me hod i we assume ha he con ols a e Lipschi z-con inuous; i.e., he e exis s a posi i e cons an Lu>0 such ha u( )−u(s)2≤Lu| −s|,(18) hen i s o de o con e gence bo h in ime and space is p o ed in [7, Theo em 6]. Theo em 2 Assume condi ions (3)–(5),(7)–(9)and (18)hold. Assume λ>L wi h L= CnL . Then, o 0≤h≤1/(2λ) he e exis posi i e cons an s C1=C1(λ, M ,Mg,L ,Lg) and C2=C2(λ, L ,Lg,Lu)such ha | (y)− h,k(y)|≤C1(h+k)+C2h,y∈. Condi ion (18) can be weakened and one can s ill ge con e gence as p o ed in [7, Theo em 7]. Assume he ollowing con exi y assump ion in oduced in [5, (A4)] and deno ed by (CA) as in [5,7], 123 19 Page 6 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 •(CA) Fo e e y y∈Rn, { (y,u), g(y,u), u∈Uad} is a con ex subse o Rn+1. Theo em 3 Assume condi ions (3),(4),(5),(7),(8),(9)and (CA) hold. Assume λ>L wi h L de ined as in Theo em 2. Then, o 0≤h≤1/(2λ) he e exis posi i e cons an s C1=C1(λ, M ,Mg,L ,Lg)and C2=C2(λ, M ,Mg,L ,Lg)such ha o y ∈ | (y)− h,k(y)|≤C1(h+k)+C2 1 (1+β)2λ2(log(h))2h1 1+β,β=√nL λ.(19) Le us obse e ha since βis smalle han 1, by weakening he egula i y equi emen s we loose a mos hal an o de in he a e o con e gence in ime o he me hod up o a loga i hmic e m. 3 POD App oxima ion o he Op imal Con ol P oblem Based on Time De i a i es In his sec ion we p esen a new app oach, simila o he second me hod in [1], bu wi h snapsho s based on ime de i a i es a di e en imes. We also pe o m a comple ely di e en e o analysis o he one appea ing in [1], inspi ed in he esul s in [7]and[15]. 3.1 POD App oxima ion Based on Time De i a i es Fo p∈Nle us choose di e en pai s (uν,yν 0)p ν=1in U×.SinceU=L2(0,∞;Rm) he con ols do no need o be cons an s, as hose aken in he nume ical expe imen s. By yν=y(uν;yν 0),ν=1,...,p, we deno e he solu ions o (1) co esponding o hose chosen ini ial condi ions and con ols. Le us ix T>0andM>0and ake =T/Mand j=j ,j=0,...,M.Fo N=M+1 we de ine he ollowing space V=span zν 1,zν 2,...,zν Np ν=1, wi h zν 1=√N yν,yν=1 N M  j=0 yν( j) zν j=τyν ( j−1), j=2,...,N, so ha V=span √N yν,τyν ( 1),...,τyν ( N)p ν=1, whe e he ac o τin on o he empo al de i a i es is a ime scale and i makes he snapsho s dimensionally co ec . In he nume ical expe imen s we ake τ=1. The co ela ion ma ix co esponding o he snapsho s is gi en by K=((ki,j)) ∈RpN×pN, wi h he en ies ki,j=1 pN zi k,zj l,k,l=1,...,N,i,j=1,...,p, 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 7 o 28 19 and whe e he e, and in he sequel, (·,·)deno es he inne p oduc in Rn o which he no m ||·||is associa ed. Le us deno e o simplici y V=span w1,w 2,...,wpN:= z1 1,...z1 N,...,zp 1,...,zp N. Following [21], we deno e by λ1≥λ2,... ≥λd>0 he posi i e eigen alues o Kand by 1,..., d∈RpN i s associa ed eigen ec o s o euclidean no m 1. Then, he (o hono mal) POD basis unc ions o Va e gi en by ϕk=1 √pN 1 √λk pN  j=1 j kwj,k=1,...,d,(20) whe e j kis he j- h componen o he eigen ec o k. The ollowing e o es ima e is known om [21, P oposi ion 1] 1 pN pN  j=1      wj−  k=1 (wj,ϕ k)ϕk     2 = d  k= +1 λk,(21) om which one can deduce o ν=1,...,p      yν−  k=1 (yν,ϕ k)ϕk     2 +τ2 M+1 M  j=1      yν ( j)−  k=1 (yν ( j), ϕk)ϕk     2 ≤p d  k= +1 λk.(22) In he sequel, we will deno e by V =span{ϕ1,ϕ 2,...,ϕ },1≤ ≤d,(23) and by P Rn→V , he o hogonal p ojec ion on o V .Then(21) can be w i en as 1 pN pN  j=1 wj−P wj  2= d  k= +1 λk. The ollowing lemma is p o ed in [15, Lemma 3.2]. Lemma 1 Le T >0, =T/M, n=n , n =0,1,...M, le X be a Banach space, z∈H2(0,T;X). Then, he ollowing es ima e holds max 0≤k≤Nzk2 X≤3z2 X+12T2 M M  n=1zn 2 X+16T 3( )2T 0z (s)2 Xds,(24) whe e z=1 M+1M j=0zj. Using Lemma 1we can p o e poin wise es ima es o he p ojec ions on o V . Lemma 2 The ollowing bounds hold o ν=1,...,p max 0≤j≤Myν( j)−P yν( j)2≤3+24 T2 τ2p d  k= +1 λk+16T 3( )2T 0yν (s)2ds. (25) 123 19 Page 8 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 P oo We a gue as in [15, Lemma 3.4]. Taking z=yν( j)−P yν( j)in (24) and applying (22) ( aking in o accoun ha (M+1)/M≤2) yields max 0≤n≤Myν( j)−P yν( j)2≤3+24 T2 τ2p d  k= +1 λk +16T 3( )2T 0yν (s)−P yν (s)2ds. Now, since P is an o hogonal p ojec ion, we ha e yν (s)−P yν (s)2≤yν (s)2and he p oo is inished  3.2 The POD Con ol P oblem To mi iga e he cu se o dimensionali y, he idea o he POD me hod is o wo k on a space o dimension wi h <n. To s a we need o in oduce some no a ion.We use a sligh ly di e en no a ion om he one used in [1]. In pa icula , as s a ed below, P c.isalwaysused o deno e coe icien s and ϕis always used o he linea combina ion based on he POD basis unc ions o he educed o de space. Mo e p ecisely: o any y∈⊂Rnle us deno e by P cy∈R he coe icien s o he p ojec ion o yon o V P cy={(y,ϕ k)} k=1.(26) Fo any y ∈R le us deno e by ϕy ∈Rn he ec o whose coe icien s in he POD basis a e he componen s o y , i.e., ϕy =  j=1 y jϕj,(27) whe e y jis he jcomponen o he ec o y . Fo and gin (1), (2)and(y ,u)∈R ×Uad we de ine (y ,u)=P c (ϕy ,u)∈R , g (y ,u)=g(ϕy ,u)∈R.(28) To ha e an inwa d poin ing condi ion on he dynamics in he educed space, analogous o (6), ollowing [1, Sec ion 4.2], we assume ha he e exis s a bounded polyhed on  ⊂R sa is ying P cy∈ ,∀y∈. (29) The ollowing lemma p o es ha he inwa d poin ing condi ion o  ollows om (29). Lemma 3 Condi ion (29)implies ha y +h (y ,u)∈ ,y =P cy,y∈, p o ided he s ep size h o P y−yis su icien ly small. P oo We ollow [1, Rema k 4.5] o he p oo . We i s obse e ha y +h (y ,u)=P cy+hP c (ϕy ,u). Adding and sub ac ing hP c (y,u)we ge y +h (y ,u)=P c(y+h (y,u)) +hP c( (ϕy ,u)− (y,u)). (30) 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 9 o 28 19 Applying condi ion (6)y+h (y,u)∈and applying (29) he i s e m on he igh -hand side o (30) e i ies P c(y+h (y,u)) ∈ . Then, we only need o show ha he second e m on he igh -hand side o (30) is small enough o ho P y−ysu icien ly small. Le us deno e by z= (ϕy ,u)− (y,u)∈Rn.SinceP cz={(z,ϕ k)} k=1∈R , aking in o accoun ha he unc ions ϕkde ine an o hono mal basis and ha P is a p ojec ion, we ha e P cz2=P z≤z. Applying he abo e inequali y oge he wi h (3), we ge P cz2 2=P c( (ϕy ,u)− (y,u))2 2≤ (ϕy ,u)− (y,u)2 ≤L2 ϕy −y2=L2 P y−y2,(31) so ha he p oo is concluded.  We can now de ine he educed o de p oblem we sol e in p ac ice. Fo and g de ined in (28) and a gi en ini ial condi ion y 0∈R le us conside he con olled nonlinea dynamical sys em ˙y ( )= (y ( ), u( )) ∈R , >0,y (0)=y 0∈R ,(32) oge he wi h he in ini e ho izon cos unc ional J (y ,u)=∞ 0 g (y ( ), u( ))e−λ d .(33) As in (10), we de ine he educed cos unc ional ˆ J (y 0,u)=J (y (y 0,u), u), ∀u∈Uad,y 0∈R ,(34) whe e y (y 0,u)sol es (32). Then, he POD op imal con ol can be o mula ed as ollows: o gi en y 0∈R we conside min u∈Uad ˆ J (y 0,u). The alue unc ion o he p oblem :R →Ris de ined as ollows: (y )=in ˆ J(y ,u)|u∈Uad,y ∈R .(35) Rema k 1 I is easy o check ha he egula i y assump ions o and g analogous o hose o and g,(3), (4), (5), (7), (8)and(9), hold om he de ini ion o and g and he p ope ies being ue o and g. To ge in p ac ice a ully disc e e app oxima ion in he educed space le us de ine S jm s j=1 a amily o simplices which de ines a egula iangula ion o  .Weassumeweha ens e ices/nodes in he iangula ion ˆy1 ,..., ˆyns ∈ and  = m s  j=1 S j,k =max 1≤j≤m s (diam S j). Le Vk be he space o piecewise a ine unc ions om  o Rwhich a e con inuous in  ha ing cons an g adien s in he in e io o any simplex S jo he iangula ion. As in [1, 123 19 Page 16 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Sec ion 1.2], he con ols could no be unique bu one can selec he con ol wi h minimum no m. Now, le us obse e ha o he disc e e alue unc ion i holds h,k(ˆyi+h (ˆyi,ui h,k)) = h,k(ˆyi)+h (yi,ui h,k)·∇ h,k(ˆyi)+O(h2). (52) Taking in o accoun ha h,k(ˆyi)=(1−λh) h,k(ˆyi+h (ˆyi,ui h,k)) +hg(ˆyi,ui h,k), (53) and inse ing (52)in o(53)wege h,k(ˆyi)=(1−λh) h,k(ˆyi)+h (yi,ui h,k)·∇ h,k(ˆyi)+O(h2) +hg(ˆyi,ui h,k). And hen λh h,k(ˆyi)=h (ˆyi,ui h,k)·∇ h,k(ˆyi)+hg(ˆyi,ui h,k)+O(h2). F om which λ h,k(ˆyi)= (ˆyi,ui h,k)·∇ h,k(ˆyi)+g(ˆyi,ui h,k)+O(h). Now, since λ ( ˆyi)= (ˆyi,ui)·∇ (ˆyi)+g(ˆyi,ui), (54) and h,k(ˆyi)→ (ˆyi), o h,k→0 we ob ain (ˆyi,ui h,k)·∇ h,k(ˆyi)+g(ˆyi,ui h,k)→ (ˆyi,ui)·∇ (ˆyi)+g(ˆyi,ui). (55) A guing as in [13, Sec ion 1.2], le us de ine L(y,u)=1 λ( (y,u)·∇ (y)+g(y,u)), and le us associa e wi h ya (unique) con ol u(y)such ha L(y,u(y)) =min u∈Uad L(y,u)= (y). Assume ∇ h,k(ˆyi)→∇ (ˆyi)(56) (which we ha e no p o ed) and ui h,k→ui o h,k→0. Then, on he one hand, om (55), L(ˆyi,ui h,k)→L(ˆyi,ui), and, on he o he L(ˆyi,ui h,k)→L(ˆyi,ui) which implies ui=uiand ui h,k→ui o h,k→0. Finally, le us obse e ha he a gumen in [13, Sec ion 1.2] had al eady p o ed ha o any ixed hand k→0 he ully disc e e con ols ui h,kcon e ge o he co esponding semi-disc e e ime con ol de ined in (13), o ha alue o h. 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 17 o 28 19 4 Nume ical Expe imen s We now p esen some nume ical expe imen s. We closely ollow hose in [1] so ha he new me hod we p opose can be compa ed wi h he me hod in [1]. The au ho s in [1] apply s a e snapsho s in he educed o de me hod ins ead o snapsho s based on ime de i a i es. We obse e ha , as explained in de ail in he in oduc ion, in he las case i is no necessa y o conside bo h, s a e snapsho s and ime de i a i es, since i has al eady been p o ed ha only wi h ime de i a i es op imal bounds can be ob ained. We obse e ha we ha e chosen he closed-loop con ol ype app oach ins ead o he open-loop con ol ype app oach in he p esen pape . Also, he heo y o he p esen pape de elops he i s app oach. We do no compa e he p esen me hod wi h me hods based on he i s app oach since ou aim is jus o p opose, analyze and check in p ac ice a new me hod ha could be be e o no (p obably depending on he examples) han o he me hods in he li e a u e. The nume ical expe imen s o his sec ion show ha ou me hod wo ks ine in p ac ice and is able o p o ide accu a e app oxima ions. We i s no ice ha due o nume ical easons we ha e o choose a ini e ime ho izon, so we selec a su icien ly la ge e>0, which, in he expe imen s ha ollow, i was ixed o e=3. As in [1], we conside he ollowing con ec ion- eac ion-di usion equa ion z −εzxx +γzx+μ(z3−z)=ub in I×(0, e), z(·,0)=z0in I, z(·, )=0in∂I×(0, e), (57) wi h ε=1/10, and whe e I=(0,a)is an open in e al, z:I×[0, e]→Rdeno es he s a e, and γand μa e posi i e cons an s. The con ols ubelong o he closed, con ex, bounded se Uad =L2(0, e,[ua,ub]), o eal alues ua<ub. The cos unc ional o minimize is gi en by  e 0 e−λ z(·, ,u)2 L2(I)+1 100 |u( )|2d ,(58) whe ewese λ=1. No ice hen ha in (58) he aim is o d i e he s a e o ze o. We use a ini e-di e ence me hod on a uni o m g id o size x=l/Nwi h N=100 on he in e al I=(0,l) o disc e ize (57) in space o ob ain a sys em o o dina y di e en ial equa ions (ODEs). To ob ain he snapsho s, he ODE sys em is in eg a ed in ime using Ma lab’s command ode15s, which uses he nume ic di e en ia ion o mulae (NDF) [25], wi h su icien ly small ole ances o he local e o s (below 10−12). The snapsho s we e ob ained on a uni o m ( ime) g id o diame e 1/20. The ime de i a i es we e ob ained by e alua ing he igh -hand-side o he sys em o ODEs. As in [1], equa ion (36) h,k(yi )=min u∈Uad (1−λh) h,k(yi +h (yi ,u)) +hg (yi ,u),i=1,...,ns, was sol ed by ixed-poin i e a ion, he s opping c i e ium being ha wo consecu i e i e a es di e in he maximum no m in less han a gi en ole ance TOL , ini ially se o TOL = 5×10−4. Fo he i s i e a e we choose a amily o cons an con ols ul,l=1,...,pand a any poin o he mesh yi (ini ial condi ion) we compu e he app oxima e solu ion o (32) co esponding o his ini ial condi ion and con ol ul. Then, we compu e he alue o he unc ional cos (33). Finally, he alue o he ini ial i e a e a yi is he minimum be ween he alues o he unc ional cos o l=1,...,p. 123 19 Page 18 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Once (36) is sol ed, he op imal con ol u h,k(yi )=a gminu∈Uad (1−λh) h,k(yi +h (yi ,u)) +hg (yi ,u),(59) is ob ained a any mesh poin yi ,i=1,...,ns. Then, he subop imal eedback ope a- o  (y)is compu ed by in e pola ion. This means ha o y∈we p ojec on o he POD space o ge P yand hen P y= ns  i=1 μiyi , (y)= ns  i=1 μiu h,k(yi ), whe e he coe icien s μisa is y 0 ≤μi≤1, ns i=1μi=1. Wi h his, he closed-loop sys em y( )= (y( ),  (y( ))), y(0)=y0,(60) is in eg a ed, again, using he NDF o mulae as implemen ed in Ma lab’s command ode15s wi h he same ole ances as in he compu a ion o he snapsho s. We will see below ha e y di e en app oxima ions o he solu ion o (60) can be ob ained wi h di e en alues o he ole ance TOL o he ixed-poin i e a ion sol ing (36)(seeFig.3), so ha we sol ed his equa ion o dec easing alues o TOL , each one 5 imes smalle han he p e ious one un il he ela i e e o be ween he solu ions o (60) co esponding o wo consecu i e alues o TOL was below 10% (i usually u ned ou o d op d ama ically om abo e 10% o less han 0.01%). He e and in he sequel, by he ela i e e o o a quan i y ˆywi h espec o y we mean y−ˆy/max(|y|,10−3). Fo he op imal HJB s a es, o e e y alue o ime o which he solu ion o (60) was compu ed, we compu ed he maximum o he ela i e e o s o he componen s o y( ). Fo Tes 2 in Sec . 4.2, due o he discon inous ini ial da um, i p o ed impossible o d i e he ela i e e o o wo op imal HJB s a es compu ed wi h wo di e en ole ances TOL below 10%, so ha we checked ha he alue o he ela i e e o s measu ed in he no m (63) below was smalle han 10%. Wi h espec o he compu a ional cos o sol ing (59) by ixed-poin i e a ion, i is ob i- ously p opo ional o he numbe o i e a ions. In he expe imen s below, hese alues we e 1382 o all alues o in Tes 1 below, 362, 263 and 242 o =2,3,4, espec i ely, in Tes 2 below, and 365, 1103 and 1539 o =3,4,5 in Tes 3 in Sec . 4.3 below. On each i e a ion, he bulk o he cos is inding he nonnega i e scala s μi j,j=1,...,ns, such ha yi +h (yi ,u)=μi 1y1 +···+μi nsyns , which was 70% o he cos o he i e a ion o =5 in Tes 3 in Sec .4.3, 79% o =4 in Tes 2, and 95% o =4 in Tes 1 in Sec . 4.1, ollowed by he cos o ob aining (yi ,u),i=1,...,ns, which was 4% o =4inTes s 1 and 2 below o 28% o =5 in Tes 3 in Sec .4.3. We no e ha cos o ob aining (yi ,u) can be subs an ially diminished using app op ia e enso s o by means o echniques like disc e e empi ical ine pola ion, which, o simplici y, we did no use in ou codes. 4.1 Tes 1: Semilinea Equa ion As in [1], we conside (57) wi h γ=0andμ=1, a=1andb(x)=z0(x)=2x(1−x). I is easy o check ha he uncon olled solu ion con e ges, as →∞ o a non-null s eady s a e (see also [1, Fig. 6.1]), and ha he null solu ion is uns able. 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 19 o 28 19 Fo he ini e-di e ence app oxima ion, we conside y:[0, e]→RN−1wi h componen s yj( )≈z(xj, ),xj=jx,j=1,...,N−1, x=1/N, solu ion o Cy =1 10 Ay +C(F(y)+uB)(61) whe e he componen s o Fand Ba e, espec i ely Fj=yj(1−y2 j),Bj=2xj(1−xj), j=1,...,N−1, and Aand Ca e (N−1)×(N−1) idiagonal ma ices ma ices gi en by A=1 (x)2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −21 1−21 ......... 1−21 1−2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,C=1 12 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 10 1 1101 ......... 1101 110 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ,(62) so ha he ini e-di e ence disc e iza ion (61) is ou h-o de con e gen . The no m we conside in RN−1is gi en by y2=x N−1  j=1 y2 j.(63) Le as obse e ha his no m is an app oxima ion o he in eg al 1 0y(x)2dx o a unc ion wi h alues yja he spa ial mesh nodes. To compu e he snapsho s, as in [1], o cons an con ols u∈Usnap ={−1,0,1},we ob ained he solu ions y(n)=y( n)o (61)e e y1/20 ime uni s, ha is, o n=n/20, n=1,...20 e, and hen he ime de i a i es y(n) we e compu ed om iden i y (61). Fo he educed spaces, we conside he cases o POD basis wi h only =2,3 and 4 elemen s, also as in [1]. The POD app oxima ion y was hen he mean o he snapsho s plus a linea combina ion o he POD basis. The con ol se Uad is gi en by 41 con ols equally dis ibu ed in [−1,1]. As in [1], o de ine he domain  , we compu e he p ojec ions o all he snapsho s. Wi h his p ocedu e we ob ain a se o poin s in R . Then, we de ine an hype cube con aining his se o poin s. The aim o his p ocedu e, in iew o Lemma 3, is ha he se  de ined in his way sa is ies he in a iance condi ion y +h (y ,u)∈ ,y ∈ ,u∈Uad.(64) The se  o =4wasgi enby  =[−0.87,0.41]×[−0.01,0.02]×[−0.01,0.01]×[−0.01,0.01]. Fo his se we checked ha condi ion (64) holds. We no ice ha ou se  is conside able smalle han he co esponding se in [1](see [1, 6.1. Tes 1]) whe e he au ho s use he s anda d euclidean no m in Rn a he han he no m (63) we use he e. Since he domain is smalle we also conside pa i ions o  smalle han hose in [1]. We ake maximum diame e k =0.01, and, as in [1], we choose h=0.1k . In Fig. 1we ha e ep esen ed on op he op imal solu ion (le ) o =4, he di e ence be ween op imal solu ion wi h 4 and 2 POD basis unc ions ( op middle) and he di e ence be ween op imal solu ion wi h 4 and 3 POD basis unc ions ( op igh ). On bo om we ha e ep esen ed he op imal con ols o =2, 3 and 4. We obse e ha we ge much be e esul s han hose in [1], al hough (apa om using a di e en se o snapsho s) in ou me hod, bo h he ini e-di e ence me hod and he ime 123 19 Page 20 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Fig. 1 Tes 1: Op imal HJB s a es compu ed wi h =4 POD basis unc ions ( op-le ), di e ence be ween op imal solu ion wi h 4 and 2 POD basis unc ions ( op-middle), di e ence be ween op imal solu ion wi h 4 and 3 POD bases ( op- igh ). Op imal HJB con ols wi h =4,3,2 (bo om). The ed c osses co espond o he alues o he con ols ha we ha e joined by a blue line Fig. 2 Tes 1: Value o he cos unc ional (58) on he op imal HJB s a es o =2,3,4. The ed c osses co espond o he alues o he cos alues ha a e joined by a pd line Fig. 3 Tes 1: Rela i e e o be ween he op imal HJB s a es wi h =4 co esponding o sol ing (36)by ixed poin i e a ion wi h ole ances TOL =5×10−4and TOL =1×10−4(le ), TOL =1×10−4 and TOL =2×10−5(cen e), and op imal HJB con ols ( igh ) in eg a o ha we use a e mo e accu a e han hose in [1]. We also no ice ha he e is li le disc epancy be ween he alues o he op imal HJB s a es o he di e en alues o ha we ied. We also compu ed he alues o he cos unc ional (58) on he op imal HJB s a es o he h ee alues o . The alues a e shown in Fig. 2. I can be seen ha he alues dec ease wi h and ha hey di e in he nin h signi ican digi . As men ioned abo e he e can be a signi ican di e ence be ween he op imal HJB s a es compu ed wi h solu ions ob ained by sol ing (36) wi h di e en ole ances TOL .InFig.3we show he ela i e e o be ween he op imal HJB s a es co esponding o ole ances TOL = 5×10−4and TOL =1×10−4(le ), and be ween his one and ha co esponding o TOL = 2×10−5(cen e). The igh plo shows he co esponding op imal HJB con ols. Figu e 3 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 21 o 28 19 Table 1 Rela i e e o s o he op imal HJB s a es and con ols o =3 compu ed wi h x=1/N,N=25 and N=50, wi h espec o hose compu ed wi h N=100 Ny Ra e  (y) a e 25 7.24 ×10−51.99 ×10−5 50 4.25 ×10−64.09 1.17 ×10−64.09 Fig. 4 Tes 1: Resul s o di e en alues o k ; ela i e e o s be ween HJB s a es ( op le ) and con ols (bo om le ) wi h espec o o k =0.005; HBJ con ols (cen e) and alues o he cos unc ional (58) ( igh ) shows he impo ance o sol ing (36) accu a ely in o de o ob ain good op imal HJB s a es, hus, jus i ying ha we compu ed he (app oxima ions o he) solu ion o (36) wi h dec easing alues o TOL un il he ela i e e o o he co esponding op imal HJB es a es was below 10%. Following he sugges ion o one o he e iewe s, we checked i he op imal con ol compu ed wi h x=1/100 s abilizes he same PDE compu ed on ine meshes. We ied his o =4andx=1/400 and 1/800, and he con olled solu ions con e ged o ze o as as as in he case x=1/100. Gi en he accu acy wi h which we had compu ed he con ols o x=1/100 and he accu acy o he disc e iza ion i sel , we did no expec o he wise. The esul s abo e sugges ha , o his p oblem, i is enough wi h =3. Fo his alue o we now check he e ec o he ini e-di e ence mesh in he op imal HJB s a es. In Table 1we show he ela i e e o s o he op imal HJB s a es compu ed wi h N=25 and N=50 wi h espec o ha compu ed wi h N=100, as well as he ela i e e o s o he co esponding con ols. Fo he con ols, we show in Table 1 he maximum o all alues o ∈{0,0.05,0.1,...,3}o he ela i e e o s, and o he s a es we show he maximum on hesame alueso o he maximum o he ela i e e o s o he s a e on all he alues o he co esponding spa ial g id. They con i m ha he ini e-di e ence disc e iza ion is o o de 4. Due o he excellen accu acy ob ained wi h x=1/50, he esul s ha ollow a e done wi h ha alue o x. We now check he e ec o di e en alues k o he diame e o he pa i ion o  .Todo his, we compa e he esul s ob ained wi h =3, x=1/50 and k =0.02,0.01,0.005. In o de no o spoil he be e accu acy ob ained wi h he smalle alue o k we ook Uad wi h 161 con ols equally dis ibu ed in [−1,1] o k =0.005, and, o simpli y compu a ions wi h only 11 con ols o k =0.02 (we also y wi h 21 and 41 con ols, bu , al hough we do no ha e a p esen an explana ion o i , using only 11 con ols wi h k =0.02 ga e somewha be e esul s). The esul s can be seen in Fig. 4. The plo s on he le show he (maximum o he 51 poin s o he spa ial g id o he) ela i e e o s o he op imal HJB s a es ( op) and hei con ols (bo om) o k =0.02 and k =0.01 wi h espec o hose o k =0.005, while he plo in he cen e shows he HJB con ols. The e o s, as expec ed, a e smalle o k =0.01 han o k =0.02, excep o he con ols o ∈[1.8,2.55]whe e hey 123 19 Page 22 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Fig. 5 Tes 1: Resul s o POD basis ex ac ed om snapsho s (x=1/50, =3); Rela i e e o s o HJB s a e (le ) and con ol (cen e) wi h espec o he case whe e POD basis is aken om ime de i a i es; HBJ con ol ( igh ) a e sligh ly la ge . We also no ice ha , o k =0.01, he ela i e e o s emain below 10% excep o ∈[1.5.2.2](whe e hey emain below 18%) in he case o he e o s in he op imal HJB s a es and ∈[1.8,3] o he con ols. No ice, howe e , ha he la ges e o s ake place whe e bo h he s a es and he con ols a e close o ze o ( ecall he plo s in Fig. 1), we e i is di icul o ob ain small ela i e e o s. Maybe his is he eason o he simila alues o he cos unc ional (58) o he h ee alues o k , which a e shown on he igh plo in Fig. 4; he ela i e e o s (wi h espec o k =0.005) o k =0.02 and k =0.01 a e 0.081% and 0.0023%, espec i ely. One may wonde wha is he esul i he snapsho s a e used o ob ain he POD basis as in [1] ins ead o he ime de i a i es as in he p esen pape . Thus, we epea ed ou compu a ions bu eplacing he ime de i a i es by he snapsho s minus hei mean. We did no ind any signi ican di e ence. In Fig. 5we show he esul s co esponding o x=1/50, =3, he se  being  =(−0.42,0.9)×(−0.01,0.02)×(−0.01,0.01). The op imal HJB con ol is shown on he igh -plo , while he o he wo plo s show he ela i e e o s o he HJB s a e (le ) and i s con ol (cen e) wi h espec o he esul s when he POD basis is aken om he ime de i a i es, =3andx=1/100. We see ha he ela i e e o s a e below 0.1%, and hus, no di e ence can be seen be ween he igh -plo in Fig. 5and he cen e plo in Fig. 1. We also no ice ha ou esul s when he POD basis is aken om he snapsho s a e be e han hose in [1]. We belie e ha his is due o he highe accu acy o ou compu a ions ( ou h-o de con e gen ini e-di e ence me hod ins ead o a second-o de con e gen one, NDF wi h small ole ances o compu e he snapsho s ins ead o implici Eule me hod, dense se s Uad o he con ol a iable, smalle ole ance TOL in he ixed poin me hod o sol e (36), e c). The ac ha e y simila esul s a e ob ained when he POD basis is aken om he snapsho s o he ime de i a i es should no be su p ising. As shown in [18], we he be e esul s a e ob ained i he POD basis is ex ac ed om he snapsho s o om hei di e ence quo ien s is case-dependen and, as shown in [16], e y simila esul s a e usually ob ained when he POD basis is aken om he ime de i a i es o he snapsho s di e ence quo ien s. The ad an age o using ime de i a i es o di e ence quo ien s o he POD basis is mo e om he heo e ical side, since i allows o p o e op imal con e gence o he POD me hods wi h less assump ions han when he POD basis is ex ac ed om he snapsho s. Mo e ecen ly, in [17], i has been p o ed ha using only snapsho s o he POD basis, i is possible o p o e e o es ima es o he co esponding POD me hods wi h con e gence a es as close o op imal as he smoo hness o he solu ion om whe e he snapsho s a e aken allows. In 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 23 o 28 19 iew o he ecen esul s in [17], he analysis in he p esen pape can be easily adap ed o co e also he case whe e POD basis is aken om he snapsho s. 4.2 Tes 2: Ad ec ion–Di usion Equa ion As in [1], we now conside (57) wi h γ=1andμ=0, I=(0,2)and z0(x)= max(0,0.5sin(π x)).We akebas he cha ac e is ic unc ion o he in e al (1/2,1).To compu e he POD basis we compu e he ime de i a i es o he s a es o cons an con ols u=−2.2,−1.1,0. The semidisc e iza ion was done wi h a s anda d ini e di e ence me hod dyj d =yj+1−2yj+yj−1 10(x)2−yj+1−yj−1 2x+b(xj), j=1,...,N−1,y0=yN=0, which is second o de con e gen in p oblems wi h su icien ly smoo h solu ions. Since he ini ial s a e z0does no possess second-o de de i a i es in L2, we no ice hen ha he ime de i a i e z blows up when →0. Fo his eason, a e he spa ial disc e iza ion by ini e di e ences, we eplaced he ime de i a i e a =0 by he di e ence quo ien (y(1)− y(0))/ o s a es a =0and = . Pe haps also o lack bounded ime de i a i es a =0 and he mo e dissipa i e na u e o he implici Eule me hod, we ound ha , in he compu a ion o he op imal HBJ s a es and con ols, be e esul s we e ob ained i he implici Eule me hod wi h  =1/20 was used ins ead o he NDF wi h small ole ances. Also, o easons ha we do no unde s and a p esen , we ound ha be e esul s we e ob ained when he POD app oxima ion was a linea combina ion o he POD basis plus he ini ial condi ion y0, ins ead o a linea combina ion o he POD basis plus he mean as in he p e ious sec ion. In he p e ious es we had an in a iance se so ha we did no need o impose any bounda y condi ion o sol ing (36). In his es we ound i impossible o ind a se  sa is ying condi ion (64) bo h when he POD basis is aken om he s a es and om hei ime de i a i es. In his las case he se  o =4 we used in he expe imen s was  =(−0.5,0.7)×(−0.3,1.5)×(−0.3,0.2)×(−0.05,0.15). To o e come he lack o in a iance o his se , whene e o a e ex yi we had yi + h (yi ,u)/∈ , we simply eplaced yi +h (yi ,u)by i s closes poin on ∂ .This esul ed in changing he alue o yi +h (yi ,u)in less han 2% in he i s wo coo dina es in he POD basis and 15% in he emaining ones, excep o some nega i e alues o he ou h coo dina e whe e e o s up o 60% we e encoun e ed. Fo example, an e o o 15% in he hi d coo dina e means ha o some alues o yi +h (yi ,u) he hi d coo dina e could be in he se [−0.345,0.23]ins ead o [−0.3,0.2]. Ne e heless, as we will see below, he esul s ob ained wi h he POD app oxima ion in his es we e excellen . Since his p oblem is linea -quad a ic, he solu ion o HJB equa ion can be compu ed by sol ing Ricca i equa ion. In Fig. 6we show he uncon olled solu ion (le ), he op imal LQR s a e (middle) and he op imal LQR con ol ( igh ). In Fig.7we ha e ep esen ed on op he op imal solu ion (le ) o =4, he di e ence be ween op imal solu ion wi h 4 and 2 POD basis unc ions ( op middle) and he di e ence be ween op imal solu ion wi h 4 and 3 POD basis unc ions ( op igh ). On he bo om pa we ha e ep esen ed he op imal con ols o =2, 3 and 4. Also in his case, he imp o emen wi h espec o he esul s in [1] is ema kable. In pa icula , he op imal con ols in Fig.7 compa e e y well wi h he op imal LQR con ol o Fig.6e en o he case wi h only =2 basis unc ions in ou POD me hod. 123 19 Page 24 o 28 Jou nal o Scien i ic Compu ing (2025) 103 :19 Fig. 6 Tes 2: Uncon olled solu ion (le ), he op imal LQR s a e (middle) and he op imal LQR con ol ( igh ) Fig. 7 Tes 2: Op imal HJB s a es compu ed wi h =4 POD basis unc ions ( op-le ), di e ence be ween op imal solu ion wi h 4 and 2 POD basis unc ions ( op middle), di e ence be ween op imal solu ion wi h 4 and 3 POD basis unc ions ( op- igh ). Op imal HJB con ols wi h =4,3,2 (bo om) To conclude, in Fig. 8(le ) we show he di e ence be ween he op imal LQR s a e and he op imal HJB s a e compu ed wi h =4 POD basis unc ions. On he igh , we show he ela i e e o s uHJB −uLQR/max(10−3,uLQR)o he op imal HJB con ols wi h espec o he op imal LQR con ol o =2,3 and 4. I can be seen a e y good ag eemen be ween HJB and LQR op imal s a es. Wi h espec o he op imal con ols, we no ice ha whe eas wi h =3and =4 POD basis unc ions he e o s do no exceed 30% and, indeed, hey s ay below 10% mos o he ime, his is no he case o =2 POD basis unc ions, whe e e o s a e abo e 100% o mo e han hal he ime in e al. Howe e , le us obse e ha we a e conside ing ela i e e o s on he igh o Fig.8and ha es ic ing ou sel es o he ime in e al in which he op imal con ol is su icien ly away om ze o he e o s o =2a e also below 35%. Again, he esul s he e (which co espond o k =0.1) compa e a ou ably wi h hose in he li e a u e. As in he p e iouis es , we checked o =4 i he op imal con ol compu ed wi h x=1/100 s abilizes he same PDE disc e ized wi h =1/400 and 1/800 wi h simila esul s as in he p e ious sec ion. 4.3 Tes 3: A Two-Dimensional Reac ion–Di usion Equa ion We ex end (57) o wo dimensions. In pa icula , we conside , 123 Jou nal o Scien i ic Compu ing (2025) 103 :19 Page 25 o 28 19 Fig. 8 Tes 2: ela i e e o s uHJB −uLQR/max(10−3,uLQR)o he op imal HJB con ols wi h espec o he op imal LQR con ol (le ) and di e ence be ween he op imal LQR s a e and he op imal HJB s a e compu ed wi h =4 POD basis unc ions ( igh ) Fig. 9 Tes 3: Uncon olled solu ion a =0,1.5,3 z −εz+(z3−z)=ub in ×(0, e), z(·,0)=z0in , z(·, )=0in∂ ×(0, e), (65) wi h ε=1/10, =[0,1]×[0,1]and z:×[0, e]deno es he s a e. The con ol u belongs o Uad =L2(0, e,[ua,ub]), wi h ua=−1andub=1. The cos unc ion is as (58) bu wi h he s a e measu ed in L2() ins ead o L2(I), ha is  e 0 e−λ z(·, ,u)2 L2() +1 100 |u( )|2d , wi h λ=1 as be o e. Simila ly o Sec .4.1,we akeb(x,y)=z0(x,y)=4x(1−x)y(1−y) and e=3. In Fig. 9we show he uncon olled solu ion a he ini ial ime, a = e/2 and = e. Fo he ini e-di e ence app oxima ion, we conside y:[0, e]→R(N−1)2wi h compo- nen s yk( )≈z(xk, ),whe e, o k=(j−1)(N−1)+i,i,j=1,...,N−1, xk=(xi,yj), and xi=ix,yj=jy,x=y=1/N, solu ion o ˆ Cy =1 10 ˆ Ay +ˆ C(ˆ F(y)+uˆ B)(66) whe e he componen s o ˆ Fand ˆ Ba e, espec i ely ˆ Fk=yk(1−y2 k),ˆ Bk=4xi(1−xi)yj(1− yj),k=(j−1)(N−1)+i,i,j=1,...,N−1, and ˆ Aand ˆ Ca e (N−1)2×(N−1)2 ma ices gi en by ˆ A=I⊗A+A⊗I,ˆ C=I⊗C+C⊗I,whe eIis he iden i y o o de N−1, ⊗ ep esen s he K onecke p oduc o ma ices, and Aand Ca e he ma ices in (62) so ha , as in Sec .4.1, he ini e-di e ence disc e iza ion (66) is ou h-o de con e gen . The 123