Hamburger Beiträge zur Angewandten Mathematik

Transcription

1 Hamburger Beiräge zur Angewanden Mahemaik Asympoic Sabiliy of POD based Model Predicive Conrol for a semilinear parabolic PDE Alessandro Alla and Sefan Volkwein r Sepember 24

2 Asympoic Sabiliy of POD based Model Predicive Conrol for a semilinear parabolic PDE Alessandro Alla Deparmen of Mahemaics, Universiy of Hamburg, 246 Hamburg, Germany Sefan Volkwein Deparmen of Mahemaics and Saisics, Universiy of Konsanz, Konsanz, Germany Absrac In his aricle a sabilizing feedback conrol is compued for a semilinear parabolic parial differenial equaion uilizing a nonlinear model predicive (MPC) mehod. In each level of he MPC algorihm he finie ime horizon open loop problem is solved by a reduced-order sraegy based on proper orhogonal decomposiion (POD). A sabiliy analysis is derived for he combined POD- MPC algorihm so ha he lenghs of he finie ime horizons are chosen in order o ensure he asympoic sabiliy of he compued feedback conrols. The proposed mehod is successfully esed by numerical eamples. Keywords: Dynamic programming, nonlinear model predicive conrol, asympoic sabiliy, subopimal conrol, proper orhogonal decomposiion. 2 MSC: 35K58, 49L2, 65K, 9C3.. Inroducion In many conrol problems i is desired o design a sabilizing feedback conrol, bu ofen he closed-loop soluion can no be found analyically, even in he unconsrained case since i involves he soluion of he corresponding Hamilon- Jacobi-Bellman equaions; see, e.g., [7, ] and [22]. Bu his approach requires he soluion of a nonlinear hyperbolic parial differenial equaion wih a highdimensional spaial variable. One approach o circumven his problem is he repeaed soluion of openloop opimal conrol problems. The firs par of he resuling open-loop inpu This auhor wishes o acknowledge he suppor obained by he ESF Gran no 46. This auhor graefully acknowledges suppor by he DFG gran VO no 658/2-. S. Volkwein is he corresponding auhor. addresses: alessandro.alla@uni-hamburg.de (Alessandro Alla), sefan.volkwein@uni-konsanz.de (Sefan Volkwein) Preprin submied o Advances in Compuaional Mahemaics Sepember 28, 24

3 signal is implemened and he whole process is repeaed. Conrol approaches using his sraegy are referred o as model predicive conrol (MPC), moving horizon conrol or receding horizon conrol. In general one disinguishes beween linear and nonlinear MPC (MPC). In linear MPC, linear models are used o predic he sysem dynamics and considers linear consrains on he saes and inpus. oe ha even if he sysem is linear, he closed loop dynamics are nonlinear due o he presence of consrains. MPC refers o MPC schemes ha are based on nonlinear models and/or consider a nonquadraic cos funcional and general nonlinear consrains. Alhough linear MPC has become an increasingly popular conrol echnique used in indusry, in many applicaions linear models are no sufficien o describe he process dynamics adequaely and nonlinear models mus be applied. This inadequacy of linear models is one of he moivaions for he increasing ineres in nonlinear MPC; see. e.g., [3, 2, 5, 24]. The predicion horizon plays a crucial role in MPC algorihms. For insance, he quasi infinie horizon MPC allows an efficien formulaion of MPC while guaraneeing sabiliy and he performances of he closed-loop as shown in [4, 3, 9] under appropriae assumpions. For he purpose of our paper we will use a differen approach since we will no deal wih erminal consrains. Since he compuaional compleiy of MPC schemes grows rapidly wih he lengh of he opimizaion horizon, esimaes for minimal sabilizing horizons are of paricular ineres o ensure sabiliy while being compuaionally fas. Sabiliy and subopimaliy analysis for MPC schemes wihou sabilizing consrains are sudied in [5, Chaper 6], where he auhors give sufficien condiions ensuring asympoic sabiliy wih minimal finie predicion horizon. oe ha he sabilizaion of he problem and he compuaion of he minimal horizon involve he (relaed) dynamic programming principle (DPP); see [6, 23]. This approach allows esimaes of he finie predicion horizon based on conrollabiliy properies of he dynamical sysem. Since several opimizaion problems have o be solved in he MPC mehod, i is reasonable o apply reduced-order mehods o accelerae he MPC algorihm. Here, we uilize proper orhogonal decomposiion (POD) o derive reduced-order models for nonlinear dynamical sysems; see, e.g., [8, 28] and [7]. The applicaion of POD is jusified by an a priori error analysis for he considered nonlinear dynamical sysem, where we combine echniques from [2, 2] and [27]. Le us refer o [4], where he auhors also combine successfully an MPC scheme wih a POD reduced-order approach. However, no analysis is carried ou ensuring he asympoic sabiliy of he proposed MPC-POD scheme. Our conribuion focusses on he sabiliy analysis of he POD-MPC algorihm wihou erminal consrains, where he dynamical sysem is a semilinear parabolic parial differenial equaion wih an advecion erm. In paricular, we sudy a minimal finie horizon for he reduced-order approimaion such ha i guaranees he asympoic sabiliy of he surrogae model. Our approach is moivaed by he work [6]. The main difference here is ha we have added an advecion erm in he dynamical sysem and uilize a POD subopimal sraegy o solve he open-loop problems. Since he minimal predicion horizon 2

4 can be large, he numerical soluion of he open-loop problems is very epensive wihin he MPC algorihm. The applicaion of he POD model reducion reduces efficienly he compuaional cos by compuing subopimal soluions. Bu we involve his subopimaliy in our sabiliy analysis in order o ensure he asympoic sabiliy of our MPC scheme. The paper is organized in he following manner: In Secion 2 we formulae our infinie horizon opimal conrol problem governed by a semilinear parabolic equaion and bilaeral conrol consrains. The MPC algorihm is inroduced in Secion 3. For he readers convenience, we recall he known resuls of he sabiliy analysis. Furher, he sabiliy heory is applied o our underlying nonlinear semilinear equaions and bilaeral conrol consrains. In Secion 4 we invesigae he finie horizon open loop problem which has o be solved a each level of he MPC algorihm. Moreover, we inroduce he POD reducedorder approach and prove an a-priori error esimae for he semilinear parabolic equaion. Finally, numerical eamples are presened in Secion Formulaion of he conrol sysem Le = (, ) R be he spaial domain. For he iniial ime R + = {s R s } we define he space-ime cylinder Q = (, ). By H = L 2 () we denoe he Lebesgue space of (equivalence classes of) funcions which are (Lebesgue) measurable and square inegrable. We endow H by he sandard inner produc denoed by, H and he associaed induced norm ϕ H = ϕ, ϕ /2 H.Furhermore,V = H () H sands for he Sobolev space V = { ϕ H ϕ () } 2 d< and ϕ() = ϕ() =. Recall ha boh H and V are Hilber spaces. In V we use he inner produc ϕ, φ V = ϕ ()φ ()d for ϕ, φ V and se ϕ V = ϕ, ϕ /2 V for ϕ V. For more deails on Lebesgue and Sobolev spaces we refer he reader o [], for insance. When he ime is fied for a given funcion ϕ : Q R, he epression ϕ() sands for a funcion ϕ(,) considered as a funcion in only. Recall ha he Hilber space L 2 (Q) canbe idenified wih he Bochner space L 2 (, ; H). We consider he following conrol sysem governed by a semilinear parabolic parial differenial equaion: y = y(, ) solves he semilinear iniial boundary value problem y θy + y + ρ(y 3 y) =u in Q, (2.a) y(, ) =y(, ) = in(, ), (2.b) y( )=y in. (2.c) 3

5 In (2.a) i is assumed ha he conrol u = u(, ) belongs o he se of admissible conrol inpus U ad ( )= { u U( ) u(, ) Uad for almos all (f.a.a.) (, ) Q }, (2.2) where U( ) = L 2 (, ; H) andu ad = {u R u a u u b } wih given u a u b. The parameers θ and ρ saisfy (θ, ρ) D ad = { ( θ, ρ) R 2 θ a θ and ρ a ρ } wih posiive θ a and ρ a. Furher, in (2.c) he iniial condiion y = y () is supposed o belong o H. A soluion o (2.) is inerpreed in he weak sense as follows: for given (,y ) R + H and u U ad( ) we call y a weak soluion o (2.) for fied (θ, ρ) D ad if y() V, y () V hold f.a.a. and y saisfies y( )=y in H as well as d d y(),ϕ H + θy ()ϕ + ( y ()+ρ(y 3 () y()) ) ϕ d = u()ϕ d (2.3) for all ϕ V and f.a.a. >. Here, y () sands for he disribuional derivaive wih respec o he ime variable saisfying [, p. 477] d d y(),ϕ H = y (),ϕ V,V for all ϕ V. The following resul is proved in [8], for insance. Proposiion 2.. For given (,y ) R + H and u U ad( ) here eiss a unique weak soluion y = y [u,,y ] o (2.) for every (θ, ρ) D ad. Le (,y ) R + H be given. Due o Proposiion 2. we can define he quadraic cos funcional: Ĵ(u;,y ):= y [u,,y 2 ]() y d 2 d + λ u() 2 H H d (2.4) 2 for all u U( ) U ad ( ), where y [u,,y ] denoes he unique weak soluion o (2.). We suppose ha y d = y d () is a given desired saionary sae in H (e.g., he equilibrium y d =)andhaλ> denoes a fied weighing parameer. Then we consider he nonlinear infinie horizon opimal conrol problem min Ĵ(u;,y ) subjec o (s..) u U ad ( ). (2.5) Suppose ha he rajecory y is measured a discree ime insances n = + nδ, n, where he ime sep Δ > sands for he ime sep beween wo measuremens. Thus, we wan o selec a conrol u U ad () such ha he associaed rajecory y [u,,y ] follows a given desired sae y d as good as possible. This problem is called a racking problem, and, if y d =holds,asabilizaion problem. Since our goal is o be able o reac o he curren deviaion of he sae y a ime = n from he given reference value y d, we would like o have he conrol in feedback form, i.e., we wan o deermine a mapping μ : H U ad ( )wih u() =μ(y()) for [ n, n+ ]. 4

6 3. onlinear model predicive conrol We presen an MPC approach o compue a mapping μ which allows a represenaion of he conrol in feedback form. For more deails we refer he reader o he monographs [5, 24], for insance. 3.. The MPC mehod To inroduce he MPC algorihm we wrie he weak form of our conrol sysem (2.) as a paramerized nonlinear dynamical sysem. For (θ, ρ) D ad le us inroduce he θ-and ρ-dependen nonlinear mapping F which maps he space V H ino he dual space V of V as follows: F(ϕ, v) = θϕ + ϕ + ρ(ϕ 3 ϕ) v for (ϕ, v) V H. Then, we can epress (2.3) as he nonlinear dynamical sysem y () =F(y(),u()) V for all >, y( )=y in H (3.) for given (,y ) R + H. The cos funcional has been already inroduced in (2.4). Summarizing, we wan o solve he following infinie horizon minimizaion problem min Ĵ(u;,y )= l ( y [u,,y ](),u() ) d s.. u U ad ( ), (P( )) where we have defined he running quadraic cos as l(ϕ, v) = ( ) ϕ y d 2 H 2 + λ v 2 H for ϕ, v H. (3.2) If we have deermined a sae feedback μ for (P( )), he conrol u() =μ(y()) allows a closed loop represenaion for [, ). Then, for a given iniial condiion y H we se =,y = y in (3.) and inser μ o obain he closed-loop form y () =F(y(),μ(y())) in V for (, ), y( )=y in H. (3.3) oe ha he infinie horizon problem may be very hard o solve due o he dimensionaliy of he problem. On he oher hand i guaranees he sabilizaion of he problem which is very imporan for cerain applicaions. In an MPC algorihm a sae feedback law is compued for (P( )) by solving a sequence of finie ime horizon problems. To formulae he MPC algorihm we inroduce he finie horizon quadraic cos funcional as follows: for (,y ) R + H and u U ad () we se Ĵ (u;,y )= l ( y [u,,y ](),u() ) d, 5

7 where is a naural number, = + Δ is he final ime and Δ denoes he lengh of he ime horizon for he chosen ime sep Δ >. Furher, we inroduce he Hilber space U ( )=L 2 (, ; H) and he se of admissible conrols U ad( )= { u U ( ) u(, ) Uad f.a.a. (, ) Q } wih Q = (, ) Q; compare (2.2). In Algorihm he mehod is presened. Algorihm (MPC algorihm) Require: ime sep Δ >, finie horizon, weighing parameer λ>. : for n =,, 2,... do 2: Measure he sae y( n ) V of he sysem a n = nδ. 3: Se = n = nδ, y = y( n ) and compue a global soluion o min Ĵ (u;,y ) s.. u U ad( ). (P ( )) We denoe he obained opimal conrol by ū. 4: Define he MPC feedback value μ (;,y )=ū (), (, +Δ] and use his conrol o compue he associaed sae y = y [μ ( ),,y ] by solving (3.) on [, +Δ]. 5: end for We sore he opimal conrol on he firs subinerval [, +Δ] =[, Δ]andhe associaed opimal rajecory. Then, we iniialize a new finie horizon opimal conrol problem whose iniial condiion is given by he opimal rajecory ȳ() = y [μ ( ),,y ]() a = +Δ using he opimal conrol μ (;,y )=ū () for (, +Δ]. We ierae his process by seing = +Δ. Of course, he larger he horizon, he beer he approimaion one can have, bu we would like o have he minimal horizon which can guaranee sabiliy [6]. oe ha (P ( )) is an open loop problem on a finie ime horizon [, + Δ] which will be sudied in Secion Dynamic programming principle (DPP) and asympoic sabiliy For he reader s convenience we now recall he essenial heoreical resuls from dynamic programming and sabiliy analysis. Le us firs inroduce he so called value funcion v defined as follows for an infinie horizon opimal conrol problem: v(,y ):= inf Ĵ(u;,y ) for (,y ) R + H. u U ad ( ) 6

8 Le be chosen. The DDP saes ha he value funcion v saisfies for any k {,...,} wih k = k + kδ: v(,y ) { k = inf l ( y [u, u U k ad (,y ](),u() ) d + v ( + kδ, y [u,,y ]( + kδ) ) } ) which holds under very general condiions on he daa; see, e.g., [7] for more deails. The value funcion for he finie horizon problem (P ( )) is of he following form: v (,y )= inf Ĵ (u;,y ) for (,y ) R + H. u U ad () The value funcion v saisfies he DPP for he finie horizon problem for + kδ, <k<: v (,y ) { +kδ = inf l ( y [u, u U k ad (,y ](),u() ) d + v k( y [u,,y ]( + kδ) ) }. ) onlinear sabiliy properies can be epressed by comparison funcions which we recall here for he readers convenience [5, Definiion 2.3]. Definiion 3.. We define he following classes of comparison funcions: K = { β : R + R+ β is coninuous, sricly increasing and β() = }, K = { β : R + } R+ β K,βis unbounded, { L = β : R + } R+ β is coninuous, sricly decreasing, lim β() =, KL = { β : R + R+ } R+ β is coninuous, β(,) K,β(r, ) L. Uilizing a comparison funcion β KLwe inroduce he concep of asympoic sabiliy; see, e.g. [5, Definiion 2.4]. Definiion 3.2. Le y [μ( ),,y ] be he soluion o (3.3) and y H an equilibrium for (3.3), i.e., we have F(y,μ(y )) =. Then, y is said o be locally asympoically sable if here eis a consan η> and a funcion β KL such ha he esimae y [μ( ),,y ]() y H β ( y y H,) holds for all y H saisfying y y H <ηand all. Le us recall he main resul abou asympoic sabiliy via DPP; see [6]. 7

9 Proposiion 3.3. Le be chosen and he feedback mapping μ be compued by Algorihm. Assume ha here eiss an α (, ] such ha for all (,y ) R + H he relaed DPP v (,y ) v ( +Δ, y [μ ( ),,y ]( +Δ) ) + α l ( y,μ (y )) ) (3.4) holds. Then we have for all (,y ) R + H: α v(,y ) α Ĵ(μ (y [μ ( ),,y ]);,y ) v (,y ) v(,y ), (3.5) where y [μ ( ),,y ] solves he closed-loop dynamics (3.3) wih μ = μ. If, in addiion, here eiss an equilibrium y H and α,α 2 K saisfying ( ) l (y )= min l(y,u) α y y u U H, ad ( ) α 2 y y H v (,y ) (3.6a) (3.6b) hold for all (,y ) R + H, heny is a globally asympoically sable equilibrium for (3.3) wih he feedback map μ = μ and value funcion v. Remark 3.4. ) Our running cos l defined in (3.2) saisfies condiion (3.6a) for he choice y d = y. Furher, (3.6b) follows from he finie horizon quadraic cos funcional Ĵ, he definiion of he value funcion v and our a-priori analysis presened in Lemma 3.6 below. Therefore, we only have o check he relaed DPP (3.4). 2) I is proved in [6] ha lim α =. Hence, we would like o find α close o one o have he bes approimaion of v in erms of v. On he oher hand, a large implies ha he numerical soluion of (P ( )) is much more involved. We will discuss he numerical compuaion of α ne. 3) By (3.5) we obain he subopimaliy esimae Ĵ(μ ( ) v (,y ) y [μ ( ),,y ]);,y α v(,y ) α ; compare [5, Secion 4.3]. In order o esimae α in he relaed DPP we require he eponenial conrollabiliy propery for he sysem. Definiion 3.5. Sysem (3.) is called eponenially conrollable wih respec o he running cos l if for each (,y ) R + H here eis wo real consans C>, σ [, ) and an admissible conrol u U ad ( ) such ha: l(y [u,,y ](),u()) Cσ l (y ) f.a.a.. (3.7) We presen an a-priori esimae for he unconrolled soluion o (3.), i.e., he soluion for u =. For a proof we refer o he Appendi A. Recall ha V is coninuously (even compacly) embedded ino H. Due o he Poincaré inequaliy [] here eiss a consan C V > such ha ϕ H C V ϕ V for all ϕ V. (3.8) 8

10 Lemma 3.6. Le (,y ) R + H and u = Ky U ad( ) wih an appropriae real consan K>. Then, he soluion y = y [u,,y ] o (3.) saisfies he a- priori esimae y() H e γ(k)( ) y H f.a.a. (3.9) wih γ(k) =γ(k; θ, ρ) =K + θ/c V ρ. Remark 3.7. ) Le K = hold. Then, for θ>ρc V we have γ>. Then, (3.9) implies ha y() H < y H for any >. Moreover, he origin y = is unsable for γ<; see[5, Eample 6.27]. 2) If K>ρ θ/c V holds, y() H ends o zero for. Le us choose y d =. Suppose ha we have a paricular class of sae feedback conrols of he form u(, ) = Ky(, ) wih a posiive consan K; see [6]. This assumpion helps us o derive he eponenial conrollabiliy in erms of he running cos l and o compue a minimal finie ime predicion horizon Δ ensuring asympoic sabiliy. Combining (3.9) wih he desired eponenial conrollabiliy (3.7) and using y d = we obain for all [6]: l(y(),u()) = 2 ( 2 y() H + λ ) u() 2 H = 2 ( + λk2 ) y() 2 H 2 C(K)e 2γ(K)( ) y 2 H = C(K)σ(K) l (y ) (3.) f.a.a. and for every (,y ) R + H, where C(K) =(+λk 2 ), σ(k) =e 2γ(K). (3.) In he following heorem we provide an eplici formula for he scalar α in (3.4). A complee discussion is given in [6]. Theorem 3.8. Assume ha he sysem (3.) and l saisfy he conrollabiliy condiion (3.7). Le he finie predicion horizon Δ be given wih and Δ >. Then he parameer α depends on K and is given by: ( α η (K) ) ( i=2 ηi (K) ) (K) = i=2 η i(k) ( i=2 ηi (K) ) (3.2) where η i (K) =C( σ i )/( σ) and he consans C = C(K), σ = σ(k) are given by (3.). Remark 3.9. ) Theorem 3.8 suggess how we can compue a minimal horizon which ensures asympoic sabiliy; see [5]. Due o (3.) we fi a small finie horizon compue a (global) soluion K o ma α (K) s.. γ(k) ε (3.3) wih <ε andη i (K) from Theorem 3.8. If he opimal value α ( K) is greaer han zero, he finie horizon guaranees asympoic sabiliy. If α ( K) < holds, we enlarge and solve (3.3) again. 9

11 K y a < y a y b = no consrains no considered y b < K u b / y b impossible y b > K min { u a /y b,u b / y a } K u a /y b Table 3.: Consrains for he feedback facor K in u(, ) = Ky(, ) considering he bilaeral conrol consrains (3.4) and he iniial condiion (3.5). 2) Since we suppose ha u U ad (), we have o guaranee he bilaeral conrol consrains u a Ky(, ) u b f.a.a. (, ) Q (3.4) wih u a u b. This leads o addiional consrains for K in (3.3). Since we deermine K in such a way ha γ(k) > is saisfied, we derive from (3.9) ha y() H y H f.a.a.. Leussupposehawehavey and y() C() y C() f.a.a.. Then, we define y a =miny (), y b =may (). (3.5) Then, K has o saisfy γ(k) ε and he resricions shown in Table 3.. Summarizing, K has always an upper bound due o he consrains u a, u b and a lower bound due o he sabilizaion relaed o γ(k) >. 4. The finie horizon problem (P ( )) In his secion we discuss (P ( )), which has o be solved a each level of Algorihm. 4.. The open loop problem Recall ha we have inroduced he final ime = +Δ and he conrol space U ( )=L 2 (, ; H). The space Y ( )=W(, )isgivenby W (, )= { ϕ L 2 (, ; V ) ϕ L 2 (, ; V ) }, which is a Hilber space endowed wih he common inner produc [, pp ]. We define he Hilber space X ( )=Y ( ) U ( ) endowed wih he sandard produc opology. Moreover, we inroduce he Hilber space Z ( )=

12 Z ( ) H wih Z ( )=L 2 (, ; V ) and he nonlinear operaor e =(e,e 2 ): X ( ) Z ( ) by e (),ϕ Z ( ),Z () = + y (),ϕ() V,V d ( θy ()ϕ()+ y ()+ρ ( y() 3 y() ) ) u() ϕ()dd, e 2 (),φ H = y( ) y,φ H for =(y, u) X ( ), (ϕ, φ) Z ( ), where we idenify he dual Z ( ) of Z ( )wihl 2 (, ; V ) H and, Z ( ),Z () denoes he dual pairing beween Z ( ) and Z ( ). Then, for given u U ( ) he weak formulaion for (2.3) can be epressed as he operaor equaion e() =inz ( ). Furher,wecanwrie(P ( )) as a consrained infinie dimensional minimizaion problem wih he feasible se min J() = l(y(),u()) d s.. F ad( ) (4.) F ad( )= { =(y, u) X ( ) e() =inz ( ) and u U ad( ) }. For given fied conrol u U ad () we consider he sae equaion e(y, u) = Z ( ), i.e., y saisfies d d y(),ϕ H + θy ()ϕ + ( y ()+ρ(y() 3 y()) ) ϕ d = u()ϕ d f.a.a. (, (4.2) ], y( ),ϕ H = y,ϕ H for all ϕ V. The following resul is proved in [29, Theorem 5.5]. Proposiion 4.. For given (,y ) R + H and u U ad () here eiss a unique weak soluion y Y ( ) o (4.2) for every (θ, ρ) D ad. If, in addiion, y is essenially bounded in, i.e., y L () holds, we have y L (Q ) saisfying y Y ( ) + y L (Q ) C( u U ( ) + y ) L () (4.3) for a C>, which is independen of u and y. Uilizing (4.3) i can be shown ha (4.) possesses a leas one (local) opimal soluion which we denoe by =(ȳ, ū ) F ad (); see [29, Chaper 5]. For he numerical compuaion of we urn o firs-order necessary opimaliy condiions for (4.). To ensure he eisence of a unique Lagrange muliplier we invesigae he surjeciviy of he linearizaion e ( ):X ( ) Z ( ) of

13 he operaor e a a given poin =(ȳ, ū ) X ( ). oe ha he Fréche derivaive e ( )=(e ( ),e 2( )) of e a is given by e ( ), ϕ Z ( ),Z () = + e 2( ), φ H = y( ),φ H y (),ϕ() V,V d ( θy ()ϕ()+ y ()+ρ ( 3ȳ () 2 ) ) y() u() ϕ()dd, for =(y, u) X ( ), (ϕ, φ) Z ( ). ow, he operaor e ( ) is surjecive if and only if for an arbirary F =(F,F 2 ) Z ( ) here eiss a pair = (y, u) X ( ) saisfying e ( )=F in Z ( ) which is equivalen wih he fac ha here eis a u U ( )anday Y ( ) solving he linear parabolic problem y θy + y + ρ(3ȳ 2 )y = F in Z ( ), y( )=F 2 in H. (4.4) Uilizing sandard argumens [] i follows ha here eiss for any u U ( ) a unique y Y ( ) solving (4.4). Thus, e ( ) is a surjecive operaor and he local soluion o (4.) can be characerized by firs-order opimaliy condiions. We inroduce he Lagrangian by L(, p, p )=J()+ e(), (p, p ) Z ( ),Z ( ) for X ( )and(p, p ) Z ( ). Then, here eiss a unique associaed Lagrange muliplier pair ( p, p ) o (4.) saisfying he opimaliy sysem y L(, p, p )y = y Y ( ) (adjoin equaion) u L(, p, p )(u ū ) u U ad( ) (variaional inequaliy), e( ), (p, p ) Z ( ),Z ( ) = ( p, p ) Z ( ) (sae equaion). I follows from variaional argumens ha he srong formulaion for he adjoin equaion is of he form p θ p p ρ ( 3(ȳ ) 2) p = y d ȳ in Q, p (, ) = p (, ) = in(, ), p ( )= in. (4.5) Moreover, we have p = p ( ). The variaional inequaliy base he form (λū p )(u ū )dd for all u U ad( ). (4.6) Using he echniques as in [3, Proposiion 2.2] one can prove ha secondorder sufficien opimaliy condiions can be ensured provided he residuum ȳ y d L2 (, ;H) is sufficienly small. 2

14 4.2. POD reduced order model for open-loop problem To solve (4.) we apply a reduced-order discreizaion based on proper orhogonal decomposiion (POD); see [7]. In his subsecion we briefly inroduce he POD mehod, presen an a-priori error esimae for he POD soluion o he sae equaion e() = Z ( ) and formulae he POD Galerkin approach for (4.) The POD mehod for dynamical sysems By X we denoe eiher he funcion space H or V. Then, for le he so-called snapshos or rajecories y k () X be given f.a.a. [, ] and for k. A leas one of he rajecories y k is assumed o be nonzero. Then we inroduce he linear subspace { } V =span y k () [, ]a.e.and k X (4.7) wih dimension d. We call he se V snapsho subspace. The mehod of POD consiss in choosing a complee orhonormal basis in X such ha for every l d he mean square error beween y k () and heir corresponding l-h parial Fourier sum is minimized on average: min k= y k () l y k 2 (),ψ i X ψ i d i= s.. {ψ i } l i= X and ψ i,ψ j X = δ ij, i, j l, X (P l ) where he symbol δ ij denoes he Kronecker symbol saisfying δ ii = and δ ij = for i j. An opimal soluion { ψ i } l i= o (Pl ) is called a POD basis of rank l. The soluion o (P l ) is given by he ne heorem. For is proof we refer he reader o [7, Theorem 2.3]. Theorem 4.2. Le X be a separable real Hilber space and y k,...,yn k X be given snapshos for k. Define he linear operaor R : X X as follows: Rψ = ψ, y k () X y k ()d for ψ X. (4.8) k= Then, R is a compac, nonnegaive and symmeric operaor. Suppose ha { λ i } i and { ψ i } i denoe he nonnegaive eigenvalues and associaed orhonormal eigenfuncions of R saisfying R ψ i = λ i ψi, λ... λ d > λ d+ =...=, λi as i. (4.9) Then, for every l d he firs l eigenfuncions { ψ i } l i= solve (Pl ). Moreover, he value of he cos evaluaed a he opimal soluion { ψ i } l i= saisfies E(l) = k= y k () l y k (), ψ 2 i X ψi d = i= X d i=l+ λ i. (4.) 3

15 Remark 4.3. In real compuaions, we do no have he whole rajecories y k () a hand f.a.a. [, ] and for k. Moreover, he space X has o be discreized as well. In his case, a discree version of he POD mehod should be uilized; see, e.g., [7] The Galerkin POD scheme for he sae equaion Suppose ha (,y ) R + H and = + Δ wih predicion horizon Δ >. For given fied conrol u U ad () we consider he sae equaion e(y, u) = Z ( ), i.e., y saisfies (4.2). Le us urn o a POD discreizaion of (4.2). To keep he noaion simple we apply only a spaial discreizaion wih POD basis funcions, bu no ime inegraion by, e.g., he implici Euler mehod. In his secion we disinguish wo choices for X: X = H and X = V. We choose he snapshos y = y and y 2 = y, i.e., we se = 2. By Proposiion 4. he snapshos y k, k =,...,, belong o L 2 (, ; V ). According o (4.9) le us inroduce he following noaions: R V ψ = R H ψ = k= k= ψ, y k () V y k ()d for ψ V, ψ, y k () H y k ()d for ψ H. To disinguish he wo choices for he Hilber space X we denoe by he sequence {(λ V i,ψv i )} i R + V he eigenvalue decomposiion for X = V, i.e., we have R V ψ V i = λ V i ψ V i for all i. Furhermore, le {(λ H i,ψh i )} i R + H in saisfy R H ψ H i = λ H i ψ H i for all i. Then, d =dimr V (V )=dimr H (H) ; see [27]. The ne resul also aken from [27] ensures ha he POD basis {ψi H}l i= of rank l build a subse of he es space V. Lemma 4.4. Suppose ha he snapshos {y k } k= belong o L2 (, ; V ). Then, we have ψi H V for i =,...,d. Le us define he wo POD subspaces V l =span { ψ V,...,ψ V l } V, H l =span { ψ H,...,ψ H l } V H, where H l V follows from Lemma 4.4. Moreover, we inroduce he orhogonal projecion operaors P l H : V Hl V and P l V : V V l V as follows: v l = PHϕ l for any ϕ V iff v l solves min ϕ w l V, w l H l v l = PV l ϕ for any ϕ V iff v l solves min ϕ w l V. w l V l (4.) 4

16 I follows from he firs-order opimaliy condiions for (4.) ha v l = P l H ϕ saisfies v l,ψ H i V = ϕ, ψ H i V, i l. (4.2) Wriing v l H l in he form v l = l j= vl j ψh j we derive from (4.2) ha he vecor v l =(v l,...,v l l ) R l saisfies he linear sysem l ψj H,ψi H V vj l = ϕ, ψi H V, i l. (4.3) j= Summarizing, v l = P l H ϕ Hl is given by he epansion l j= vl j ψh j,wherehe coefficiens {v l j }l j= saisfy he linear sysem (4.3). For he operaor Pl V : V V l we have he eplici represenaion P l V ϕ = l ϕ, ψ i V ψ i for ϕ V. (4.4) i= We conclude from (4.) ha k= y k () P l V y k () 2 V d = Le us define he linear space X l V as X l =span { ψ,...,ψ l }, d i=l+ λ V i. (4.5) where ψ i = ψi V in case of X = V and ψ i = ψi H in case of X = H. Hence, X l = V l and X l = H l for X = V and X = H, respecively. ow, a POD Galerkin scheme for (4.2) is given as follows: find y l () X l f.a.a. [, ] saisfying d d yl (),ψ H + y l ( ),ψ H = y,ψ H θy()ψ l + ( y()+ρ(y l l () 3 y l ()) ) ψ d = u()ψ d f.a.a. (, ], (4.6) for all ψ X l. I follows by similar argumens as in he proof of Proposiion 4. ha here eiss a unique soluion o (4.6). If y L (Q ) holds, y l saisfies he a-priori esimae y l Y ( ) + yl L (Q ) C( y L () + u U ( )), (4.7) where he consan C>isindependen of l and y.lep l denoe PV l in case of X = V and PH l in case of X = H. The ne resul is proved in Appendi B. 5

17 Theorem 4.5. Suppose ha (,y ) R + L (), = + Δ wih predicion horizon Δ >. Furher, le u U ad () be a fied conrol inpu. By y and y l we denoe he unique soluion o (4.2) and (4.6), respecively, where he POD basis of rank l is compued by choosing =2, y = y and y 2 = y. Then, d y l ( ) PV l y 2 H + λ V i, X = V, y y l 2 Y ( ) C i=l+ d y l ( ) PHy l 2 H + λ H i ψi H PHψ l i H 2 V, X = H i=l+ for a C> which is independen of l. Inparicular,lim l y y l Y ( ) = The Galerkin POD scheme for he opimaliy sysem Suppose ha we have compued a POD basis {ψ i } l i= of rank l by choosing X = H or X = V. Suppose ha for u U ad () he funcion y l is he POD Galerkin soluion o (4.6). Then he POD Galerkin scheme for he adjoin equaion (4.5) is given as follows: find p l X l =span{ψ,...,ψ l } f.a.a. [, ] saisfying d d pl (),ψ H + θp l ()ψ ( p l ()+ρ( 3y l () 2 ) ) p l ()ψ d ( = yd y l () ) ψ d = f.a.a. [, ), (4.8) p l ( ),ψ H = for all ψ X l. A-priori error esimaes for he POD soluion p l o (4.8) can be derived by variaional argumens; compare [26] and [7, Theorem 4.5]. If p l is compued, we can derive a POD approimaion for he variaional inequaliy (4.6): (λu p l )(ũ u)dd for all ũ U ad( ). (4.9) Summarizing, a POD subopimal soluion,l = (ȳ,l, ū,l ) X ad () o (P ( )) saisfies ogeher wih he associaed Lagrange muliplier p,l Y ( ) he coupled sysem (4.6), (4.8) and (4.9). The POD approimaion of he finie horizon quadraic cos funcional (4.) reads Ĵ,l (u;,y )= l ( y[u, l,y ] (),u()) d, where y[u, l,y ] is he soluion o (4.6). In Algorihm 2 we se up he POD discreizaion for Algorihm. Due o our POD reduced-order approach an opimal soluion o (P,l ( )) can be compued much faser han he one o (P ( )). In he ne subsecion we address he quesion, how he subopimaliy of he conrol influences he asympoic sabiliy. 6

18 Algorihm 2 (POD-MPC algorihm) Require: ime sep Δ >, finie conrol horizon, weighing parameer λ>, POD olerance τ pod >. : Compue a POD basis {ψ i } l i= saisfying (4.) wih E(l) τ pod. 2: for n =,, 2,... do 3: Measure he sae y( n ) V of he sysem a n = nδ. 4: Se = n = nδ, y = y( n ) and compue a global soluion o min Ĵ,l (u l ;,y l ) s.. u l U ad( ). (P,l ( )) We denoe he opimal conrol by ū,l and he opimal sae by ȳ,l. 5: Define he MPC feedback value μ,l (;,y )=ū,l () and use his conrol o compue he associaed sae y = y [μ,l ( ),,y ] by solving (3.) on [, +Δ]. 6: end for 4.3. Asympoic sabiliy for he POD-MPC algorihm In his subsecion we presen he main resuls of his paper. We give sufficien condiions ha Algorihm 2 gives a sabilizing feedback conrol for he reducedorder model. Due o Definiion 3.5 we have o find an admissible conrol u U ( ) for any so ha he soluion o (3.) saisfies (3.7). In (3.2) we have inroduced our running quadraic cos. As in Secion 3.2 we choose y d = y =. Suppose ha y l is he reduced-order soluion o (4.6) for he conrol u l = Ky l.ifk saisfies appropriae bounds (see Remark 3.9-2)), we can ensure ha u l U ad () holds. Analogously o (3.9) ands (3.) we find y l () 2 H σ(k) y 2 H f.a.a. (4.2) and l ( y l (),u l () ) C(K) y l () 2 H 2. (4.2) wih he same consans C(K) andσ(k) as in (3.). Le y [ul,,y ] be he (fullorder) soluion o (4.6) for he same admissible conrol law u = u l. Uilizing he Cauchy-Schwarz inequaliy we ge l ( y [u l,,y ](),u l () ) 2 y [u l,,y ]() y l () 2 + l( y l (),u l () ) H + y [ul,,y ]() y l () H y l () H. (4.22) If y holds, we infer ha y l () H is posiive for all [, ]. Then, we conclude from (4.2), (4.22) and (4.2) ha he eponenial conrollabiliy condiion (3.7) holds for he admissible conrol law u l = Ky l : l ( y [u l,,y ](),u l () ) ( ) Err(; l) 2 + C(K)+2Err(; l) y l () 2 H 2 2 Cl (K) σ(k) y 2 H = Cl (K) σ(k) l (y ) 7

19 wih he error erm Err(; l) = y [u l,,y ]() y l () H y l () H (4.23) and he consan C l (K) =C(K)+2Err(; l)+err(; l) 2 C(K). (4.24) Thus, he consan C l (K) akes ino accoun he approimaion made by he POD reduced-order model. In he following heorem we provide an eplici formula for he scalar α,l which appears in he relaed DPP. The noaion α,l inends o sress ha we are working wih POD surrogae model. We summarize our resul in he following heorem. Theorem 4.6. Le he consan C l be given by (4.24) and Δ denoe he finie predicion horizon wih and Δ >. Then he parameer α,l is given by he eplici formula: ( η l (K) ) ( η l i (K) ) α,l (K) = i=2 i=2 ηl i (K) i=2 ( η l i (K) ) (4.25) wih η l i (K) =Cl (K)( σ i (K))/( σ(k)) and σ(k) as in (3.). Remark 4.7. ) If Err(; l) is small, Theorem 4.6 informs we can compue he consan α,l α basically in he same way of he full-model, replacing he consans C, η wih C l, η l, respecively, aking ino accoun he POD reduced-order modelling. Then, (3.5) implies ha a subopimaliy esimae holds approimaely; see Remark 3.4. To obain he minimal horizon which ensures he asympoic sabiliy of he POD- MPC scheme we maimize (4.25) according o he consrains α,l >, K>ma(,ρ θ/c V ) and o he consrains in Table 3.. 2) Due o (4.2) and u l = Ky l he norm u l () H is bounded independen of l. By Theorem 4.5 and (B.2) we have lim l y [u l,,y ]() y l () H = holdsforall [, ]. Thus, if we choose l sufficienly large we can ensure ha Err(; l) is small enough provided he denominaor saisfies y l () H C wih a posiive consan C which is independen of l. 3) In Algorihm 2 we compue he conrol law ū,l insead of Ky l.therefore, one can replace Err(; l) by Ẽrr(; l) = y [ū,l ( ),,y ]() y,l () H y,l () H ha can be evaluaed easily, since y,l () andy [ū,l,,y ] are known from Algorihm 2, seps 4 and 5, respecively. I urns ou ha for our es eamples boh error erms lead o he same choices for he predicion horizon, for he posiive feedback facor K and for he relaaion parameer α,l (, ]. 8

20 5. umerical ess This secion presens numerical ess in order o show he performance of our proposed algorihm. All he numerical simulaions repored in his paper have been made on a MacBook Pro wih CPU Inel Core i5 2.3 Ghz and 8GB RAM. 5.. The finie difference approimaion for he sae equaion For we inroduce an equidisan spaial grid in by i = iδ, i =,..., +, wih he sep size Δ = /( + ). A = and + = he soluion y is known due o he boundary condiions (2.). Thus, we only compue approimaions yih () for y(, i ) wih i and [, f ]. We h define he vecor y h () = (yh (),..., y ()) R of he unknowns. Analoh h h gously, we define u = (u,..., u ) R, where uhi approimaes u(i, ) for i. Uilizing a classical second-order finie difference (FD) scheme and an implici Euler mehod for he ime inegraion we derive a discree approimaion of he parabolic problem. In Figure 5. he discree soluions are ploed for = 99, for [, 2] and wo differen iniial condiions Figure 5.: FD sae y for y =. sgn(.3) (lef plo) and y =.2 sin π (righ plo) wih u =, (θ, ρ) = (., ) and = 99. As we see from Figure 5., he unconrolled soluions do no end o zero for, indeed i sabilizes a one POD-MPC eperimens In our numerical eamples we choose yd, i.e., we force he sae o be close o zero, and λ =. in (2.4). A finie horizon open loop sraegy does no seer he rajecory o he zero-equilibrium (see Figure 5.2). Therefore, sabilizaion is no guaraneed by he heory of asympoic sabiliy. oe ha we are no dealing wih erminal consrains and he erminal condiion of he adjoin equaion (4.5) is zero. In our ess, he snapshos are compued aking he unconrolled sysem, e.g. u, in (2.) and he corresponden adjoin equaion (4.5). Several hins for he compuaion of he snapshos in he cone of MPC 9

21 Figure 5.2: Open-loop soluion y for y =. sgn(.3), (θ, ρ) = (., ), = 99, f = 2 (lef plo) and y =.2 sin π, (θ, ρ) = (., ), = 99, f = 2 (righ plo). are given in [4]. The nonlinear erm is reduced following he Discree Empirical Inerpolaion Mehod (DEIM) which is a mehod ha avoid he evaluaion of he full model of he nonlinear par building new basis funcions upon he nonlinear erm; compare [9] for more deails. oe ha, in our simulaions, he opimal predicion horizon is always obained from Theorem 4.6. Run 5. (Unconsrained case wih smooh iniial daa). The parameers are presened in Table 5.. According o he compuaion of α in (3.2) relaed T Δ Δ θ ρ y () ua ub K sin(π) 2.46 Table 5.: Run 5.: Seing for he opimal conrol problem, minimal sabilizing horizon and feedback consan K. o he relaed DPP, he minimal horizon ha guaranes asympoic sabiliy is =. Even in he POD-MPC scheme he asympoic sabiliy is achieved for =, provided ha Err(; l) 3 for all. oe ha he horizon of he surrogae model is compued by (4.25). In Figure 5.3 we show he conrolled sae rajecory compued by Algorihm aking = 3 and =. As we can see, we do no ge a sabilizing feedback for = 3, whereas = leads o a sae rajecory which ends o zero for. oe ha we plo he soluion only on he ime inerval [,.5] in order o have a zoom of he soluion. Furher, in Figure 5.3 he soluion relaed o u = Ky is presened. As we can see, he MPC conrol sabilized o he origin very soon while he conrol law u = Ky requires a larger ime horizon. This is due o he fac ha MPC sabilizes in an opimal way, in conras o he conrol law u = Ky. In Table 5.2 we presen he error in L2 (, T ; H)-norm considering he soluion coming from he Algorihm as he ruh soluion (in our case he finie difference soluion denoed by y F D ). The eamples are compued wih Err(, l) 3. The CPU ime for he full2

22 Figure 5.3: Run 5.: MPC sae wih = 3 (lef plo), wih = (middle plo) and wih u = Ky (righ plo) Ĵ ime K y FD y L 2 (,T ;H) Soluion wih u = Ky Alg..5 49s Alg. 2 (l =3, l DEIM = 5).6 8s.47 Alg. 2 (l =3, l DEIM = 2).6 6s.58 Table 5.2: Run 5.: Evaluaion of he cos funcional, CPU ime, subopimal soluion. model urns ou o be 49 seconds, whereas he POD-subopimal approimaion wih only hree POD and wo DEIM basis funcions requires 6 seconds. We can easily observe an impressive speed up facor eigh. Moreover he evaluaion of he cos funcional in he full model and he POD model provides very close values. We have no considered he CPU ime in he subopimal problem since i did no involve a real opimazion problem. As soon as we have compued K, wihin an offline sage, we direcly approimae he equaion wih he conrol law u = Ky. Run 5.2 (Consrained case wih smooh iniial daa). In conras o Run 5. we choose u a =.3 andu b =. As epeced, he minimal horizon increases compared o Run 5.; see Table 5.3. As one can see from Figure 5.4 he MPC T Δ Δ θ ρ y () u a u b K.5...2sin(π) Table 5.3: Run 5.2: Seing for he opimal conrol problem, minimal sabilizing horizon and feedback consan K. sae wih = 4 ends faser o zero han he sae wih u = Ky. The soluion coming from he POD model is in he middle of Figure 5.4. oe ha E(l =3)=., E(l = 3) =, and Err(; l) 3 for any l and. Indeed, 2

23 Figure 5.4: Run 5.2: MPC sae wih = 4 (lef plo), POD-MPC sae wih =4 (middle plo) and sae wih u = Ky (righ plo) Table 5.4 presens he evaluaion of he cos funcionals for he proposed algorihms and he CPU ime which shows ha he speed up by he reduced order approach is abou 6. oe ha K in Run 5.2 is smaller compared o Run 5. Ĵ ime K y FD y L2 (,T ;H) Soluion wih u = Ky Alg s Alg. 2 (l = 3, l DEIM = 5).32 5s.54 Alg. 2 (l =3,l DEIM = 2).33 4s.55 Table 5.4: Run 5.2: Evaluaion of he cos funcional, CPU imes, subopimal soluion. due o he consrain of he conrol space. Furher, he error is presened in Table 5.4. To sudy he influence of Err(; l) we presen in Figure 5.5, on he lef, how he opimal predicion horizon changes according o differen olerance. The blue line corresponds o he opimal predicion horizon in Run 5., and he red one o Run 5.2. I urns ou ha, as long as Err(; l) 3, we can work eacly wih he same horizon we had in he full model in boh eamples. In he middle plo of Figure 5.5 here is a zoom of he funcion α wih differen values of Err(; l) wih respec o Run 5.2. The righ plo of Figure 5.5 shows herelaiveerrorerr(; l) for.5 wihl = 3. One of he big advanages of feedback conrol is he sabilizaion under perurbaion of he sysem. The perurbaion of he iniial condiion is a ypical eample which comes from many applicaions in fac, ofen he measuremens may no be correc. For a given noise disribuion δ = δ() we consider a perurbaion he following form: y () = ( +δ() ) y () for. The perurbaion is applied only a every iniial condiion of he MPC algorihm (see (P ( )) in Algorihm ) and i is random wih respec o he spaial variable. The sudy of he asympoic sabiliy does no change: we can compue 22

24 Figure 5.5: Run 5.2: Opimal horizon and α,l according o differen Err(; l) = 3, Influence of he relaive error Err(; l) = 3 for l =3. he minimal predicion horizon as before. As we can see in Figure 5.6 he POD- MPC algorihm is able o sabilize wih a noise of δ() 3% Figure 5.6: Run 5.2: POD-MPC sae wih 3% noise (lef plo); Run 5.3: MPC sae wih = 3 (middle plo) and POD-MPC sae wih = 3, l = l DEIM = 6 (righ plo). Run 5.3 (Consrained case wih smooh iniial daa). ow we decrease he diffusion erm and, as a consequence, he predicion horizon increases; see Table 5.5 and middle plo of Figure 5.6. Even if he horizon is very large, T Δ Δ θ ρ y () u a u b K.5.. / 2.2sin(π) 3 5 Table 5.5: Run 5.3: Seing for he opimal conrol problem. he proposed Algorihm 2 acceleraes he approimaion of he problem. The decrease of θ may give some roubles wih he POD-model since he dominaion of he convecion erm causes a high-variabiliy in he soluion, hen a few basis 23

25 funcions will no suffice o obain good surrogae models (see [, 2]). oe ha, in our eample, he diffusion erm is sill relevan such ha we can work wih only 2 POD basis funcions. The CPU ime in he full model is 84 seconds, whereas wih a low-rank model, such as l = 2 we obained he soluion in five seconds and an impressive speed up facor of 6. Even wih a more accurae POD model we have a very good speed up facor of nine. The evaluaion of he cos funcional is given in Table 5.6. In he righ plo of Figure 5.6 he POD- Ĵ ime K y FD y L2 (,T ;H) Subopimal soluion (u = Ky) Algorihm.6 84s Algorihm 2 (l = 6, l DEIM = 6).7 9s.92 Algorihm 2 (l =2,l DEIM = 3).8 5s.93 Table 5.6: Run 5.3: Evaluaion of he cos funcional and CPU ime. MPC sae is ploed for l = 6 POD basis and l DEIM = 6 DEIM ansaz funcions. The error beween he MPC sae and he POD-MPC sae is less han.. Run 5.4 (Consrained case wih no-smooh iniial daa). In he las es we focus on a differen iniial condiion and differen conrol consrains. The parameers are presened in Table 5.7. The minimal horizon which ensures T Δ Δ θ ρ y () u a u b K.5.. /2 5. sgn(.3) Table 5.7: Run 5.4: Seing for he opimal conrol problem. asympoic sabiliy is = 43. Table 5.8 emphazises again he performance of he POD-MPC mehod wih an acceleraion 2 imes faser han he full model. Ĵ ime K y FD y L2 (,T ;H) Soluion wih u = Ky 4.7e Alg. 4.e-4 5s Alg. 2 (l = 7, l DEIM = 9) 4.4e-4 2s.34 Alg. 2 (l =3,l DEIM = 4) 4.4e-4 4s.35 Table 5.8: Run 5.4: Cos funcional, CPU ime and subopimal soluion. The evaluaion of he cos funcional gives he same order in all he simulaion we provide. In Figure 5.7 we presen he MPC sae for =43(lef 24

26 plo), he POD-MPC sae wih = 43, l =3,l DEIM = 4 (middle plo) and he increase of he opimal horizon according o he perurbaion Err(; l). The error beween he MPC sae and he POD-MPC sae is.35 when

27