Alessandro Alla Maurizio Falcone Stefan Volkwein

Transcription

1 Universität Konstanz Error anaysis for POD approximations of infinite horizon probems via the dynamic programming approach Aessandro Aa Maurizio Facone Stefan Vokwein Konstanzer Schriften in Mathematik Nr. 339, September 5 ISSN Konstanzer Onine-Pubikations-System KOPS) URL: Fachbereich Mathematik und Statistik Universität Konstanz Fach D 97, Konstanz, Germany

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3 ERROR ANALYSIS FOR POD APPROXIMATIONS OF INFINITE HORIZON PROBLEMS VIA THE DYNAMIC PROGRAMMING APPROACH A. ALLA, M. FALCONE, AND S. VOLKWEIN Abstract. In this paper infinite horizon optima contro probems for noninear high-dimensiona dynamica systems are studied. Noninear feedback aws can be computed via the vaue function characterized as the unique viscosity soution to the corresponding Hamiton-Jacobi-Beman HJB) equation which stems from the dynamic programming approach. However, the botteneck is mainy due to the curse of dimensionaity and HJB equations are ony sovabe in a reativey sma dimension. Therefore, a reduced-order mode is derived for the dynamica system and for this purpose the method of proper orthogona decomposition POD) is used. The resuting errors in the HJB equations are estimated by an a-priori error anaysis, which suggests a new samping strategy for the POD method. Numerica experiments iustrates the theoretica findings. Key words. Optima contro, noninear dynamica systems, Hamiton-Jacobi Beman equation, proper orthogona decomposition, error anaysis AMS subject cassifications. 35K, 49L, 49L5, 49J, 65N99. Introduction. The Dynamic Programming approach to the soution of optima contro probems driven by dynamica systems in R n offers a nice framework for the approximation of feedback aws and optima trajectories. It suffers from the botteneck of the computation of the vaue function since this requires the approximation of a noninear partia differentia equation in dimension n. This is a very chaenging probem in high dimension due to the huge amount of memory aocations necessary to work on a grid and to the ow reguarity properties of the vaue function which is typicay ony Lipschitz continuous even for reguar dynamics and running costs). Despite the number of theoretica resuts estabished for many cassica contro probems via the dynamic programming approach see e.g. the monographies by Bardi and Capuzzo-Docetta [8] on deterministic contro probems and by Feming and Soner [9] on stochastic contro probems) this has aways been the main obstace to appy this nowadays rather compete theory to rea appications. The curse of dimensionaity has been mitigated via domain decomposition techniques and the deveopment of rather efficient numerica methods but it is sti a big obstace. Athough a detaied description of these contributions goes beyond the scopes of this paper, we want to mention [8] for a domain decomposition method with overapping between the subdomains and [] for simiar resuts without overapping. It is important to note that in these papers the method is appied to subdomains with a rather simpe geometry see the book by Quarteroni and Vai [3] for a genera introduction to this technique) in order to appy transmission conditions at the boundaries. More recenty another way to decompose the probem has been proposed by Krener and Navasca [3] who have used a patchy decomposition based on A brekht method. Later in the paper [] the patchy idea has been impemented taking into account an approximation of the underying optima dynamics to obtain subdomains which are amost Department of Mathematics, Universität Hamburg, Bundesstrasse 55, D-46 Hamburg, Germany, aessandro.aa@uni-hamburg.decorresponding author) Diparnto di Matematica, SAPIENZA - Università di Roma, P.e Ado Moro, I-85 Rome, Itay, facone@mat.uniroma.it Department of Mathematics and Statistics, Universität Konstanz, Universitätsstraße, D Konstanz, Germany, Stefan.Vokwein@uni-konstanz.de

4 invariant with respect to the optima dynamics, ceary in this case the geometry of the subdomains can be rather compex but the transmission conditions at the interna boundaries can be eiminated saving on the overa compexity of the agorithm. In genera, domain decomposition methods reduce a huge probem into subprobems of manageabe size and aows to mitigate the storage imitation distributing the computation over severa processors. However, the approximation schemes used in every subdomain are rather standard. Another improvement can be obtained using efficient acceeration methods for the computation of the vaue function in every subdomain. To this end one can use Fast Marching methods [34, 35] and Fast Sweeping methods [37] for specific casses of Hamiton-Jacobi equations. In the framework of optima contro probems an efficient acceeration technique based on the couping between vaue and poicy iterations has been recenty proposed and studied by Aa, Facone and Kaise in [3, 4]. Finay, we shoud mention that the interested reader can find in [7] a number of successfu appications to optima contro probems and games in rather ow dimension. In parae to these resuts severa mode reduction techniques have been deveoped to dea with high dimensiona dynamics in a rather economic way. These techniques are reay necessary when deaing with optima contro probems governed by partia differentia equations. Despite the vast iterature concerning the anaysis and numerica approximation of optima contro probems for PDEs, the amount of works devoted to the synthesis of feedback controers is rather sma. In this direction, the appication of the dynamic programming principe DPP) is a powerfu technique which has been appied mainy to inear dynamics, quadratic cost functions and unbounded contro space, the so-caed inear quadratic reguator LQR) contro probem. For this probem an expicit feedback controer can be computed by means of the soution of an agebraic Riccati equation. However if the underying structura assumptions are removed, the feedback contro has to be obtained via the approximation of a Hamiton-Jacobi-Beman equation defined over the state space of the system dynamics. As we mentioned, this is a major botteneck for the appication of DPP-based techniques in the optima contro of PDEs, as the natura approach for this cass of contro probems is to consider a semi-discretization in space) via finite eements or finite differences of the abstract governing equations, eading to an inherenty highdimensiona state space. However, in the ast years severa steps have been made to obtain reduced-order modes for compicated dynamics and by the appication of these techniques it is now possibe to have a reasonabe approximation of arge-scae dynamics using a rather sma number of basis functions. This can open the way to the DPP approach in high-dimensiona systems. Reduced-order modes are used in PDE-constrained optimization in various ways; see, e.g., [, 4, 33] for a survey. However, the main stream for the optima contro of PDEs is sti reated to open-oop contros based on the Pontryagin Maximum Principe an extensive presentation of this cassica approach can be found in the monograph [3, 38]). Let us refer to [5, 7, 7, 8, 9], where it has been observed that modes of reduced order can pay an important and very usefu roe in the impementation of feedback aws. More recenty, the Proper Orthogona Decomposition POD) has been proposed for PDE contro probems in order to reduce the dynamics to a sma number of state variabe via a carefu seection of the snapshots. This technique, couped with the Dynamic Programming approach, has been deveoped mainy for inear equations starting from the heat equation where one can take advantage of the reguarity of the soutions to reduce the dimension [7] and then attacking more

5 difficut probems as the advection-diffusion equation [,, 6, 5], Burgers s equation [7, 8] and Navier-Stokes [5]. The aim of this paper is to study the interpay between reduced-order dynamics, the associated dynamic programming equation, the resuting feedback controer and its performance over the high-dimensiona system. In our anaysis we wi derive some a-priori error estimates which take into account the and space discretization parameter t and x as we as the dimension of the POD basis functions used for the reduced mode. The paper is organized as foows: in Section we reca some basic facts about the approximation of the infinite horizon probem via the dynamic programming approach. Section 3 is devoted to present in short the POD technique and the basic ideas behind the construction of the reduced mode. In Section 4 we present the main resuts and our a-priori estimates for the numerica approximation of the reduced mode. These a-priori estimates have been aso used in the construction of the agorithm which is described in detai in Section 5. Some numerica tests are presented and anayzed in Section 6 and finay we draw some concusions in Section 7.. Optima contro probem. In this section we wi reca the Dynamic Programming approach and its numerica approximation for the soution of infinite horizon contro probem... The infinite horizon probem. For given noninear mapping f : R n R m R n and initia condition y R n et us consider the foowing controed noninear dynamica systems.) ẏt) = f yt), ut) ) R n for t >, y) = y R n together with the infinite horizon cost functiona.) Jy, u) = g yt), ut) ) e λt dt In.) we assume that λ > is a given weighting parameter and g maps R n R m to R. We ca y the state and u the contro. The set of admissibe contros has the form U ad = { u U ut) U ad for amost a t }, where we set U = L, ; R m ) and U ad R m denotes a compact, convex subset. Let M R n n denote a symmetric, positive definite mass) matrix with smaest and argest positive eigenvaues λ min and λ max, respectivey. Then, we introduce the foowing weighted inner product in R n : y, ỹ M = y Mỹ for y, ỹ R n, where stands for the transpose of a given vector or matrix. By M =, / M we define the associated induced norm. Reca that we have Then, y soves.) if λ min y y M λ max y for a y R n..3) ẏt) fyt), ut)), ϕ M = for a ϕ R n and for amost a t >, y) y, ϕ M = for a ϕ R n 3

6 We ca.3) the variationa formuation of the dynamica system. Let us suppose that.) has a unique soution y = yu; y ) Y = H, ; R n ) for every admissibe contro u U ad and for every initia condition y R n ; see, e.g., [8, Chapter III]. Thus, we can define the reduced cost functiona as foows: Ĵu; y ) = Jyu; y ), u) for u U ad and y R n, where yu; y ) soves.) for given contro u and initia condition y. Then, our optima contro can be formuated as foows: for given y R n we consider P) min Ĵu; y )... The Hamiton-Jacobi-Beman equation and its discretization. We define the vaue function of the probem v : R n R as foows: vy) = inf { Ĵu; y) u Uad } for y R n. This function gives the best vaue for every initia condition, given the set of admissibe contros U ad. It is characterized as the unique viscosity soution of the Hamiton- Jacobi-Beman HJB) equation corresponding the infinite horizon.4) λvy) + sup { fy, u) vy) gy, u) } = for y R n. In order to construct the approximation scheme as in [5]) et us consider first a discretization where h is a stricty positive step size. A dynamic programming principe for the discrete probem hods true giving the foowing semi-discrete scheme for.4).5) v h y) = min { λh)vh y + hfy, u)) + hgy, u) } for y R n. Throughout our paper we suppose the foowing hypotheses. Assumption. ) The right-hand side f : R n R m R n is continuous and gobay Lipschitzcontinuous in the first argument, i.e., there exists an L f > satisfying fy, u) fỹ, u) L f y ỹ for a y, ỹ R n and u U ad Furthermore, fy, u) = max i n f i y, u) is bounded by a constant M f for a y Ω and u U ad. ) The running cost g : R n R m R n is continuous and gobay Lipschitzcontinuous in the first argument with a Lipschitz constant L g >. Moreover, gy, u) M g for a y, u) Ω U ad with M g >. If Assumption and λ > L f hod, the function v h is Lipschitz-continuous satisfying.6) vh y) v h ỹ) L g y ỹ λ L f for a y, ỹ Ω and h [, /λ); see [6, p. 473]. Let us reca the foowing resut [5, Theorem.3]: Theorem.. Let Assumption and λ > max{l g, L f } hod. Let v and v h be the soutions of.4) and.5), respectivey. Suppose the semiconcavity assumptions.7) fy + ỹ, u) fy, u) + fy ỹ, u) C f ỹ, gy + ỹ, u) gy, u) + gy ỹ, u) Cg ỹ 4

7 for a y, ỹ, u) R n R n U ad. Then, there is a constant C satisfying sup vy) v h y) Ch y R n for any h [, /λ). Remark.. The constant C in Theorem. can be bounded by see [5, Remark ]. { Lg C max λ, M f λ, L } f λ, M f + L ) g ; λ L f.3. The arge-scae approximation of the HJB equations. For the numerica reaization we have to restrict ourseves to a bounded subset of R n. Suppose that there exists a bounded) poyhedron Ω R n such that for sufficienty sma h >.8) y + hfy, u) Ω, for a y Ω and u U ad. We want to point out that the above invariance condition is used here to simpify the probem and focus on the main issue of the a-priori error estimate. If.8) is not satisfied one can appy state constraints to the probem and use appropriate boundary conditions provided at avery point of the boundary there exists at east one contro point inside Ω for this and even more genera state constraints boundary conditions the interested reader can find in [7] some hints and additiona references). Let {S j } m S j= be a famiy of simpices which defines a reguar trianguation of the poyhedron Ω see, e.g., []) such that Ω = m S j= S j and k = max j m S diam Sj ). Throughout this paper we assume that we have n S vertices/nodes y,..., y ns in the trianguation. Let V k be the space of piecewise affine functions from Ω to R which are continuous in Ω having constant gradients in the interior of any simpex S j of the trianguation. Then, a fuy discrete scheme for the HJB equations is given by {.9) v hk y i ) = min λh)vhk yi + hfy i, u) ) + hgy i, u) } for any vertex y i Ω. Ceary, a soution to.5) satisfies.9). Let us reca the foowing resut [5, Coroary.4] and [6, Theorem.3]: Theorem.3. Assume that Assumption,.7) and.8) hod. Let v, v h and v hk be the soutions of.4),.5) and.9), respectivey. For λ > L f we obtain.) sup v h y) v hk y) L f k λλ L f ) h for any h [, /λ). For λ > max{l f, L g } we have.) sup vy) vhk y) Ch + L g λ L f k h for any h [, /λ). 5

8 Coroary.4. Assume that Assumption,.7) and λ > L f hod. Let v hk be the soution of.9). Then, we have for h [, /λ) vhk y) v hk ỹ) C k h + C y ỹ for a y, ỹ Ω, where C = L f /λλ L f )) and C = L g /λ L f ). Proof. Suppose that v h is the soution to.5). Then, we derive from.6) and.) that vhk y) v hk ỹ) vhk y) v h y) + vh y) v h ỹ) + vh ỹ) v hk ỹ) which gives the caim. L f k λλ L f ) h + L g y ỹ λ L, f 3. The POD method and reduced-order modeing. The focus of this section is the construction of surrogate modes by means of the Proper Orthogona Decomposition POD). Here we reca the basics of the method and appy the POD method to optima contro probems. 3.. POD for parametrized noninear dynamica systems. For p N et us choose different pairs contros {u ν, y ν )} p ν= in U ad Ω. By y ν = yu ν ; y ν ) Y, ν =,..., p, we denote the soution to.). We introduce the snapshot subspace as V = span { y ν t) t [, ) and ν p } R n. For every {,..., d}, with dimension d n, a POD basis of rank is defined as a soution to the minimization probem see, e.g., []) p min y ν t) y ν t), ψ P i M ψ i dt ) ν= M i= such that {ψ i } i= R n and ψ i, ψ j M = δ ij, i, j, where δ ij is the Kronecker symbo satisfying δ ii = and δ ij = for i j. It is we-known that a soution to P ) is given by a soution to the eigenvaues probem Rψ i = λ i ψ i for λ λ... λ λ d > with the inear, bounded, symmetric integra operator R : R n V Rψ = p ν= y ν t), ψ M y ν t) dt for ψ R n compare, e.g., [,, 36]). If {ψ i } i= is a soution to P ), we have the approximation error 3.) p ν= y ν t) y ν t), ψ i M ψ i i= M dt = d i=+ In rea computations, we do not have the whoe trajectory yt) for a t [, ). For that purpose we choose T sufficienty arge and define a grid in [, t e ], where 6 λ i.

9 t e T, by = t < t <... < t nt = t e. Let yj ν yν t j ) R n denote approximations for the introduced trajectories {yj ν}p ν= at the instance t j for j =,..., n T. We set V n T = span {yj ν j n T, ν p} with d n T = dim V n T minn, n T p). Then, for every {,..., d n T } we consider the minimization probem p n T min α n T j yj ν yj ν, ψ n T i M ψ n T i P n T ) ν= j= such that {ψ n T i i= M } i= R n and ψ n T i, ψ n T j M = δ ij, i, j, instead of P ). In P n T ) the α j s stand for the trapezoida weights α n T = t t, α n T j = t j t j for j n T, α n T n T = t n T t nt The soution to P n T ) is given by the soution to the eigenvaue probem [, ] R n T ψ n T i = λ n T i ψ n T i for λ n T λ n T... λ n T λ dnt > with the inear, bounded, symmetric and nonnegative operator R n T ψ = n T n T ν= j= Anaogous to 3.) a soution to P n T ) satisfies p n T ν= j= α n T j y ν j α n T j y ν j, ψ M y ν j for ψ R n. i= y ν j, ψ n T i M ψ n T i = M d n T i=+ λ n T i. The reationship between P ) and P n T ) is investigated in [, 6]. 3.. Reduced-order modeing for the state equation. We introduce the POD coefficient matrix Ψ = [ ψ... ψ ] R n and the subspace V = span {ψ,..., ψ } R n. In particuar, the matrix M = Ψ MΨ R is the identity matrix. The reduced-order mode for.3) is derived as foows: we repace the vector yt) R n by its POD approximation Ψy t) R n with the unknown dependent coefficients y t) R and choose ϕ = ψ i for i =,...,. It foows that 3.) ẏ t) = f y t), ut) ) R for t >, y ) = y R, where we have set y = Ψ My R and f y, u) = Ψ MfΨy, u) R for y, u) R U ad, i.e. no discrete interpoation method is used at the moment compare, e.g., [9, 3]). Suppose that 3.) possesses a unique soution y = y u; y ) Y = H, ; R ) for any admissibe contro u U ad. Let us introduce the inear, orthogona projection P : R n V as P y = ΨΨ My = y, ψ i M ψ i for y R n. i= 7

10 We note that the error of a soution to.) and 3.) on a finite horizon can be estimated; see [39, Theorem. and Remark.]. Proposition 3.. For given y Ω, u U ad and t e > et y = yu; y ) H, t e ; R n ) be the unique soution to.) on the finite interva [, t e ]. In P ) we choose p =, y = y and y = ẏ. Suppose that y H, t e ; R n ) is the unique soution to 3.) for {,..., d}. Then, it foows that te yt) y t) M dt Ĉ d i=+ λ i for a constant Ĉ > Reduced-order modeing for the optima contro probem. Next we introduce the POD reduced-order mode for P). For given u, y ) U ad Ω et y = y u; y ) Y denote the unique soution to 3.). Then, the reduced POD cost is given by Ĵ u; y ) = Jy u; y ), u) = Jy, u) = g Ψy t), ut) ) e λt dt = g y t), ut) ) e λt dt, where we have set g y, u) = gψy, u) for y, u) R U ad. Then, the POD approximation for P) reads as foows: for given y Ω we consider P ) min Ĵ u; y ) such that u U ad. 4. A-priori error for the HJB-POD approximation. In this section we present the a-priori error anaysis for the couping between the HJB equation and the POD method. Our first a-priori error estimate is better from a theoretica point of view, whereas for the numerica reaization the second a-priori error estimate is much more appropriate. In the first estimate we assume to work in R on a number of vertices which have been obtained mapping the y i nodes of R n into R. Even if the maximum distance between the y i neighbouring nodes is bounded by k, this ceary produces a non uniform grid where the distance between the neighbouring nodes can not be predicted a-priori since it depends on Ψ. The second error estimate takes into account a uniform grid of size K in R. 4.. First a-priori error estimate. We introduce two different POD approximations for the HJB equation. The first one is based on.9), where we project a vertices {y i } n S i= into R by setting y i = Ψ My i for i =,..., n S. Here we assume that yi y j hods for i, j {,..., n S} with i j. Then, a POD discretization of.9) is given by 4.) vhky i { ) = min λh)v hk y i + hf yi, u) ) + hg yi, u) } for i n S. We define the mapping ṽ hk : Ω R by ṽ hky) = v hkψ My) for a y Ω with Ψ My Ω. 8

11 Using P = ΨΨ M R n n we have ṽhky i ) = vhkψ My i ) = vhky i ) { = min λh)v hk y i + hf yi, u) ) + hg yi, u) } { = min λh)v hk Ψ My i + hfp y i, u) ) + hgp y i, u) } { = min λh)ṽ hk yi + hfp y i, u) ) + hgp y i, u) } for i n S. Thus, 4.) can be equivaenty expressed as 4.) ṽhky { i ) = min λh)ṽ hk yi + hfp y i, u) ) + hgp y i, u) } for i n S. The foowing resut measures the error between a soution to.5) and a soution to 4.). The proof is simiar to the proof of Theorem.3 in [6] and requires an invariance condition which wi be discussed ater in Remark 4.. Proposition 4.. Assume that Assumption,.8) and λ > L f hod. Let v h and be the soutions of.5) and 4.), respectivey and et the invariance condition ṽ hk 4.3) y i + hfp y i, u) Ω for i =,..., n S and for a u U ad be satisfied. Then, there exist two constants Ĉ, Ĉ such that sup vh y) ṽhky) k Ĉ h + Ĉ ns / y i P y i ) for any h [, /λ). i= Proof. For any y Ω there are rea coefficients µ i = µ i y), i n S, of the convex combination representation of y satisfying n S y = µ i y i, µ i and i= n S i= µ i =. Since ṽhk is piecewise affine, we obtain ṽ hk y) = n S i= µ iṽhk y i). Thus, we have 4.4) vh y) ṽhky) n S µ i vh y) v h y i ) ) n S + µ i vh y i ) ṽhky i ) ). i= From y Ω we infer that there exists an index j with y S j Ω. Let I j = {i,..., i k } {,..., n S } denote the index subset such that y i S j hods for i I j. Then, µ i = hods for a i I j. Moreover, n S i= µ i = i I j µ i = and y y i k for any i I j. From.6) we have 4.5) n S i= i= µ i v h y) v h y i ) = i I j µ i v h y) v h y i ) L g λ L f k for h [, /λ). Using 4.) and.5) we have 4.6) v h y i ) ṽhky i ) v h y i ) λh)ṽhk yi + hfp y i, ū,i hk )) + hgp y i, ū,i hk ) λh) v h yi + hfy i, ū,i hk )) ṽhk yi + hfp y i, ū,i hk ))) + h gy i, ū,i hk ) gp y i, ū,i hk )), 9

12 where ū,i hk U ad is defined as 4.7) ū,i hk = argmin { λh)ṽ hk yi + hfp y i, u) ) + hgp y i, u) }. Appying.6) again we deduce that vh yi + hfy i, ū,i hk )) v h yi + hfp y i, ū,i hk )) hl g L f λ L f y i P y i for i n S and h [, /λ). Hence, from 4.3) it foows v h yi + hfy i, ū,i hk )) ṽhk yi + hfp y i, ū,i hk )) v h yi + hfy i, ū,i hk )) v h yi + hfp y i, ū,i hk ))) + v h yi + hfp y i, ū,i hk )) ṽhk yi + hfp y i, ū,i hk ))) hl gl f y i P y i λ L + sup vh y) ṽhky) f for i n S and h [, /λ). Using the inequaity h gy i, ū,i hk ) gp y i, ū,i hk )) hl g y i P y i we derive from 4.6) v h y i ) ṽhky i ) C h y i P y i + λh) sup v h y) ṽhky) for i n S and h [, /λ) with C = L g L f /λ L f ) + ). By interchanging the roe of v h and ṽhk in 4.6) we derive 4.8) vh y i ) ṽhky i ) C h y i P y i + λh) sup v h y) ṽhky) for i n S and h [, /λ). Note that n S i= µ i n S i= µ i = hods for the coefficients in the convex combination representation. Inserting 4.5) and 4.8) into 4.4) we find vh y) ṽhky) λh) sup vh y) ṽhky) + C k + C h λh) sup vh y) ṽhky) + C k + C h for h [, /λ) with C = L g /λ L f ), which impies n S i= ns µ i y i P y i y i P y i i= sup vh y) ṽhky) k Ĉ h + ns Ĉ y i P y i for h [, /λ) with Ĉi = C i /λ, i =,. i= ) / ) /

13 Remark 4.. Let us give sufficient conditions for 4.3). First, we observe that for any i {,..., n S } and u U ad we have y i + hfp y i, u) = y i + hfy i, u) + h fp y i, u) fy i, u) ). To ensure 4.3) we repace.8) by the stronger assumption y + hfy, u) int Ω for a y Ω and for a u U ad, where int Ω stands for the open) interior of the set Ω, we have y i + hfy i, u) int Ω for any i {,..., n S }. Moreover, Assumption -) impies that fp y i, u) fy i, u) L f P y i y i hods. Consequenty, if the mesh size h or if P y i y i are sufficienty sma, the norm of the vector hfp y i, u) fy i, u)) can be made sufficienty sma so that y i + hfp y i, u) Ω. From Theorem. and Proposition 4. we derive the foowing a-priori error estimate. Theorem 4.3. Assume that Assumption,.8) and 4.3) hod. Suppose that f, g satisfy the semiconcavity conditions.7). Let v and ṽhk be the soutions of.4) and 4.), respectivey. If λ > max{l f, L g }, then there exists constants c, c, c such that 4.9) sup vy) ṽ hk y) k c h + c h + c ns y i P y i i= ) / for any h [, /λ). Remark 4.4. The a-priori error estimate presented in Theorem 4.3 is natura, because it combines the discretization error between v and v hk compare.)) with the POD approximation quaity for the finite many) vertices {y i } n S i=. In particuar, if we determine the POD basis by soving n S min y i y i, ψ j ψ j i= j= such that {ψ i } i= R n and ψ i, ψ j = δ ij, i, j, we get the a-priori error estimate sup vy) ṽ hk y) k c h + c h + c ns ) / λ i. i=+ However, the POD grid points {yi }n S i= are not we distributed in genera, which is disadvantageous for the numerica reaization. 4.. Second a-priori error estimate. From a numerica point of view 4.) is not appropriate, because in genera the grid points {yi }n S i= are not uniformy distributed in R and their distribution wi strongy depend on Ψ. Therefore, we define a second POD discretization of the HJB equations where we have an expicit bound on the distance between the neighbouring nodes. Ceary in this case we wi need an interpoation operator defined on the grid tipicay, this wi be a piecewise inear

14 interpoation operator). With.8) hoding we assume that there exists a bounded poyhedron Ω R satisfying 4.) Ψ My int Ω for a y Ω. Remark 4.5. Let y Ω, u U ad be arbitrariy chosen and set y = Ψ My R. Then, y + hf y, u) = Ψ My + hψ MfΨΨ My, u) = Ψ M y + hfy, u) ) + hψ M fψψ My, u) fy, u) ). We infer from.8) that z = y + hfy, u) Ω hods. Hence, by 4.), we have Ψ Mz int Ω. Furthermore, we derive from hψ M fψψ My, u) fy, u) ) hl f Ψ M ΨΨ My y and span {ψ,..., ψ n } = R n that y + hf y, u) Ω hods for step size h or ΨΨ My y sufficienty sma. If M = Id R n n hods, we have Ψ M =. Let {S j }m S j= be a famiy of simpices which defines a reguar trianguation of the poyhedron Ω such that Ω = m S j= S j and K = max j m S diam S j ). Let V K be the space of piecewise affine functions from Ω to R which are continuous in Ω having constant gradients in the interior of any simpex S j of the trianguation. Then, we introduce the foowing POD scheme for the HJB equations 4.) vhky i) { = min λh)v hk y i + hf yi, u) ) + hg yi, u) } for any vertex y i Ω. Throughout this paper we assume that we have n S vertices y,..., y n S. We set y i = Ψy i for i n S and define ṽ hky) = v hkψ My) for a y Ω Reca that 4.) ensures Ψ My int Ω for any y Ω. Moreover, y i = Ψy i and Ψ MΨ = Id R impies that Using 4.) we obtain ṽ hky i ) = v hkψ My i ) = v hky i) for i n S. ṽhky i ) = vhky i) { = min λh)v hk y i + hf yi, u) ) + hg yi, u) } { = min λh)v hk Ψ MΨyi + hfψyi, u)) ) + hgψyi, u) } { = min λh)v hk Ψ My i + hfy i, u)) ) + hgy i, u) } = min { λh)ṽ hk yi + hfy i, u) ) + hgy i, u) } for i n S.

15 Thus, 4.) can be written as 4.) ṽ hky i ) = min { λh)ṽ hk yi + hfy i, u) ) + hgy i, u) } for i n S. Proposition 4.6. Assume that Assumption,.8) and 4.) hod. Let v h and be the soutions of.5) and 4.). Let ṽ hk 4.3) Ψ M y + hfp y, u) ) Ω for a y Ω and for a u U ad. be satisfied. For λ > L f there exists constants c, c such that sup vh y) ṽhky) K c Ψ h + c sup y P y h for any h [, /λ). Proof. Let y Ω be chosen arbitrariy. We set y = Ψ My. By 4.) we have y in Ω. Then, there are rea coefficients µ i = µ i y ), i n S, of the convex combination representation of y satisfying y = n S i= µ iy i, µ i and n S i= µ i =. Since v hk is piecewise affine we have v hky ) = n S i= µ i v hk y i ). Using y i = Ψy i, we have 4.4) vh y) ṽhky) vh y) v h P y) n S + µ i vh P y) v h y i ) ) + n S i= i= µ i vh y i ) v hky i) ). By.6) the first term on the right-hand side of 4.4) can be bounded as foows: 4.5) vh y) v h P y) L g y P y λ L for a h [, /λ). f Furthermore, there exists an index j with y S j Ω. Let I j = {i,..., i k } {,..., n S } denote the index subset such that y S j hods for i I j. Then, µ i = hods for a i I j. Moreover, n S i= µ i = i I j µ i = and y y i K for any i I j. Reca that P y = ΨΨ My = Ψy hods. Again using.6) we find 4.6) n S i= µ i vh P y) v h y i ) = µ i vh Ψy ) v h Ψyi) L g Ψ K λ L f i I j for h [, /λ). Using 4.) and.5) we have v h y i ) vhky i) = v h y i ) ṽhky i ) v h y i ) λh)ṽhk yi + hfy i, ū,i hk )) + hgy i, ū,i hk )} λh) v h yi + hfy i, ū,i hk )) ṽhk yi + hfy i, ū,i hk ))) λh) sup v h ỹ) ṽhkỹ), ỹ Ω 3

16 where ū,i hk U ad is defined as ū,i hk = argmin { λh)ṽ hk yi + hfy i, u) ) + hgy i, u) }. By interchanging the roe of v h and vhk we find 4.7) v h y i ) vhky i) λh) sup v h ỹ) ṽhkỹ). ỹ Ω Inserting 4.5), 4.6) and 4.7) into 4.4) we find 4.8) sup v h y) ṽ hky) c h sup ) y P K y + Ψ h with c = L g /λλ L f )). Remark 4.7. ) In Remark 4.5 we have showed that 4.3) can be ensured provided 4.) hods and the step size h or y P y are sufficienty sma. ) We shoud baance K sup y P y in order to get a rate K/h for the convergence. Combining Theorem. and Proposition 4.6 we obtain the foowing resut. Theorem 4.8. Assume that Assumption,.8), 4.) and 4.3) hod. Suppose that f, g satisfy the semiconcavity conditions.7). Let v and ṽhk be the soutions of.4) and 4.), respectivey. If λ > max{l f, L g }, then there exists constants c, c, c such that 4.9) sup vy) ṽhky) K c h + c Ψ h + c sup y P y h for any h [, /λ). Remark 4.9. Let us comment on the differences between the a-priori error estimates 4.9) and 4.9). First of a, both estimates invove the terms depending on h and on k/h or K/h. Note that Ψ = if we choose M = Id in the computation of the POD basis. However, the POD approximation errors have different impacts. In 4.9) there is no factor /h. Moreover, in 4.9) the term y P y has to be sma for a y Ω, whereas in 4.9) this is need ony for the vertices y,..., y ns Ω. In order to get convergence from 4.9) one has to guarantee that K = oh) and sup y P y = oh) but, as we wi see in our numerica exampes, the method seems to be rather efficient aso for arger K. 5. Practica impementation of the agorithm. In this section we present an agorithm for the HJB equation based on the POD a-priori anaysis presented in Section4.. Estimate in Theorem 4.8 suggests the foowing steps. ) Time discretization. First the infinite horizon [, ) has to be repaced by a finite one. Thus, we choose t e sufficienty arge and define a possiby non equidistant) grid in [, t e ] by = t < t <... < t nt = t e. 4

17 ) Snapshots computation. Let us suppose that the set of admissibe contros U ad is given by choosing a discrete set U ad = {u,..., u p } U = R. To sove.) we appy the impicit Euer method on the grid {t j } n T j=. By yν j yν t j ) R n, j n T and ν p, we denote the computed impicit Euer approximation of the soution to.) at instance t j for the contros u ν t) = u ν for a t [, t e ] and ν p. 3) Rank of the POD basis. In P n T ) we choose M as the identity matrix = for j =,..., n T. The rank of the POD basis {ψ n T i } i= is chosen such and α n T j that we can show the decay of the error when we increase. In this paper we choose {, 3, 4}. 4) Reduction of the poyhedron Ω. We first define a the POD grid points y i = Ψ y i R and choose the hypercube Ω = [a, b ] [a, b ] R, where we set a j = min { y ) j,..., y m y ) j }, bj = max { y ) j,..., y m y ) j } for j and y i ) j stands for the j-th component of the vector y i R. It foows that y i Ω for i m y. Then, when the domain Ω is obtained, we buid an equidistant grid with step size computed as expained in Remark 4.7. We note that Ω shoud be arge enough in order to contain a possibe trajectories, this is aso the reason we compute severa snapshots in order to have a sufficienty accurate overview of the probem. 5) Computation of the vaue function vhk. The piecewise inear vaue function vhk is determined on the vertices y i for i n S of the domain Ω. Since the reduced-order approach yieds a sma <, we are abe to perform a standard fixed point iteration method, e.g. the vaue iteration method. We refer the reader aso to the faster agorithm introduced in [4] and the references therein. 6) Feedback aw and cosed-oop contro. We compute the vaue function ṽ hk y) = v hkψ y) satisfying 4.) at each grid point y = y i for i m y. At any grid point y i we store the associated optima contro ū,i hk U ad soving u,i { hk := argmin λh)ṽ hk y i + hfp y i, u)) + hgp y i, u) }. Then, the suboptima) feedback operator Φ : Ω U ad is defined as m y Φ y) = µ i ū,i hk i= for y Ω, where the coefficients {µ i } my i= are given by the convex combination m y y = µ i y i, µ i, i= Now the cosed-oop system for.) is m y µ i =. 5.) ẏt) = f yt), Φ yt)) ) R n for t > y) = y R n. Equation 5.) is soved by a semi impicit Euer scheme, where the second argument Φ yt)) is evauated at the previous step. We note that every step t i we 5 i=

18 pug the suboptima contro u,i h,k into 5.) and then project into the POD space in order to have the next initia condition. The agorithm is summarized beow. Agorithm 5. HJB-POD feedback contro Require: distance K, step size h, fina t e, grid {t j } n T j= set U ad = {u,..., u p } R; : Set Y = [] and = K; : for ν =,..., p do, discrete contro 3: Compute approximation {y ν j }n T j= for the soution to.) with u u ν; 4: Set Y = [Y y ν,..., y ν n T ] R n νn T ) ; 5: end for 6: Set m y = pn T and Y = [y,..., y my ] R n my ; 7: Determine a POD basis of rank by soving P n T ); 8: Compute Ω and the reduced vaue function ṽ hk ; 9: Compute the suboptima contro and the optima trajectory; 6. Numerica Tests. In this section we present our numerica tests. First et us describe the optima contro probem in detai. The governing equation is given by 6.) w t εw xx + γw x + µw w 3 ) = bu in ω, t e ), w, ) = w in ω, w, t) = in ω, t e ), where ω = a, b) R is an open interva, w : ω [, t e ] R denotes the state, and the parameters ε, γ and µ are rea positive constants. The contros are eements of the cosed, convex, bounded set U ad = {u L, t e ; R) ut) U ad for t } with U ad = {u R u a u u b } with given u a, u b R. Later, we wi consider U ad as a discrete set in the approximation of the HJB equation. The initia vaue and the shape function are denoted, respectivey, by w and b. Note that we dea with zero Dirichet boundary conditions. Equation 6.) incudes, e.g., the inear heat equation µ =, γ = ), inear advection diffusion equation µ = ) and a semi-inear paraboic probem with a reaction term µ ). As expained in the previous section we need to choose t e big enough to have an accurate approximation of the infinite horizon probem. The cost functiona we want to minimize is given by 6.) Ĵu; w ) = te w, t; u) w L ω) + α ut) ) e λt dt, where w, t; u) is the soution to 6.) at t, w is the desired state, α R + hods and λ > is the discount factor. The optima contro probem can be formuated as 6.3) min Ĵu; w ) such that u U ad. Existence and uniqueness resuts for 6.3) can be found in [3] for finite horizons. We spatiay discretize the state equation 6.) by the standard finite difference method. This approximation eads to the foowing semi-discrete system of ordinary differentia equations: 6.4) My t εay + γhy + µfy) = Bu in, t e ], y) = y, 6

19 where y : [, t e ] R n is an approximation for the soution w, t) to 6.) at n spatia grid points, A, H R n n, B, y R n and F : R n R n is given by y y 3 Fy) =. for y = y,..., y n ) R n. y n yn 3 In genera, the dimension n of the dynamica system 6.4) is rather arge i.e., n > ) so that we can not sove the HJB equations numericay. Therefore, we appy the POD method in order to reduce the dimension of optima contro probem and sove it by HJB equations. This probem fits into our probem P) and therefore we can appy Agorithm 5. to sove our optima contro probem. Next subsections wi present our numerica tests, in particuar we draw our attention on the estimate presented in Theorem 4.8. In order to check the quaity of the computed suboptima contro u we wi pug it into the fu mode yu ) and into the surrogate mode y u ). Moreover we evauate the cost functiona and compute the error with respect the true soution where is known. 6.. Test : Advection-diffusion equation. Our first test concerns the inear advection-diffusion equation, we set in 6.): t e = 3, ε =, γ =, µ =, ω =, ) and w x) =.5 sinπx). The shape function b is the characteristic function over the subset.5, ) ω. In 6.) we choose λ =, and w =. To compute the POD basis we determine soutions to the state equation for contros in the set Usnap = {.,., }. In 4.) we consider K {.,.5}, h =.K and the optima trajectory is obtained with a stepsize of.5 for the impicit Euer method. The set U ad of admissibe contros is, then, given by 3 contros equay distributed from -. to. In the eft pot of Figure 6. we show the soution of the uncontroed equation 6.), i.e. for u. Since our probem is inear-quadratic, the soution of the Fig. 6.. Test : Uncontroed soution eft), LQR optima soution midde), LQR optima contro right). HJB equation can be computed by soving the we-known Riccati s equation for LQR approach see [4]). Then, the optima LQR state is presented in the midde of Figure 6., whereas the optima LQR contro is potted on the right of Figure 6.. Then, we show the controed soution computed by means of Agorithm 5. on the eft of Figure 6.. Since it is hard to visuaize differences from the optima soutions we pot the difference between the optima soution obtained with 4 POD basis functions and POD basis in the midde of Figure 6. and with 3 POD basis on the right side. Nevertheess, one can even have a ook at the error anaysis in Figure 6.3. In order to anayze our numerica approximation we consider the evauation of the cost functiona Ĵu ; y ), the distance between yu ) and y u ) and the error between the 7

20 Fig. 6.. Test : Optima HJB states computed with Agorithm 5. with = 4 POD basis functions top-eft), difference between optima soution with 4 POD basis and POD basis topmidde), difference between optima soution with 4 POD basis and 3 POD basis top-right), optima HJB contros with =, 3, 4 bottom)..6.5 K=. K= k=. k= K=. K= Number of POD basis 3 4 Number of POD basis 3 4 Number of POD basis Fig Test : Evauation of the cost functiona eft), L error for yu ) and y u ) midde) and L -error between LQR soution and yu ). The bue ine refers to the approximation with K =. in HJB, whereas the red one to K =.5. truth soution and the suboptima yu ). The error anaysis is shown in Figure 6.3. On the eft we show the decay of the cost functiona when we increase the number of POD basis functions. In the midde we compute the L error between the optima reduced soution y u ) and the suboptima soution yu ). Even in this case the error decays when increases and K decreases. This error measures the quaity of the surrogate mode, since we want to check whether the suboptima contro fits into the non-reduced probem. Finay, on the right, we compute the error between the optima soution and the suboptima yu ). As expected, increasing the number of basis function and decreasing the step size K remember h and K are inked) for the approximation of the vaue function the optima soution is improved. The decay of the singuar vaues is presented in Figure 6.4. As we can see they do not decay reay fast with respect to the right pot which refers to the next exampe where the convection term is not dominated. Finay, we want to give an idea of the term sup y P y /h in the error estimate 4.9). It is cear we do not know Ω, but we chose severa randomy contro sequences in the set of admissibe contros. in order to have an approximation of the set. Now, 8

21 we can compute the aforementioned error term. The decay is shown on the right of Figure h=. h=.5 h= Number of POD basis functions Fig Decay of the singuar vaues for the snapshots set associated with Test eft) and Test right). 6.. Test : Semi-inear equation. The second test concerns the semi-inear equation. In 6.) we set t e = 3, ε =, γ =., µ =, ω =, ) and w x) = x x ). The shape function b is equa to the initia condition w. In 6.) we choose λ =, and w =. To compute the POD basis we determine soutions to the state equation for contros in the set Usnap = {,, } with a semi-impicit finite difference scheme with step of.5 and space step of.. In 4.) we consider K {.,.5}, h =.K. The optima trajectory is obtained with a step size of.5. The contro set is given by contros equay distributed from - to. The shape function b is equa to the initia condition w. The uncontroed soution is shown on the eft of Figure 6.5. As we can see the K=. K=.5 K=. K= Number of POD basis Number of POD basis Fig Test : Uncontroed soution eft), evauation of the cost functiona midde) and distance between yu ), y u ) right). semi-inear part does not aow to stabiize to zero the soution. Our goa is to steer the soution to the origin. The optima soutions and its correspondent optima contros are shown in Figure 6.6. Moreover we pot the differences between the computed soutions pease note the different scaing of the pictures). As we can see the difference decreases when the number of POD basis functions increase. The quaity of our approximation is confirmed by Figure 6.5 where we can see from the evauation of the cost functiona and consistency of the suboptima contro. In this case the error decays much faster than in the previous exampe. This depends on the decay of the singuar vaue of the snapshots set as shown in Figure Test 3: Semi-inear equation with uniform noise. In this test we dea with the semi-inear equation discussed in the previous exampe but we negect the convection term γ = ) and we add noise to the optima trajectory. The uncontroed 9

22 x Fig Test : Optima HJB states computed with Agorithm 5. with = 4 POD basis functions top-eft), difference between optima soution with 4 POD basis and POD basis topmidde), difference between optima soution with 4 POD basis and 3 POD basis top-right), optima HJB contros with =, 3, 4 bottom). soution is shown on the eft of Figure 6.7, the optima trajectory and contro computed by means of Agorithm 5. are in the midde and the right side of Figure Fig Test 3: Uncontroed soution eft), Optima state midde) and corresponding optima HJB contro right). The goa is to show the stabiization of the feedback contro under strong perturbations of the system. We note that in this case the vaue function is stored from the system without perturbation, but the reconstruction of the feedback contro is affected by uniform noise ηx) between [, ] in every step: yx, ) = + ηx))yx, ). Figure 6.8 shows optima soution and contro corresponding to different noise eves ηx) 5% top) and ηx) 9 % bottom)). In this exampe we can see the power of the feedback contro, and in particuar, the importance of the knowedge of the vaue function. In both exampes, the trajectory is stabiized cose to the origin. If we have a ook at the optima contro input we can observe a strong chattering. In both cases the optima contro jumps often from - to. In particuar, in the second case, it is possibe to observe a stronger chattering due to the high disturbances. Nevertheess, the feedback contro is abe to stabiize the perturbed system.

23 Fig Test 3: Optima HJB-POD state eft) and corresponding optima contro right) with ηx) 5% top) and ηx) 9% noise bottom). 7. Concusion and Remarks. In this paper we present a new a-priori error anaysis for the couping between the HJB equation and the POD method. The proposed estimate is presented for the infinite horizon contro probems with inear and noninear dynamica systems but this approach coud be aso appied to other optima contro probems provided one has a priori estimates on the approximation based on the HJB equation. The convergence of the method is guaranteed under rather genera assumptions on the optima contro probem and some technica assumptions on the dynamics and on the POD approximation. For the atter, it is cear that a cever choice of the snapshots set can pay a crucia roe in the estimate in order to reduce the contribution of the POD approximation in the a-priori estimate. Severa choices are possibe based on greedy techniques or on a previous open-oop approximation, these choices wi be investigated in a future paper. At present, the numerica tests iustrated in the ast section confirms our theoretica findings and show the robustness of the Beman s approach aso under strong disturbances of the dynamica system. REFERENCES [] A. Aa and M. Facone. An adaptive POD approximation method for the contro of advectiondiffusion equations, in K. Kunisch, K. Bredies, C. Cason, G. von Wincke, eds) Contro and Optimization with PDE Constraints, Internationa Series of Numerica Mathematics, 64, Birkhäuser, Base, 3, -7. [] A. Aa and M. Facone. A -adaptive POD method for optima contro probems, in the Proceedings of the st IFAC Workshop on Contro of Systems Modeed by Partia Differentia

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