Optimal Control of Dynamical Systems

Governed by Partial Differential Equations:

A Perspective from Real-life Applications ⋆

Hans Josef Pesch∗

∗ Chair of Mathematics in Engineering Sciences, D-95440 Bayreuth,Germany (e-mail: hans-josef.pesch@uni-bayreuth.de)

Abstract: This survey article summarizes some ideas of the two principle procedures for solvingoptimal control problems governed by partial differential (algebraic) equations: the indirectmethod or first-optimize-then-discretize approach based on first-order necessary conditions, andthe direct method or first-discretize-then-optimize approach based on non-linear programmingsoftware. The focus of this paper is to discuss some pros and cons when applications come fromreal-life problems. It also glances at the main method which currently may have the best chancefor online applications in PDE constrained optimization. The demonstrator example deals withthe optimal control of a certain type of high temperature fuel cell system.

Keywords: Optimal control, partial differential equations, numerical methods, real-time, fuelcell systems

1. INTRODUCTION

When analyzing mathematical models for complex dynam-ical systems, their analysis and numerical simulation aregenerally only first steps. Thereafter, one often wishes tocomplete these investigations by an optimization step toexploit inherent degrees of freedom for optimizing a desiredperformance index with the dynamical system as sidecondition. This generally leads to optimization problems ofextremely high complexity, particularly if the underlyingsystem is described by (time dependent) partial differentialequations (PDEs) or, more generally, by a system of partialdifferential algebraic equations (PDAEs).

In this survey paper we will summarize the main ideas ofoptimization subject to PDE constraints and in particularexhibit some of the challenges we are facing and have tocope when solving real-life problems.

For the purpose of motivation three problems from engi-neering applications shall be examplarily described first:

(1) Hot cracking is a common risk in welding of alu-minium alloys. According to a Russian patent [Shu-milin et al. (1980)] this risk can be avoided by ap-plying so-called multi-beam laser welding techniques.By two additional laser beams the thermal stressintroduced by the main welding laser can be com-pensated, if the additional laser beams are optimallyplaced and sized, while they must not melt on thematerial. Mathematically we obtain a semi-infinite

⋆ Part of this work was funded by the Federal Ministry of Education

and Research of the Federal Republic of Germany from May 2002

until December 2005 in cooperation with the Max Planck Institute

for Dynamics of Complex Technical Systems, Magdeburg, the Chair

for Process Systems Engineering, University of Magdeburg, the IPF-

Heizkraftwerksbetriebsgesellschaft mbH, Magdeburg, and the MTU

Onsite Energy GmbH, Ottobrunn.

Fig. 1. A three-beam laser welding technique for avoidinghot cracks

optimization problem with PDE and non-standardstate variable inequality constraints, a special caseof an optimal control problem for PDEs; see Pet-zet (2008) and Figs. 1 and 2. The model is brieflysketched in Appendix A.

(2) Future concepts for intercontinental flights of passen-ger aircraft envisage aircraft which are able to flighat hypersonic speeds. Due to such high velocities thethermal heating of the aircraft is an issue which hasto be taken into account. This multi-physics problemleads to an optimal control problem for a system of or-dinary differential equations (ODEs) where the aero-thermic heating of the aircraft’s body is modelled bya quasilinear heat equation with nonlinear boundaryconditions. The temperature of the thermal protec-tion system constitutes a complicated state variableinequality constraint in an ODE-PDE optimal controlproblem; see Chudej et al. (2008) and Witzgall andChudej (2012). Figure 3 shows the coupling structurebetween ODE- and PDE-subsystem. In Fig. 4 the

Fig. 2. Three-beam laser welding technique: results ofoptimization; cf. Petzet (2008). The more evenlydistributed temperature avoids hot crack initiation asits indicator shows

Fig. 3. Challenge in ODE-PDE optimal control: minimumfuel intercontinental hypersonic flights under a limi-tation of aero-thermal heating

Fig. 4. Numerical results of minimum fuel trajectory opti-mization without (dashed) and with temperature con-straint (solid); cf. Chudej et al. (2008). The boundaryarc of the state constraint can be seen (lower line).Note that the state constraint is not observed in theuncritical region in order to save computer time. Theyare indicated by the horizontal lines of the graphs

effect of the state constraint on the trajectory isdepicted and the appearance of a boundary arc of theheating constraint can be seen. The model is brieflysketched in Appendix B.

Fig. 5. Schematical exposition of the gas streams througha single fuel cell

Fig. 6. The Hotmodule under construction at MTU OnsiteEnergy GmbH, Ottobrunn: the stack of about 300 fuelcell is just pushed into the module

(3) The main example, which will be discussed in moredetail in this paper, is concerned with the optimalcontrol of certain fuell cell systems for an environ-mentally friendly production of electricity. Reaction-advection equations describing the electro-chemicalreactions in the gas streams of anode and cathodefor each compound, a heat equation describing theheating-up of the solid caused by the reactions, as wellas additional ordinary, resp. algebraic, resp. integrodifferential equations sum up to a huge coupled sys-tem of up to 28 instationary spatially two-dimensionalPDAEs of extremely high complexity. The inflow datainto the anode inlet and into the catalytic burner,and the amount of exhaust gas that is fed backfrom the cathode outlet to the burner are the controlvariables, cp. Fig. 5. A complete compendium of theresearch within this project is given in Sundmacher etal (2007), whereas Chudej et al. (2008) focus on themathematics behind the project. Figure 6 shows aphoto of the type of stationary high temperature fuelcell module under construction. The stack of cells isjust positioned into the Hotmodule, which has beeninvestigated in this project and which is particularlysuitable for the highly efficient power-heat coupling.

After this motivation a glimpse on the mathematical the-ory of optimal control problems for one elliptic equation isgiven to describe the amount of preparatory work that hasto be invested when solving optimal control problems byso-called adjoint based methods, i. e. methods which arebased on first order necessary conditions. Thereafter twonumerical concepts, namely first optimize then discretizeand first discretize then optimize are sketched and dis-cussed with respect to their pros and cons when applyingto real-life problems. In addition an overlook over thepresent mathematical toolbox from literature is given toshow some current trends of research in PDE optimalcontrol. A glimpse on a certain model reduction technique,which may help in case of real-time demands, finalizes thispaper.

2. A GLIMPSE ON THE THEORY

2.1 A linear-quadratic elliptic optimal control problem

Let us consider the following elliptic model optimal controlproblem of so-called tracking type for a distributed control:

minu∈UadJ(y, u) :=

1

2

∫

Ω

(y − yd)2 dx +

λ

2

∫

Ω

u2 dx

subject to the side conditions

−∆y + y = u in Ω ⊂ RN ,

∂ny = 0 on Γ = ∂Ω .

Here, let Ω be a bounded Lipschitz domain, on whichthe state variable y and the control variable u ∈ Uad :=u ∈ L2(Ω): umin ≤ u ≤ umax a. e. live where the set ofadmissble controls is defined by functions umin , umax ∈L∞(Ω). Finally the function yd to be tracked is assumedto be of class L2(Ω) and the regularisation parameter λshall be positive, λ > 0.

According the the Theorem of Lax-Milgram the ellpiticboundary value problem has a unique solution y ∈ H1(Ω)so that there exist a linear and continuous solution op-erator S: Uad → H1(Ω) mapping each control u to itsassociated state y. Herewith, we formally arrive the so-called reduced functional

minUadf(u)

with f : Uad → R via f(u) := J(S u, u). Hence we obtainan optimization problem in the Hilbert space L2(Ω). Dueto the convexity of this problem it has a unique optimalsolution which must fulfill the variational inequality

(

S (S u∗ − yd) + λu∗ , u − u∗)

L2(Ω)≥ 0 ∀ u ∈ Uad ,

where (·, ·)L2(Ω) denotes the scalar product in L2(Ω) and S

the adjoint solution operator. The asterix ( )∗ indicatesoptimal variables. Because of the convexity of the problem,this necessary condition is here sufficient, too.

By obeying S u∗ − yd = y∗ − yd and replacing p :=S (y∗ − yd), the above variational inequality can be rewrit-ten as

(

p + λu∗ , u − u∗)

U≥ 0 ∀ u ∈ Uad

with the adjoint state variable p satisfying the adjointequation

−∆p + p = y∗ − yd in Ω ⊂ RN ,

∂np = 0 on Γ = ∂Ω .

Moreover, the variational inequality can be replaced bythe pointwise control law

u∗(x) = PUad

−1

λp(x)

with the projection operator PUadonto the set of admissi-

ble controls.

The original boundary value problem for the state vari-able y ∈ H1(Ω) and the boundary value problem for theadjoint state p being also of class H1(Ω) constitute to-gether with the optimal control law the so-called first-orderoptimality or Karush-Kuhn-Tucker system. This is a well-defined coupled system for two elliptic equations admittinga unique solution under reasonable assumptions. For moredetails it is referred to the most recent textbook of thisfield concerning theory, the book of Troltzsch (2010).

Such optimality systems can be easily set up by theformal Lagrange technique [cf. Troltzsch (2010)]. Definethe Lagrangian L and integrate twice by parts

L(y, u, p)

:= J(y, u) −

∫

Ω

(−∆y + y − u) p dx −

∫

∂Ω

∂ny p ds

= J(y, u) −

∫

Ω

(−∆p + p) y − u p dx −

∫

∂Ω

∂np y ds .

Then a differentiation in the direction of h ∈ H1(Ω) yields

DyL(y∗, u∗, p)h = 0 ∀h ∈ H1(Ω)

=⇒ −∆p + p = y∗ − yd , ∂np = 0 ,

DuL(y∗, u∗, p) (u − u∗) ≥ 0 ∀u ∈ Uad

=⇒(

p + λu∗ , u − u∗)

U≥ 0 ∀ u ∈ Uad .

In addition there holds f ′(u) = DuL(y∗, u∗, p).

2.2 A semilinear-quadratic elliptic optimal control problem

The generalisation to semilinear equations for distributedas well as boundary controls v ∈ Vad, resp. u ∈ Uad for anobjective function of type

minu∈Uad,v∈VadJ(y, v, u) :=

1

2

∫

Ω

(y − yΩ)2 dx

+1

2

∫

Γ

(y − yΓ)2 ds +λ

2

∫

Ω

u2 dx +µ

2

∫

Γ

v2 ds

leads, under suitable assumptions on the non-linearities band d, to the following optimality system

−∆y + y + b(y) = u in Ω ⊂ RN ,

∂ny + y + d(y) = v on Γ = ∂Ω ,

−∆p + p + b′(y) p = y − yΩ in Ω ⊂ RN ,

∂np + p + d′(y) p = y − yΓ on Γ = ∂Ω ,

u(x) = PUad

−1

λp(x)

, v(x) = PVad

−1

µp(x)

.

For details, see again Troltzsch (2010).

In this book also general nonlinear functionals as well asinstationary partial equations such as parabolic equationsare investigated. Note that for this model problem theadjoint equation has the same main part of the differentialoperator as the original state equation while their non-linearities are linearized. In case of linear or semilinearparabolic equations with time t ∈ (0, tf ) the adjointequation is retrogressive in time, i. e. the term yt passesover to a term −pt with initial conditions given in t = tfinstead of t = 0 as for the state variable y.

Necessary conditions even for quite general nonlinear prob-lems can be easily set up by the formal Lagrange tech-nique, even in cases, where existence proofs fail, however,without a theoretical justification. Nevertheless, this tech-nique often provides the only possiblity to proceed towardsadjoint-based numerical methods.

Since PDE models are mostly not nonlinear in the mainpart of their differential operator, except for quasilinearequations, the possibly severest nonlinearities are oftengiven by the source terms b and d. They may then betreated separately by symbolic or automatic differenti-ation. We will demonstrate this for the MCFC modelmentioned in the introduction.

3. A GLIMPSE ON THE NUMERICS

Let us write a general problem of optimal control for PDEsin abstract form as follows:

(P ) minu∈Uad , y∈Yad

J(y, u) s.t. PDE(y) = B(u) .

Here PDE stands for an arbitrary differential equation inthe state y and B is an operator acting on the control u.The aim in solving those problems efficiently is to keep theratio of the amount of computation for the optimizationversus the amount of computation for the simulation of theproblem below a small constant, which obviously cannotundershoot 2 in view of the above theoretical results.

The two essential numerical procedures are known asdirect, resp. first discretize then optimize-method, andindirect, resp. first optimize then discretize-method. Adirect method can be described as follows:

(Ph) minuh∈Uh

ad, yh∈Y h

ad

Jh(yh, uh) s.t. PDEh(yh) = Bh(uh)

where the index h indicates discrete variables, resp. dis-cretized operators. Crucial are the choices of the finitedimensional spaces Uh

ad and Y had for the state, resp. control

variable as well as the discretization of the PDE. Thisapproach leads to a generally huge nonlinear program-ming problem for which even standard software for largescale constrained nonlinear optimization may today beapplicable only for problems of moderate size unless corsediscretizations are accepted, e. g., using the combinationof Ampl [The AMPL Company (2012)] and Ipopt [Lairdand Wachter (2012)].

In contrast, indirect methods exploit the first order nec-essary conditions: The minimization of the reduced func-tional f(u) := J(S u, u) requires the solution of an op-timality system of the form f ′(u) = 0 which then hasto be discretized, say as Fh(uh) = 0. Here appropriatefinite dimensional ansatz spaces for the adjoint state and

generally also for certain multipliers are to be chosenadditionally.

In summary, first discretize then optimize means: replaceall quantities of the infinite dimensional optimization prob-lem by finite dimensional substitutes and solve an NLP:

min f(u) = J(S(u), u) =⇒ min fh(uh) = Jh(Sh(uh), uh) ,

while first optimize then discretize means: derive optimal-ity conditions of the infinite dimensional system, discretizethe optimality system and find a solution of the discretizedoptimality system:

f ′(u) = 0 =⇒ Fh(uh) = 0 .

In ideal manner, one should strive for a discrete concept forwhich both approaches commute; see Vexler et al (2011).

A rich mathematical toolbox — not neccessarily com-plete — has been developed so far consisting, e. g., of one-shot-iterations based on the structure of the optimalitysystem (cf. Griewank, Schulz), of non-smooth solutiontechniques for state constrained problems (cf. Ito, Hin-termuller, Kunisch, M. Ulbrich), of multigrid methods (cf.Borzi, Schulz), of taylored discrete concepts (cf. Hinze,Meyer, Rosch), of penalty or barrier methods for the relax-ation of constraints (cf. Hintermuller, Kunisch, Schiela),of methods for state-constrained optimal control problemsvia set optimal control problems using shape calculus (cf.Frey, Bechmann, Pesch, Rund), of various concepts ofadaptive algorithms (cf. Becker, Rannacher; Vexler; Hin-termuller, Hoppe; Hinze, Gunther, and many others), ofsurrogate models for the PDE system in the optimalitysystem (cf. Hinze et. al., Sachs et. al., Kunisch, Troltzsch,S. Ulbrich, Volkwein), of shape optimization methods (cf.Sokolowski, Zolesio; Gauger, Schulz; Hintermuller, Ring;M. Ulbrich, S. Ulbrich), of automatic differentiation toolsproviding adjoints automatically (cf. Griewank, Walther).

Most of the associated references can be found in thetextbooks focussing on numerical methods for PDE con-strained optimization of Hinze, Pinneau, M. Ulbrich, andS. Ulbrich (2008) and Borzi and Schulz (2011), resp. inthe survey article of Herzog and Kunisch (2010), resp. inarticles of the aforementioned authors the references ofwhich can be found on their homepages, resp. which canbe downloaded from there.

4. A REAL-LIFE APPLICATION: OPTIMIZATIONOF MOLTEN CARBONATE FUEL CELLS

Fuel cells are apparatuses for an enviromentally friendlyenergy production in which a controlled oxyhydrogen re-action goes on. Fuel cell systems can be classified accordingto their range of application: mobil or stationary, and totheir operating temperature: low, moderate, or high.

Here, we exploit two models of a hierarchy of mathe-matical models for so-called Molten Carbonate Fuel Cells(MCFCs), a prototype of which is the Hotmodule of MTUOnsite Energy GmbH, Ottobrunn; see Fig. 6. These mod-els have been developed by Heidebrecht (2004) and latersimplified by Rund and Chudej (2011). The results, whichare summarized for this review, were, in great part, fundedby the Federal Ministry of Education and Research ofthe Federal Republic of Germany from May 2002 untilDecember 2005.

Fig. 7. Functionality of a molten carbonate fuel cell, herefor the 1D counter flow design

4.1 Some aspects of modelling MCFC

First, the functionality of MCFCs shall be describedbriefly. MCFCs are high temperature fuel cells that canproduce the necessary hydrogen from its fuel gas, e. g.,from methane CH4 because of its high operating tempera-ture. This technique is known as internal reforming and isone of the advantages of MCFCs, since naturally occurringelemental hydrogen is relatively rare on Earth, despite itis the most abundant chemical element.

The two reduction reactions in the anode gas channel (seeFig. 7) are endothermic, hence they consume energy. Theanode exhaust gas is completely oxidized in the burningchamber by inserting air. After passing a mixer two ox-idation reactions take place in the cathode gas channelwhich sum up to the second reduction reaction shown inFig. 7. This reaction is exothermic. Part of the exhaust gasof the cathode can be refed into the catalytic burner; seeagain Fig. 7. This figure emblematizes the simplest modelfor a MCFC by a spatially 1D counter flow configuration.In contrast, Fig. 8 depicts a spatially 2D cross flow con-figuration, which is realized in the Hotmodule mentionedabove.

In summary, seven chemical compounds indexed by k (cp.Fig. 8) appear: CH4, H2O, H2, CO, CO2, O2, and N2.Electricity is produced if the anode is provided by hy-drogen and the cathode by oxygen. The two electrodesare separated by a special electrolyte, which only let pass

carbonate ions CO2−

3 , cp. Fig. 7.

By the way, the exhaust gas at the cathode outlet can bedeclared as air according to the German Federal PollutionControl Act.

The variables appearing in the PDAE model due to Hei-debrecht (2004) can be obtained from Fig. 8. They can begrouped according to their time scales into slow, fast, andvery fast variables. The complete system consists of a semi-linear heat equation for the solide temperature θs, a setof quasilinear hyperbolic reaction advection equations forthe molar fractions χ of all chemical compounds indexedby k, the anode and cathode gas temperatures θa and θc

as well as for the molar flow densities γ the equation ofwhich degenerate, i. e. do not depend of partial derivativeswith respect to time τ . Here the time is scaled due to a

Fig. 8. Notation of variables, here for the 2D cross flowdesign

transformation of all variables to dimensionless quantities.The spatial variables are denoted by z ∈ [0, 1] for the1D counter flow design (Fig. 7) and by ζ1 and ζ2 for the2D cross flow design (Fig. 8). The potentials Φ and the cellcurrent Ucell are given by ordinary differential algebraicequations (DAEs) in each spatial point the nonlinear termsof which depend on integral terms which in turn dependon all other variables. Depending on the complexity of themodel the chemical reactions ϕ in the pores of the elec-trodes are modelled by additional algebraic equations. Thetransition from the anode outlet through burner and mixerto the cathode inlet is described by additional algebraicand ordinary differential algebraic equations. Completemodels can be found for the 2D cross flow model beingof dimension up to 28 in Heidebrecht (2004) and Chudejet al. (2008) and for the 1D counter flow model beingof dimension 21 in Rund and Chudej (2011) and Rund(2012).

This dynamical system can be controlled by up to sevencontrols for the configuration of the inflowing gas streamsat the anode inlet and into the burning chamber eitherfrom outside or from the cathode outlet via the recir-culation device. Finally, Icell is an input variable to bechosen for load changes. Optimal load changes will bethe major task for the optimization of those complicatedPDAE models. In doing so large variations of the solidtemperature θs must be avoided to get around materialcorrosion induced by hot spots.

4.2 Numerical results for the 2D cross flow configurationvia the first-discretize-then-optimize approach

In her thesis Sternberg (2007) has investigated modelsfor the 2D cross flow configuration. In particular shehas computed optimal trajectories for fast and safe loadchanges including sensitivity studies. Her first-discretize-then-optimize approach is based on standard discretizationschemes, including conservative upwind schemes for thehyperbolic transport equations. Thereby the PDAE modelis transferred via semi-discretization in space into a largescale system of DAEs, known as method of lines. Thenthe code Nudocccs of Buskens (2002) was applied tosolve the DAE optimal control problem. Inside Nudocccs

the sparse NLP solver SNOPT has been employed, the

Fig. 9. Fast load change by tracking the cell current usingeither the full set of controls (top) or the molar flowdensity at anode inlet only [Sternberg (2007)]

Fig. 10. Solid temperature θs at final time [Sternberg(2007)]. Note the gas flow directions: in the anodefrom ζ1 = 0 to ζ1 = 1, in the catode from ζ2 = 1to ζ2 = 0. A hotspot occurs at the anode outlet, resp.at the cathode inlet

latest version of which can be found in Gill, Murray, andSaunders (2012).

Due to the complexity of the system and the long processtime (about 3 hrs.), but mainly due to the non-efficient ap-plication of numerical differentiation for the computationof the gradient of the objective function with respect tothe discretized control variables only corse discretizationscould be handled. Nevertheless, the numerical results weresatisfactory at that time thanks to the smoothing influenceof the heat equation. However, the amount of computationwent up to days.

Figure 9 shows the optimization of a fast load change fromIcell = 0.7 to Icell = 0.6. Hereby, the stationary value of thecell current Ucell associated with the new load is tracked inthe L2-norm. Because of this type of functional the entiretime interval could be partioned in a pseudo-logarithmicscale to compensate for the dynamical behaviour of fastvariables at the beginning of the process. The optimizationwas then performed on the resulting subintervals in asuboptimal manner. Since the cell current reacts relativelyfast, the new stationary point of the MCFC could beachieved in less than 1 sec. Moreover it turned out thatthe molar flow densitiy alone is sufficient for controlingfast load changes. Figures 10, 11, and 12 depict the solidtemperature θs and the gas temperatures in the anode andcathode channel, θa, resp. θc.

Fig. 11. Anode gas temperature θa at final time [Sternberg(2007)]. The anode reaction is firstly endothermic,then exothermic

Fig. 12. Cathode gas temperature θc at final time [Stern-berg (2007)]. The cathode reaction is uniformly en-dothermic

More results can be found in Sternberg (2007) and Chudejet al. (2008). In the latter fast and safe load changes havebeen obtained by modifying the objective functional

min

τf∫

τ0

(

1 − ε(τ)) (

Ucell(τ) − U to−becell

)2

+ε(τ)

∫

Ω

(

θs(ζ, τ) − θref(ζ, τ))2

dx

dτ

with

ε = 0 on

[0.0,0.1][0.1,1.1][1.1,11.1][11.1,111.1]

and ε = 1 on [111.1, 1111.1] .

Herewith the fast reacting cell current Ucell is initially tobe tracked towards U to−be

cell and, at the end, the functionalswitches to the slowly varying solid temperature θs. Thischoice avoids too large and therefore risky variations inthe solid temperature without employing a pointwise stateconstraint on it or even a constraint on the gradient of thesolid temperature which would be quite a challenge forsuch a complicated model; see Fig. 13 for the numericalresults.

Fig. 13. Solid temperature θs at final time [Chudej et al.(2008)]. Due to the special choice of the objectivefunctional pointwise state constraints can be avoided(right) in contrast to the choice ε = 0 in the entiretime interval

4.3 Numerical results for the 1D counter flow configurationvia the first-optimize-then-discretize approach

Rund (2012) recently succeeded in applying the first-optimize-then-discretize approach to the 1D counter flowdesign for a modified model where all superfluous variableswere ommitted. To give an impression of the complexityof this simplified model its differential operator part ispresented here in detail while the nonlinear source termsare abbreviated.

Equations prevailing in the gas channels and the solid:

∂χk,a

∂t+ γa ϑa

∂χk,a

∂z= fχk,a , k ∈ I

∂χk,c

∂t− γc ϑc

∂χk,c

∂z= fχk,c , k ∈ I

∂ϑa

∂t+ γa ϑa

∂ϑa

∂z= fϑa

∂ϑc

∂t− γc ϑc

∂ϑc

∂z= fϑc

∂(γa ϑa)

∂z(z, t) = fγaϑa

−∂(γc ϑc)

∂z(z, t) = fγcϑc

cp,s∂ϑs

∂t−

1

Pe

∂2ϑs

∂z2= fϑs

Equations prevailing in burner and mixer:

χk,b(t) = fχk,b , k ∈ I

ϑb(t) = fϑb

γb(t) = fγb

γair(t) = fγair

dχk,m

dt(t) = fχk,m , k ∈ I

dϑm

dt(t) = fϑm

γm(t) = fγm

Equations describing the cell current and the potentialfields:

Ia(t) =

1∫

0

ia(z, t) dz , ia(z, t) = f ia

Ic(t) =

1∫

0

ic(z, t) dz , ic(z, t) = f ic

Ie(t) =

1∫

0

ie(z, t) dz , ie(z, t) = f ie

i(z, t) = f i

dUcell

dt(t) = fUcell

∂ΦLa

∂t(z, t) = fΦL

a

∂ΦLc

∂t(z, t) = fΦL

c

Except the nonlinearity in the two degenerated transportequations for the molar flow densities γ, which, by the way,can be transformed by introducing auxiliary variables v :=γ θ, the entire nonlinearity of the system is contained inthe f⋄-terms. The full model can be found in Rund (2012).This property seems to be typical for complex multi-physics models and alleviates the derivation of the adjointequations by applying the formal Lagrange technique withthe Lagrangian

L(...) = J

−∑

k∈I

∫∫

Q

(

∂χk,a

∂t+ va

∂χk,a

∂z− fχk,a

)

χk,adzdt

−

tf∫

0

∑

k∈I

(

χk,a(0, t) − χk,a,in

)

χk,adt

−∑

k∈I

∫∫

Q

(

∂χk,c

∂t− vc

∂χk,c

∂z− fχk,c

)

χk,cdzdt

−

tf∫

0

∑

k∈I

(

χk,c(1, t) − χk,m

)

χk,cdt

−

∫∫

Q

(

∂ϑa

∂t+ va

∂ϑa

∂z− fϑa

)

ϑadzdt

−

tf∫

0

(

ϑa(0, t) − ϑa,in

)

ϑadt

−

∫∫

Q

(

∂ϑc

∂t− vc

∂ϑc

∂z− fϑc

)

ϑcdzdt

−

tf∫

0

(

ϑc(1, t) − ϑm

)

ϑcdt

−

∫∫

Q

(

∂va

∂z− fva

)

vadzdt −

tf∫

0

(

va(0, t) − va,in

)

vadt

−

∫∫

Q

(

−∂vc

∂z− fvc

)

vcdzdt −

tf∫

0

(

vc(1, t) − γmϑm

)

vcdt

−

∫∫

Q

(

cp,s∂ϑs

∂t−

1

Pe

∂2ϑs

∂z2− fϑs

)

ϑsdzdt

−1

Pe

tf∫

0

[

∂ϑs

∂zϑs

]1

0

dt

−∑

k∈I

tf∫

0

(

χk,b − fχk,b

)

χk,bdt −

tf∫

0

(

ϑb − fϑb

)

ϑbdt

−

tf∫

0

(

γb − fγb

)

γbdt −

tf∫

0

(

γair − fγair

)

γairdt

−∑

k∈I

tf∫

0

(

dχk,m

dt− fχk,m

)

χk,mdt −

tf∫

0

(

γm − fγm

)

γmdt

−

tf∫

0

(

dϑm

dt− fϑm

)

ϑmdt

−

∫∫

Q

(

i − f i)

idtdz −∑

k∈a,c,e

∫∫

Q

(

ik − f ik

)

ikdtdz

−∑

k∈a,c,e

tf∫

0

Ik −

1∫

0

ikdz

Ikdt

−

tf∫

0

(

dUcell

dt− fucell

)

Ucelldt

−

∫∫

Q

(

∂ΦLa

∂t− fΦL

a

)

ΦLa dtdz

−

∫∫

Q

(

∂ΦLc

∂t− fΦL

c

)

ΦLc dtdz .

Here the tilde and the hat indicate Lagrange multipliersassociated with state variables governed by PDEs or othertypes of equations, resp. multipliers for initial conditions.Then partial integrations with respect to time and spaceyield the differential operators for the adjoint equations.This can be relatively easily done analytically. For thelinearization of the nonlinear terms f⋄ however — cp.the adjoint equation of the semilinear model problem inSubsection 2.2 — one should employ symbolic or au-tomatic differentiation. Further partial integrations mayalleviate to read the adjoint system out of the so modifiedLagrangian by means of standard variational arguments;cp. the example in Subsection 2.1.

Rund (2012) has developed an efficient discretizationscheme for the integration of the optimality system basedon an implicit time adaptive integrator, which can handlethe different time scales, and, e. g. on upwind schemesfor the advection-reaction equations. The two subsystems,state and adjoint system, are integrated successively dueto the coupling structure shown in Fig. 14 similar to thewell-known backward sweep method of the early days ofODE optimal control. This method is efficient even for

Fig. 14. Coupling structure of state and adjoint system;cf. Rund (2012). Note that the controls enter boththe anode inlet and the burner (top). Due to the timereversal of the adjoint system, the anode “outlet” aswell as the burner “oulet”, as seen from the viewpointof the adjoint state, enter the state system (bottom) inturn twofold via complicated variational inequalities.This closes the loop in the backward sweep method

Fig. 15. Optimal solutions for the anode, cathode, andsolid temperature for a load change started at 0.1 sec;cf. Rund (2012)

many time steps which are to be performed because of thedifferent time scales.

The two subsystems are connected by a set of nonlinearvariational inequalities caused by the twofold entry of thecontrols into the fuel cell; cp. Fig. 14. The iteration isperformed by a Quasi-Newton method with BFGS updatesnot relying on second order information. SQP methodsseem to be hardly applicable due to their neccessity ofproviding second order information. For more details seeRund (2012). The amount of computation for the 1Dcounter flow configuration can thus be reduced to minutes,if not less, e. g., using N = 81 lines, 160 time steps chosenby the code ode15s (Matlab R2010b) on Intel64 3.3 GHzDC, 8GB: 5 sec CPU. — In principle this approach is alsoadaptible for 2D models.

Figure 15 shows the optimal solutions for the anode,cathode, and solid temperature for a prescribed loadchange.

4.4 Some aspects of online control using model reductiontechniques

In order to apply those computed optimal solutions foronline control purposes, the complexity of the model mustbe reduced. Common approaches are known as modelreduction techniques such as the Karhunen-Loeve decom-position (1946, 1955), known under various names, todaypreferably called POD (proper orthogonal decomposition);see, e. g., the survey paper of Hinze and Volkwein (2005).

The basic idea can be discribed as follows. Let T stand fora state variable of a dynamical system governed by a PDEof type

∂T

∂t(t, y, z) = D T (t, y, z) + f(T (t, y, z))

with the differential operator

D = ay

∂

∂y+ az

∂

∂z+ by

∂2

∂y2+ bz

∂2

∂z2.

Using the ansatz

T (t, y, z) ≈

N∑

i=1

Ti(t)ϕi(y, z)

with precomputed basis functions ϕi(y, z) reflecting themain properties of the solution of T . Such basis functionscan, for example, be obtained from a simulation with a pre-scribed control programme to be expected “close enough”to the optimal one. Then take “snapshots” at differenttime instants ti, i. e. define ϕi(y, z) := T (ti, y, z). Or-thogonalize these basis functions, e. g. by a singular valuedecomposition. The PDE model can then be reduced, viathe method of weighted residuals∫∫

Ω

(

∂T

∂t− D T − f(T )

)

ϕj dy dz = 0 , j = 1, . . . , N ,

to an ordinary differential (algebraic) system

dTi

dt= gi(T1, T2, . . . , TN ) , i = 1, . . . .N .

Generally, N can be chosen much smaller than the numberof discretization points for the semi-discretization in spaceaccording to the method of lines.

This approach has been applied in the aforementionedproject by Mangold (2006) for the construction of a stateestimator from the measurements that are available inthe MCFC system (see Fig. 16). Figure 17 shows thehigh accuracy that can be obtained despite a drasticallyreduced number of ODEs from the POD ansatz com-pared to the number of ODEs from the method-of-linesapproximation. Figure 17 finally depicts the results of thestate estimater for the solid temperature and its gradientdeveloped by Mangold (2006). This state estimator wasactually implemented into the Hotmodule fuel cell systemat the powerplant of the University Hospital, Magdeburg.

For more details for this approach applied to fuel cellonline-control, see Mangold (2006).

Recent results on error estimates for POD approximationscan be found, e. g., in Sachs and Volkwein (2010). Errorestimates, if available at all at that time, were not takeninto account in Mangold (2006).

Fig. 16. Control requirement for a state estimator frommeasurements; cf. Mangold (2006)

Fig. 17. Comparision between full discretization and PODapproximation according to Mangold (2006)

Fig. 18. Performance of the state estimator of Mangold(2006).

5. CONCLUSION

The first-discretize-then-optimize approach lacks under akind of curse of dimension and is applicable only for prob-lems of modest size since the number of variables in NLPproblems which can be treated by today’s best solvers aretoo limited for PDE constrained optimization problems.Important is to use NLP solvers which can extract exactfirst and second-order gradient information out off thediscretization schemes by automatic differentiation, e. g.,if they can be linked with the mathematical-modelling-language tool Ampl of The AMPL Company (2012), e. g.

the interior point solver IPOPT of Laird and Wachter(2012), or one uses directly discretization schemes wherethe two approaches commute.

The first-optimize-then-discretize approach requires themost skilled user and lacks under its difficult handling ifnon-standard problems like the ones presented here are tobe optimized.

In cases where dynamical systems are controlled aroundtheir stationary state POD methods seem to be a promis-ing method for online applications. Highly non-linear dy-namical behaviour however cannot be treated so far.

ACKNOWLEDGEMENTS

The author is greatfully indebted to Professor Kai Sund-macher, Max Planck Institute for Dynamics of ComplexTechnical Systems, Magdeburg, who headed the challeng-ing and inspiring MCFC project that was sponsored by theFederal Ministry of Education and Research of the FederalRepublic of Germany. The author greatfully acknowledgethe deceased mentor of this project, Dr.-Ing. h. c. JoachimBerndt, who was a visionary entrepreneur with far reach-ing ideas in an age where most of the people already enjoytheir retirement. The author would like to thank also allthe young members of this group, especially Dr.-Ing. PeterHeidebrecht for his art in modelling, Dr. Kati Sternberg,the author’s former PhD student, who was undeterredby the huge PDE system and has computed satisfyingapproximations for the optimal solutions of the 2D crossflow configuration, Michael Mangold for his remarkablecontributions to POD applications in MCFC control, andArmin Rund, the author’s most recent PhD student, whocomputed precise approximations of optimal solutions forthe 1D counter flow configuration by means of adjointbased methods which the author never has thought to beaccomplishable for an MCFC model.

The author’s thank also includes his collaborators of manyyears, Professores Christof Buskens, University of Bremen,Kurt Chudej, University of Bayreuth, Matthias Gerdts,University of the Armed Forces, Munich, and RolandHerzog, University of Chemnitz, who supported not onlysome of the difficult applications discussed here.

Concerning the welding project mentioned in the intro-duction the author is indebted to Professor Victor A.Karkhin from St. Petersburgh State Polytechnical Univer-sity of Technology, Professor Vasily Ploshikhin, Universityof Bremen, and Dr.-Ing. Andrey Prikhodovsky from NeueMaterialien Bayreuth GmbH, and last but not least hisformer PhD student Dr. Verena Petzet.

Concerning the hypersonic aircraft problem mentioned inthe introduction the author would like to thank ProfessorEmeritus Gottfried Sachs, University of Technology, Mu-nich, and Dr.-Ing. Markus Wachter, German Institute ofScience and Technology, Singapore, for providing us thehypersonic aircraft model as well as Florent Le Bras, aformer internship student, today with the Laboratoire derecherche en balistiques et aerodynamiques of the frenchDelegation Generale de l’armemen, for some of the com-putations presented in the introduction.

REFERENCES

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A. Borzi and V. Schulz. Computational Optimizationof Systems Governed by Partial Differential Equations.SIAM, Philadelphia, 2011.

C. Buskens. Echtzeitoptimierung und Echtzeitoptimal-steuerung parametergestorter Probleme. Habilitations-schrift, University of Bayreuth, Germany, 2002.

K. Chudej, H. J. Pesch, K. Sternberg. Optimal Control ofLoad Changes for Molten Carbonate Fuel Cell Systems:A Challenge in PDE Constrained Optimization. SIAMJ. on Applied Mathematics , 70(2):621–639, 2009.

K. Chudej, H. J. Pesch, M. Wachter, G. Sachs, F. Le Bras.Instationary heat constrained trajectory optimization ofa hypersonic space vehicle by ODE-PDE constrained op-timal control. In A. Frediani, G. Butazzo, editors, Vari-ational Analysis and Aerospace Engineering, pages 127–144. Springer, Berlin, 2009.

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P. Heidebrecht. Modelling, Analysis and Optimisation ofMolten Carbonate Fuel Cell with Direct Internal Re-forming. PhD thesis, University of Magdeburg, Ger-many, 2004.

R. Herzog and K. Kunisch. Algorithms for PDE-con-strained optimization. GAMM-Mitteilungen, 33(2):163–176, 2010.

M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich. Optimizationwith PDE constraints . Mathematical Modelling: Theoryand Applications, volume 23, Springer, 2008.

M. Hinze, S. Volkwein. Proper Orthogonal DecompositionSurrogate Models for Nonlinear Dynamical Systems: Er-ror Estimates and Suboptimal Control. In P. Benner, V.Mehrmann, D. Sorensen, editors, Dimension Reductionof Large-Scale Systems , Lecture Notes in Computationaland Applied Mathematics, volume 45, pages 261–306.Springer, Berlin, Germany, 2005.

C. Laird and A. Wachter. An Interior Point OPTi-mizer for large-scale nonlinear optimization. URL:https://projects.coin-or.org/Ipopt, 2012.

M. Mangold. Rechnergestutzte Modellierung, Analyse undFuhrung von Membranreaktoren und Brennstoffzellen.Habilitationsschrift, Otto-von-Guericke-UniversitatMagdeburg, Germany, 2006.

A. Miele. Flight Mechanics I. Theory of Flight Paths.Adison-Wesley, Reading, MA, 1962

N. O. Okerblom, V. P. Demyantsevich, and I. P. Baykova.Design of Technology of Welded Structures. Sudprom-giz, Leningrad, 1963 (in Russian).

V. Petzet. Mathematische Optimierung und Sensitivi-tatsanalyse des Mehrstrahlschweißverfahrens zur Ver-hinderung der Heißrissbildung. PhD thesis, Universityof Bayreuth, Germany, 2008; Fortschritt-Berichte VDI ,Reihe 8, Nr. 1153, 2008.

V. Ploshikhin, A. Prikhodovsky, and H. W. Zoch. ZumMechanismus der Heißrissbildung beim Schweißen vonAl-Legierungen. Journal of Heat Treatment and Mate-rials , 56(6):357–362, 2003.

A. Rund. Beitrage zur Optimalen Steuerung partiell-differential algebraischer Gleichungen. PhD thesis, Uni-versity of Bayreuth, Germany, 2012.

A. Rund and K. Chudej. Optimal control for a simplified1D fuel cell model. Mathematical and Computer Mod-elling of Dynamical Systems, accepted.

E. W. Sachs and S. Volkwein. POD Galerkin approx-imations in PDE-constrained optimization. GAMM-Mitteilungen, 33:194–208, 2010.

V. G. Shumilin, V. A. Karkhin, M. I. Rakhman, K.Gatovsky. A technique of arc welding. Patent, No.1109280, USSR, 1980.

K. Sternberg. Simulation, Optimale Steuerung und Sen-sitivitatsanalyse einer Schmelzkarbonat-Brennstoffzellemithilfe eines partiell differential-algebraischen Glei-chungssystems . PhD thesis, University of Bayreuth,Germany, 2007.

T. Richter, A. Springer, and B. Vexler. Efficient numericalrealization of discontinuous Galerkin methods for tem-poral discretization of parabolic problems. Submitted ,2011.

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M. Wachter. Optimalflugbahnen im Hyperschall unterBerucksichtigung der instationaren Aufheizung. PhDthesis, University of Technology, Munich, Germany,2004.

M. Witzgall and K. Chudej. Flight Path Optimizationsubject to Instationary Heat Constraints Submitted ,2012.

Appendix A. MODEL OF THE LASER WELDINGPROBLEM

To give an impression of the model for the multi-beamlaser welding techique we firstly have to understand howthe micro structure, particularly of an aluminium alloy,develops during the solidification. In the solid-liquid re-gion, called mushy zone, between the liquidus and solidusisotherms, crystals grow dendritically until they formgrains. During this process the liquid and the solid phase ofmelt coexist. Simultaneously a segregation takes place, i. e.the different compounds of the alloy solidify at differentsolidus temperatures. Moreover tensile strains and stressesoccur which may initiate an opening displacement, thepossible source for a hot crack; see Fig. A.1.

The following thermo-mechanical model is based on theso-called strip expansion technique of Okerblom et al.(1963); cp. Fig. A.2. Let us assume that the stripsdiscretizing vertically the mushy zone can move fric-tionlessly and independently from each other and thatthey are deformable only in y-direction. Then, accordingto Ploshikhin, Prikhodovsky, and Zoch (2003), the openingdisplacement uod is given by

uod(xS) = 2 α

B∫

0

[T (xL, y) − T (xS , y)] dy ,

Fig. A.1. Schematic depiction of the nascency of a hotcrack, according to Ploshikhin, Prikhodovsky, andZoch (2003)

Fig. A.2. Strip expansion technique

where T denotes the temperature, xS < 0 and xL < 0 arethose abcissae of the solidus, resp. liquidus isotherms, i. e.T = TS, resp. T = TL, lieing behind the position of themain laser at (x = 0, y = 0), and α > 0 (thermal expensioncoefficient) as well as B > 0 (width of workpieces) areconstants; cp. Fig. A.2. This opening displacement mustbe either minimized or, at least, limited.

The optimization parameters describing position and sizeof the auxiliary laser beams are described by Fig. A.3.

These parameters can be grouped by p = (pI , pII)⊤

,

pI := (p1, p2, p3, p4)⊤

, and pII := (p5, p6)⊤

, where theparameters p5 := xS and p6 := xL are implicitly definedby the equations TS = T (p5, 0), resp. TL = T (p6, 0).These two parameters act more like state variable, sinceonly the parameters pI actually are design parameters tobe chosen optimally. Obviously, the temperature dependson these parameters, T (x, y) = T (x, y; pI). Hence, theascissae p5 := xS(pI) and p6 := xL(pI) also depend onthese design parameters. It turns out that these abscissaehave to be determined as precise as possible; see Petzet(2008). Thus, they have to be optimized while the openingdisplacement is bounded by a critical value ucrit

od .

Fig. A.3. Optimization variables pi for position and sizeof the auxiliary laser beams (left) and area relateddensity h of output power of the auxiliary laser beams(right)

The following semi-infinite optimization problem allowsfor this requirement [see Petzet (2008)]:

minp∈Pad

1

2

(T (p6, 0; pI) − TL)2

T 2L

+1

2

(T (p5, 0; pI) − TS)2

T 2S

subject to

−∆T (x, y; pI) −v

a

∂T

∂x(x, y; pI) =

h (x, y; pI)

λ sin Ω ,

T (x, y; pI) = Tamb on ∂Ω ,

max(x,y)∈D1(pI)

T (x, y; pI)≤ TS ,

uod (p5)≤ ucritod

with the admissible set of parameters

Pad :=

p ∈ R6 : p1 ≤ 0 , p5 ≤ 0 , p6 ≤ 0 , 0 ≤ p4 ≤ q ,

0.1 ≤ p3 ≤ p2 , p2 + p3 ≤ B , p5 ≤ p1 + p3 ,

p1 − p3 ≤ p6 , 0 ≤ g(p3) ≤ p3

.

Here, the constants v, a, λ, s, Tamb, and q denote velocityof the main laser, thermal diffusivity, thermal conductivity,thickness of plate, ambient temperature, and output powerof main laser. The model functions h and g describe thearea related density of power output of all lasers, resp. asmoothing function for modelling the influence region ofthe two auxiliary laser beams. See Fig. A.3 and for moredetails again Petzet (2008).

The temperature distribution is modelled by a quasi-stationary heat equation in a moving reference frame. Theobjective function is a point-functional. The inequalityconstraint constitutes a state constraint for avoiding ad-ditional weld pools under the auxiliary laser beams whichobviously can be transformed to a standard pointwise stateconstraint acting only on a subregion of Ω.

Note that the PDE allows a semi-analytical solution whichturns out to be numerically inferior to a finite elementapproximation; see Petzet (2008).

Appendix B. MODEL OF THE HYPERSONICAIRCRAFT PROBLEM

The equations of motion for a flight along a great circleover a rotational Earth are given by the following ODEsystem [cf., e. g., Miele (1962)]:

v =1

m

[

T (v, h; α, δT) cos(α + σT) − D(v, h; α)

]

−g(h) sin γ + ω2E r(h) sin γ ,

γ =1

m v

[

T (v, h; α, δT) sin(α + σT) + L(v, h; α)

]

+ cosγ

[

v

r(h)−

g(h)

v+

ω2E r(h)

v

]

+ 2 ωE ,

h = v sin γ ,

ζ = v cos γ ,

m = −βT(v, h; α, δT) .

Here v, γ, h, ζ, and m are the state variables denotingvelocity, flight path angle, altitude over Earth, path length,and mass. The control variables are the angle of attack αand the throttle setting δT . Thrust T , drag D, lift L aswell as instantaneous fuel consumption β are complicatedmodel functions describing special characteristics of thehypersonic aircraft. The functions g and r denote the ac-celeration of gravity, resp. the distance of the aircraft fromthe geocenter. The time interval [0, tf ] is unprescribed. Allother variables not mentioned are constants.

Minimum fuel trajectories are the goal of optimization,which is equivalent to maximize the mass at the unspeci-fied terminal time:

max(α, δT )∈Uad

m(tf ) .

Besides appropriate boundary conditions the typical in-equality constraints of flight path optimization are fur-ther side conditions: box constraints for the two controlsdefining the set Uad of admissible controls, mixed control-state constraints for the load factor, 0 ≤ n(v, h, m; α) ≤ 2,and a pure state constraints for the dynamic preasure,10 ≤ q(v, h) ≤ 50 [kPa]. For details see Wachter (2004).

Due to the hypersonic flight regime the aero-thermalheating Θ of the thermal protection system (TPS) is animportant issue, yielding an additional constraint of type

Θ ≤ Θmax = 1000 [K]

where the aero-thermal heating Θ is governed by a quasi-linear heat equation with nonlinear boundary conditions(because of the Stefan-Bolzmann law)

TPS cp(Θ)∂Θ

∂t− div

[

λ(Θ) gradΘ]

= 0 in Ω × (0, tf ) ,

Θ(x, 0) = Θ0(x) = 300 [K] in Ω ,

∂Θ

∂n(x, t) = qconv(v, h, α) − qrad(Θ, Θair(h))

on ∂Ω × (0, tf ) .

The ODE system and the PDE are unilaterally coupledhere, but the heat constraint feeds the information fromthe PDE back to the ODEs.

For more details it is refered to Wachter (2004), Chudejet al. (2008), and Witzgall and Chudej (2012).