Multi-objective optimal control with carbon emission and temperature constraints: for achieving a low-fossil-fuel economy

Mau e , Helmu ; Semmle , Willi
A icle — Published Ve sion
Mul i-objec i e op imal con ol wi h ca bon emission and
empe a u e cons ain s: o achie ing a low- ossil- uel
economy
Cen al Eu opean Jou nal o Ope a ions Resea ch
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Mau e , Helmu ; Semmle , Willi (2025) : Mul i-objec i e op imal con ol wi h
ca bon emission and empe a u e cons ain s: o achie ing a low- ossil- uel economy, Cen al
Eu opean Jou nal o Ope a ions Resea ch, ISSN 1613-9178, Sp inge , Be lin, Heidelbe g, Vol. 33, Iss.
2, pp. 449-471,
h ps://doi.o g/10.1007/s10100-025-00970-3
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/323286
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Cen al Eu opean Jou nal o Ope a ions Resea ch (2025) 33:449–471
h ps://doi.o g/10.1007/s10100-025-00970-3
Mul i‑objec i e op imal con ol wi hca bon emission
and empe a u e cons ain s: o achie ing alow‑ ossil‑ uel
economy
Helmu Mau e 1· WilliSemmle 2,3,4
Accep ed: 26 Feb ua y 2025 / Published online: 13 Ap il 2025
© The Au ho (s) 2025
Abs ac
In his pape we p opose mul i-objec i e con ol o deal wi h clima e change and cli-
ma e isks and he ansi ion o a low ca bon economy. Ex ending ou p e ious col-
labo a i e wo k as in A olia e al. (Ma h Con ol Rela ed Fields, 13:583–604, 2023),
we again build on he No dhaus ype DICE model o include a ious op imal mac o-
economic policies such as mi iga ion, adap a ion and clima e- ela ed in as uc u e
in es men s udying he dynamics o he deca bonizing o he economy. Based on a
ini e ho izon model ha includes he h ea s o clima e disas e s a ising om
CO2
emissions and empe a u e ise, we deal wi h p e en i e measu es such as adap a-
ion educing disas e e ec s. Ou op imal con ol p oblem o ini e ho izon is con-
sis ing o a dynamical sys em wi h i e-dimensional s a e ec o ep esen ing s ocks
o p i a e capi al, g een capi al, public capi al, s ock o b own ene gy in he g ound,
ca bon emissions, and empe a u e. The objec i e unc ion cap u es p e e ences o e
consump ion bu is also impac ed by a mosphe ic
CO2
, clima e isks e en s and by
mi iga ion and adap a ion policies. Gi en he nume ous challenges o clima e change
policies wi h mul iple objec i es he con ol ec o is eigh -dimensional including
mi iga ion, adap a ion and in as uc u e in es men . The op imal con ol p oblem is
s udied unde a ious s a e cons ain s. In wo scena ios we compu e he Pa e o on
o a bi-objec i e con ol p oblem. Op imiza ion o e he Pa e o on p o ides us
wi h sui able weigh s o he wo objec i es. In pa icula we explo e he ole o
CO2
cons ain s, as he Kyo o P o ocol has sugges ed, and empe a u e cons ain s, as he
Copenhagen–Pa is ag eemen s ha e p oposed.
Keywo ds Clima e change model· Mi iga ion· Op imal con ol· Disc e iza ion
me hods· Tu npike solu ion· Fiscal policy
Ma hema ics Subjec Classi ica ion P ima y 49N90· 49K15· 49M37; Seconda y
91-08
H. Mau e and W. Semmle ha e con ibu ed equally o his wo k.
Ex ended au ho in o ma ion a ailable on he las page o he a icle
450
H.Mau e , W.Semmle
1 In oduc ion
The Pa is, Decembe 2015, COP 20 ag eemen on clima e change is aiming a
educing he empe a u e inc ease o below
2oC
ela i e o p e-indus ial le el. This
implies ha e ec i e mi iga ion policies need o be pu sued ha no only p e en
he
CO2
emission om ising u he bu should educe he annual emission sub-
s an ially. The Pa is ag eemen is de ailed in he IPCC (2018) epo ha demon-
s a es highe p obabili y o limi ing global wa ming o
1.5oC
will only be ob ained,
i a signi ican educ ion o
CO2
ne emissions om 2020 o 2040 will be achie ed.
F om hen on he uppe bound o
CO2
should no be exceeded anymo e.
Since hose uppe limi s c ea e g ea policy challenges we p opose he e a mod-
eling s a egy ha , a emp s o answe h ee ques ions coming up in his con ex :
Fi s , wha a e he bes s a egies o keep he
CO2
emission bounded by a p ede ined
uppe bound, and, co espondingly, how can one s ee down he
CO2
emission i i
al eady has eached oo high a le el. Second, how can clima e policies be scaled up
and wha esou ces should be alloca ed o mi iga ion and adap a ion e o s, espe-
cially o he la e , in pa icula , when clima e isk, due o a lack o emission educ-
ion, is ising and u u e economic, social, and ecological damages can be expec ed.
A hi d issue is o how he e o s o mi iga ion and adap a ion a e unded and how
he unds should dynamically be alloca ed be ween adi ional in as uc u e in es -
men , mi iga ion and adap a ion e o s—and in wha sequence.
A numbe o hose issues ha e been s udied in In eg a ed Assessmen Models
(IAMs) o a ious kind using scien i ic modelling o link he economy wi h he bio-
sphe e and he a mosphe e. This b oad class o IAMs ocusing on economy-clima e
in e ac ion, use a ious scien i ic modeling and es ima ion me hods. In ou pape
we mo e speci ically ocus on he seminal wo k by No dhaus (2008, 2017) on he
economy-clima e link. This wo k speci ically in oduces he economy-clima e in e -
ac ion in ma hema ical o mula ions o an economic g ow h model, in eg a ing in o
he model o ca bon emission om indus ial p oduc ion, damages om i a ec ing
ou pu , and an op imal mi iga ion policy. No dhaus calls his majo wo k a Dynamic
In eg a ed Model o Clima e and he Economy, in sho a DICE model. Fo de ails
o he DICE model, see No dhaus and Boye (2000) and No dhaus (2008).
We ex end he la e No dhaus ype model o include beside mi iga ion, op imal
policies o adap a ion and in as uc u e in es men explo ing he dynamics o he
ansi ion o a low ossil- uel economy. Since mi iga ion policy is mainly aiming a
phasing in o enewable ene gy we also explo e wha amoun o adi ional ossil
ene gy is allowed o be ex ac ed when se ing some ca bon emission and empe a-
u e cons ain s. Whe eas No dhaus employs as objec i e unc ion p e e ences o e
consump ion, based on s anda d g ow h heo y, ou objec i e unc ion cap u es mul-
iple a ge s—p e e ences o e consump ion, bu is also impac ed by a mosphe ic
CO2
as well as he mi iga ion and adap a ion policies. Ou dynamic model, as i
includes he phasing in o enewable ene gy along wi h he issues men ioned abo e,
can be conside ed an ex ension o he DICE ype models.
We p esen a dynamic global model wi h eedback con ol, ep esen ing an
op imal con ol, ha allows us o conside he speci ic policies o in as uc u e
451
Mul i‑objec i e op imal con ol wi hca bon emission and…
in es men , mi iga ion and adap a ion. The model is mic o- ounded in he sense
ha we employ a p oduc ion echnology which uses (p i a e) physical capi al and
ene gy as inpu s. Labo inpu is supp essed o simplici y as i is supplied inelas-
ically. The e a e wo sou ces o ene gy: non- enewable, b own ene gy p oduced
by an ex ac i e esou ce sec o and enewable, g een ene gy p oduced wi h (p i-
a e physical) g een capi al. The emissions om b own ene gy use a e a sou ce
o nega i e ex e nali y ha di ec ly en e s he (ins an aneous) elici y unc ion.
No e ha we do no make damages o households dependen on he empe a u e,
bu a he on he s ock o emissions. The eason is ha he ime se ies da a on
empe a u e is e y he e ogeneous ac oss egions and qui e ola ile o e ime.
In ou model he go e nmen le ies lump-sum axes o aise e enues, a po -
ion o which p o ides di ec u ili y, ano he po ion is in es ed in public (physi-
cal) capi al, and emaining pa is adminis a i e expense. Fo models wi h o he
sou ces o capi al, such as o example bond inancing, see Bonen e al. (2016),
and O lo e al. (2018). The adi ional use o public capi al is o se e as in a-
s uc u e in es men ha augmen s he p oduc i i y o he p oduc ion p ocess.
This in as uc u e in es men can be conside ed ep esen ing adi ional as well
as clima e- ela ed in as uc u e. In ou se up, he go e nmen can also use public
capi al o adap a ion and o mi iga ion and chooses he spli be ween hese h ee
compe ing uses op imally.
Fo mally, he model gi es ise o an op imal con ol p oblem o ini e ho izon
consis ing o a dynamic sys em wi h i e-dimensional s a e ec o ep esen ing
he s ocks o p i a e capi al, g een capi al, public capi al, s ock o b own ene gy
in he g ound, and emissions. The con ol ec o is eigh -dimensional, since i also
comp ises he spli o public capi al in o mi iga ion, adap a ion and in as uc u e
as ime-dependen con ol unc ions excluding he choice o spli o public capi al
men ioned ea lie .
We cha ac e ize he op imal ax and in es men policies o he go e nmen and
examine he esul an pa hs o impo an mac oeconomic a iables, pa icula ly, o
hose ela ed o ene gy ansi ion, CO2 emission, and esou ce ex ac ion. The com-
plexi y o he p oblem, howe e , necessi a es bo h an analy ical app oach as well as
he use o nume ical me hods.
Sol ing such a model o ini e ho izon poses he challenge o show ha he u n-
pike p ope ies a e no iola ed and he ajec o ies o he ini e ho izon model can
app oxima e he solu ion o he in ini e ho izon case. The u npike p ope y usu-
ally ollows om imposing e minal s a e condi ions which ep esen he s a iona y
solu ion o he necessa y op imali y condi ions. In ou case he u npike p ope y
ollows om addi ional bound cons ain s o he capi al asse s. Fo gene ic s udies
o u npike p ope ies o such models, see also Faulwasse e al. (2020) and G üne
e al. (2021).
These nume ical solu ions allow us o in es iga e he op imal sequence o cli-
ma e policy decisions wi h espec o in as uc u e, mi iga ion and adap a ion. In
all cases conside ed in he pape , nume ical solu ions om ou ini e-ho izon se up
eco e he u npike p ope y ha is cha ac e is ics o he in ini e ho izon models.
In e ms o clima e policy esponse, in ou model, we ind ha he op imal policy
can keep he
CO2
emission bounded o a wide- ange o ini ial condi ions o capi al
452
H.Mau e , W.Semmle
s ocks and
CO2
le els. Speci ically, we conside a scena ios wi h a high le el o cap-
i al s ocks and high le el o
CO2
.
The emaining pa o he pape is o ganized as ollows. Sec ion2 desc ibes he
op imal con ol model o clima e change. In Sec .3, we in oduce he empe a u e
dynamics which is no con ained in he basic model. In Sec .4 we discuss he neces-
sa y op imali y condi ions o he con ol p oblem wi h con ol and s a e cons ain s.
Sec ion5 p esen s he nume ical solu ion me hod and epo s he esul s om se -
e al scena ios conce ning, in pa icula , bounds on he
CO2
emission. In Sec .6, we
conside wo objec i es a ising om wo di e en elas ici y exponen s in he wel-
a e unc ional. We compu e he Pa e o on in wo scena ios. Op imiza ion o e
he Pa e o on p o ides us wi h sui able weigh s o he wo objec i es. Sec ion7
concludes.
2 Op imal con ol model o clima e change
We ex end he No dhaus DICE ype model o include he ad e se e ec s o clima e
change wi h a iew o s udy he op imal policies o mi iga ion and adap a ion o
clima e change un il a ansi ion o ossil- uel- ee g een ene gy is success ul. The
g een ene gy capi al is a pe ec subs i u e o ossil uel in p oduc ion. The clima e
change is modeled as an ad e se e ec o inc ease in a mosphe ic
CO2
concen a ion
(M) on u ili y. The mi iga ion e o s educe he p opo ion o ca bon in ossil uel
bu ned ha escapes in o he a mosphe e as
CO2
. In con as , adap a ion alle ia es
he ha m ul e ec s o highe a mosphe ic
CO2
le els.
The go e nmen aises e enue (
ep
) which is used o di ec , u ili y-enhancing
se ices and p o ision o public (physical) capi al/in as uc u e (G), wi h he pos-
sibili y o some was age. To analyze he issue o clima e change, besides i s adi-
ional use o enhancing p oduc i e e iciency in he economy, we allow go e nmen
o use public capi al o mi iga ion and adap a ion.
The ou pu o he p oduc ion p ocess is gi en by
wi h
A,Ag,Au>0
,𝛼,𝛽,𝜁>0
, and
𝛼+𝛽+𝜁<1
.
Kg
is he s ock o g een capi al,
Kp
is he s ock o (p i a e) physical capi al, and
𝜈1∈(0, 1]
, as men ioned abo e, is
he ac ion o public capi al (G) used o he adi ional pu pose o enhancing p o-
duc i e e iciency. Finally, u is he amoun o ossil uel esou ce ex ac ed and used,
measu ed in e ms o i s ca bon (
CO2
) con en .
The elici y (u ili y) unc ion depends on ou a gumen s (i) pe -capi a consump-
ion C; (ii) he pe -capi a amoun o ax e enue (
𝛼2eP,𝛼2∈[0, 1]
) used o di ec
wel a e enhancemen (e.g., heal hca e); (iii) a mosphe ic concen a ion o
CO2
(M)
abo e he long- un sus ainable le el, he indus ial le el; and (i ) he pe -capi a
amoun o public capi al expendi u e (
𝜈2G,𝜈2∈[0, 1)
) alloca ed o clima e change
adap a ion. The op imal con ol model so de ined hen has i e s a e a iables:
Kp
: p i a e physical capi al pe capi a
(1)
Y
=(𝜈
1
G)
𝛽
A(A
g
K
g
+A
u
u)
𝛼
(K
p
)
𝜁

453
Mul i‑objec i e op imal con ol wi hca bon emission and…
Kg
: p i a e g een capi al pe capi a
G: public capi al pe capi a
M:
CO2
(GHG) concen a ion in he a mosphe e
R: non- enewable esou ce ( ossil ene gy)
and he i e basic con ol a iables a e
ip
: in es men in physical capi al
ig
: in es men in g een capi al
ep
: go e nmen ’s ne ax e enue
u: ex ac ion a e om he non- enewable esou ce
C: pe capi a consump ion
In addi ion, we conside he ollowing h ee alloca ions o public capi al as con-
ol unc ions:
•
𝜈1
: s anda d in as uc u e
•
𝜈2
: adap a ion
•
𝜈3
: mi iga ion
We emphasize ha hese alloca ions a e no jus pa ame e s o be op imized bu
a e conside ed as ime-dependen con ol unc ions
𝜈k=𝜈k( ),k=1, 2, 3
. The s a e
and con ol a iables a e deno ed by
The ime ho izon
>0
(yea s) is ini e. The dynamic sys em in
[0, ]
o he global
model o clima e change is gi en by
wi h ini ial condi ions:
ha will be speci ied la e . The con ol cons ain o he ex ac ion a e u is gi en by
X=(K
p
,K
g
,G,R,M)∈ℝ
5
,U=(i
p
,i
g
,e
p
,u,C)∈ℝ
5
,𝜈=(𝜈
1
,𝜈
2
,𝜈
3
)∈ℝ
3.
(2)

Kp
=i
p
−(𝛿
p
+n)K
p,
(3)

Kg
=i
g
−(𝛿
g
+n)K
g,
(4)

G
=𝛼
1
e
p
−(𝛿
G
+n)G
,
(5)

M=𝛾u−c(M−𝜅�
M)−𝜃(𝜈3
⋅
G)
𝜙
,
(6)

R=−u,
(7)
X(0)=X0
(8)
0≤u( )≤umax ∀ ∈[0, ].
454
H.Mau e , W.Semmle
The e a e h ee po en ial uses o go e nmen e enues, as men ioned ea lie .
The amoun
𝛼1ep
is in es ed in public capi al,
𝛼2ep
p o ides di ec u ili y, and
(1−𝛼1−𝛼2)ep
is adminis a i e expense/was e. O he o al public capi al, G, a ac-
ion
𝜈1
is he usual/ adi ional public capi al ha augmen s he p oduc i i y o he
p oduc ion p ocess. Ano he ac ion
𝜈2
is used o adap a ion. The emaining ac-
ion
𝜈3
is used o mi iga ion. Hence, he in as uc u al and clima e o ien ed alloca-
ions o public capi al sa is y he cons ain s:
Mo eo e , he sys em is subjec o se e al con ol-s a e and pu e s a e cons ain s.
Fo de ining a esou ce cons ain we in oduce he scala unc ion
and impose he mixed con ol-s a e equali y cons ain
No e ha he mixed cons ain does no depend on he alloca ions
𝜈
. In la e com-
pu a ions we shall ealize ha we also need s a e cons ain s on he capi al asse s
Kp,Kg,G
and he
CO2
concen a ion M. We p esc ibe lowe bounds o he capi al
asse s as s a e inequali y cons ain s,
and an uppe bound o he
CO2
concen a ion
To handle hese s a e cons ain s in a mo e con enien o m we in oduce he
unc ion
and conside he ollowing s a e inequali y cons ain
Finally, no e ha ou dynamics (5) o he e olu ion o ca bon emission, gene a -
ing he s ock o ca bon in he a mosphe e, is sligh ly di e en om he No dhaus’
DICE model as p oposed in No dhaus (2017). No dhaus de ines emission dynamics
as ime a ying ac ion o ne ou pu (a e damages) ha causes a ca bon emission
low, pa ly abso bed by he ocean, bu augmen ed by land ca bon emission, ha
p oduces a s ock o a mosphe ic ca bon. The la e , called also ca bon budge , in
u n is he main d i e o he global empe a u e, c ea ing in u n he economic dam-
ages as a ac ion o ou pu . In ou case we see he emission dynamics d i en by he
(9)
𝜈k( )≥0, 𝜈1( )+𝜈2( )+𝜈3( )=1∀ ∈[0, ].
(10)
c(X,U)=Y−C−i
p
−i
g
−e
p
−u𝜓R−𝜏
−𝜒p
2
(
ip
K
p
−𝛿p−n
)
2
Kp−𝜒g
2
(
ig
K
g
−𝛿g−n
)
2
K
g
(11)
c(X( ),U( )) = 0∀ ∈[0, ].
(12)
K
p( )
≥
K
min
p
,Kg( )
≥
K
min
g
,G( )
≥
G
min
∀0
≤
≤
,
(13)
M(
)≤
M
max ∀
0
≤
≤
.
(14)
s
(X)=(K
min
p
−Kp,K
min
g
−Kg,G
min
−G,M−M
max
)
∗
∈ℝ
4
(15)
s(X( )) ≤0∀ ∈[0, ].
455
Mul i‑objec i e op imal con ol wi hca bon emission and…
equa ion (5) wi h
𝜅>0
a pa ame e allowing o a s a iona y s ock o a mosphe ic
ca bon,
𝜅
M
, some equilib ium le el o ca bon concen a ion, a pa ame iza ion ha
also o he li e a u e has used. I now he quan i y o emission is a policy a ge , as
was in he Kyo o P o ocol, hen equa ion (5) would imply a s a iona y solu ion. On
he o he hand i empe a u e is a a ge , as in he Pa is ag eemen , hen ou emis-
sion dynamics could be ansla ed in o a global empe a u e ia equa ion (25) using
ha as a a ge ; see below. The wo base cases o policy a ge s can also be ound in
No dhaus (2008,Chap e V), gi ing ise o a s a iona y beha io o he a mosphe ic
s ock o ca bon.
Le us now in oduce he wel a e unc ional. Recall, he elici y (u ili y) unc-
ion depends on (i) pe -capi a consump ion C; (ii) he pe -capi a ax e enue
(
𝜈2eP,𝛼2∈[0, 1]
); (iii) a mosphe ic concen a ion o
CO2
(M); and (i ) he pe -cap-
i a expendi u e on adap a ion (
𝜈2G,𝜈2∈[0, 1)
). The p e e ences o he ep esen a-
i e household (o he policy make ) a e
whe e

M
is he p eindus ial le el o a mosphe ic
CO2
and

M>𝜅�
M
is a high
CO2
le el, wi h
𝜅
M
being he le el ha would no need any adap a ion and is he long-
un sus ainable le el. We ha e chosen he alue

M=4.5
in Table1.
𝜌>0
is he ime
a e o p e e ence,
n>0
is he a e o popula ion g ow h,
𝜎>0
is he in e se o he
elas ici y o in e empo al subs i u ion and
𝜂∈[0, 1]
,
𝜀∈[0, 1]
,
𝜉>0
,
𝜔∈[0, 1]
,
and
𝜅>0
a e o he pa ame e s. The es ic ions on pa ame e s ensu e ha social
expendi u es and adap a ion a e u ili y enhancing wi h diminishing ma ginal u il-
i y and ca bon emission ha inc ease M educe u ili y wi h inc easing ma ginal dis-
u ili y. As isible in ou objec i e unc ion we ha e no aken global empe a u e
o measu e he e ec on wel a e, bu a he he s ock o a mosphe ic ca bon which
appea s o be easie o measu e as d i ing a iable o damages, see he simula ions
below. No e ha o
𝜎≥1
, we only need
𝜂,𝜀>0
. Pa ame e alues a e gi en in
Table1.
This app oach di e s om o he models ha map emissions o empe a u e
changes and hen o educed p oduc i i y-cum-ou pu , see No dhaus and Boye
(2000). The di ec disu ili y app oach be e cap u es he wide anging impac s o
clima e change ha may include heal h impac s, ecological loss and heigh ened
unce ain y, in addi ion o educed p oduc i i y. Finally, no e ha he discoun ac o
adjus s o he popula ion g ow h a e n om he pu e discoun a e
𝜌
as all alues
a e no malized by he popula ion pe capi a.
The op imal con ol p oblem (OCP) now consis s in maximizing he wel a e unc-
ional (16) subjec o he dynamical cons ain s (2)-(7), he con ol cons ain s (8),
(9), he mixed con ol-s a e cons ain (11) and he pu e s a e cons ain s (12) and
(13). To ob ain a mo e compac o m o he op imal con ol p oblem we use he ec-
o o s a e and con ol a iables
(X,U,𝜈)
in oduced abo e o w i e he dynamical
sys em (2)–(6) in he o m
(16)
∫
0
e−(𝜌−n) 1
1−𝜎
{[
C
(
𝛼2ep
)
𝜂
(
1−exp
(
−𝜉
(
𝜈2G
)
𝜔
)
M−𝜅�
M

M−𝜅�
M
)𝜀]1−𝜎
−1
}
d
,
456
H.Mau e , W.Semmle
Fu he mo e, le us deno e he in eg and o he wel a e unc ional by
(17)

X
( )= (X( ),U( ),𝜈( )),X(0)=X
0.
(18)
0(X,U,𝜈)= 1
1−𝜎
{[
C
(
𝛼2ep
)
𝜂
(
1−exp
(
−𝜉
(
𝜈2G
)
𝜔
)
M−𝜅�
M

M−𝜅�
M
)𝜀]1−𝜎
−1
}.
Table 1 Pa ame e alues
Pa ame e Value De ini ion
𝜌
0.03 Pu e discoun a e
n0.015 Popula ion G ow h Ra e
𝜂
0.1 Elas ici y o ans e s and public spending in u ili y
𝜖
1.1 Elas ici y o
CO2
concen a ion in (dis)u ili y
𝜔
0.5 Elas ici y o public capi al used o adap a ion in u ili y
𝜎
2 In e empo al elas ici y o ins an aneous u ili y
A1 To al ac o p oduc i i y
Ag
1 E iciency index o g een capi al
Au
100 E iciency index o he non- enewable esou ce
𝛼
0.05 Ou pu elas ici y o inpu s,
(AgKg+Auu)𝛼
𝛽
0.1 Ou pu elas ici y o public in as uc u e,
(𝜈1G
)
𝛽
𝜓
0.1 Scaling ac o in ma ginal cos o esou ce ex ac ion
𝜏
2 Exponen ial ac o in ma ginal cos o esou ce ex ac ion
𝛿p
0.1 Dep ecia ion a e o physical capi al
𝛿g
0.05 Dep ecia ion a e o p i a e capi al
𝛿G
0.05 Dep ecia ion a e o public capi al
Ωp
∈[5, 15]
q-elas ici y o in es men spending on p i a e capi al
Ωg
∈[5, 15]
q-elas ici y o in es men spending on public capi al
𝜒p
1
(
𝛿
p
+n)Ω
p
𝜒g
1
(
𝛿
g
+n)Ω
g
𝛼1
0.3 P opo ion o ax e enue alloca ed o new public capi al
𝛼2
0.7 P opo ion o ax e enue alloca ed o ans e s and public consump ion

0.07 Wo ld in e es a e (paid on public deb )

M
2.5 Equilib ium concen a ion o
CO2
𝜅
1.2 A mosphe ic concen a ion s abiliza ion a io ( ela i e o

M
)

M
4.5 Value in disu ili y e m in wel a e (16)
𝛾
0.9 F ac ion o g eenhouse gas emissions no abso bed by he ocean
c0.01 Decay a e o g eenhouse gases in a mosphe e
𝜅
1.2 A mosphe ic concen a ion s abiliza ion a io ( ela i e o

M
)
𝜃
0.01 E ec i eness o mi iga ion measu es
𝜙
0.9 Exponen in mi iga ion e m
(
𝜈
3
G)
𝜙
463
Mul i‑objec i e op imal con ol wi hca bon emission and…
The con ol and s a e ajec o ies a e displayed in Fig.2.
The capi al asse s
Kp,Kg,G
exhibi a u npike beha io and s ay on he bound-
a y o he s a e cons ain s(37) o mos o he ime. Also, he
CO2
concen a ion
M has a bounda y a c
M( )=3.3
o
≥32
excep o a small e minal in e al.
The empe a u e se les a
T( )=292.5
o
≥50
. The high empe a u e mo i a es
us o en o ce a mo e es ic i e bound o M( ) a leas on he e minal pa o he
planning pe iod.
5.4 Solu ion o s a e cons ain
M( )
≤
3.0
o 
∈[40, 200]
We equi e ha he
CO2
concen a ion s ay below he alue
M=3.0
o
≥40
.
We ge he ollowing nume ical esul s o he wel a e and e minal s a e
a iables:
(38)
W(U,𝜈)=−20.96 ∶K
p
(
)=2.2, K
g
(
)=0.3, G(
)=0.8,
M(
)=3.3, R(
)=0.8624, T(
)=
292.5,
Fig. 2 S a e and con ol ajec o ies o e minal ime
=200
, ini ial s a es (35), s a e cons ain s (37)
and s a e cons ain
M( )
≤
3.3
. Top ow: (le ) physical capi al
Kp
, g een capi al
Kg
and go e nmen cap-
i al G, (middle)
CO2
concen a ion M, ( igh ) esou ce R. Middle ow: (le ) in es men s
ip
and
ig
and ax
e enue
ep
, (middle) empe a u e T, ( igh ) ex ac ion a e u. Bo om ow: (le ) consump ion C and p o-
duc i i y Y, (middle) in as uc u e
𝜈1
and adap a ion
𝜈2
, ( igh ) mi iga ion
𝜈3

464
H.Mau e , W.Semmle
Figu e3 shows he con ol and s a e ajec o ies.
The
CO2
concen a ion M s ays on he bounda y
M( )=3.0
o
≥40
excep
on a small e minal in e al. The empe a u e comes down o
T( )<=292
a e
a sho o e sho . To each he goal o a smalle
CO2
concen a ion he ex ac ion
a e u( ) dec eases signi ican ly so ha he esou ce R is a om being exhaus ed.
Ne e heless, he wel a e
W(X,U,𝜈)
is no much smalle han in he p e ious
cases.
5.5 Solu ion o s a e cons ain
M( )
≤
2.6, ∀ ∈[50, 200]
This e y es ic i e cons ain leads o he solu ion shown in Fig. 4. Nume ical
esul s o he wel a e and he e minal s a e a iables a e
(39)
W(U,𝜈)=−24.93 ∶K
p
(
)=2.2, K
g
(
)=0.3, G(
)=0.8,
M(
)=3.0, R(
)=0.9112, T(
)=
291.87.
Fig. 3 S a e and con ol ajec o ies o e minal ime
=200
, ini ial s a es (35), s a e cons ain s (37)
and s a e cons ain
M( )
≤
3.0
. Top ow: (le ) physical capi al
Kp
, g een capi al
Kg
and go e nmen cap-
i al G, (middle)
CO2
concen a ion M, ( igh ) esou ce R. Middle ow: (le ) in es men s
ip
and
ig
and ax
e enue
ep
, (middle) empe a u e T, ( igh ) ex ac ion a e u. Bo om ow: (le ) consump ion C and p o-
duc i i y Y, (middle) in as uc u e
𝜈1
and adap a ion
𝜈2
, ( igh ) mi iga ion
𝜈3
465
Mul i‑objec i e op imal con ol wi hca bon emission and…
The
CO2
concen a ion is s ee ed down o
M( )<=2.6
on [50,200]. This has he
e ec ha he empe a u e emains below
T=291
on [50,200] which is only 1 deg.
highe hen he ini ial empe a u e. Howe e , i can no be a oided ha he empe a-
u e eaches he high alue
T( )=292.0
al eady a
=20
. The exhaus ion a e u( )
is nea ly ze o on [0,50] and emains on a small le el which causes he high le el
R( )=0.9236
o he e minal esou ce. The small exhaus ion a e is also espon-
sible o he low p oduc ion and consump ion o
≤50
. Mi iga ion measu es a e
dominan on [0,50] and push adap a ion and in as uc u e aside.
I is in e es ing o no e ha we ob ain a simila solu ion by imposing he ollowing
s a e cons ain on he empe a u e:
(40)
W(U,𝜈)=−40.99 ∶K
p
(
)=2.2, K
g
(
)=0.3, G(
)=0.8,
M(
)=2.6, R(
)=0.9236, T(
)=
291.87,
(41)
T( )≤Tmax =291 ∀50 ≤ ≤200.
Fig. 4 S a e and con ol ajec o ies o e minal ime
=200
, ini ial s a es (35), s a e cons ain s (37)
and s a e cons ain
M( )
≤
2.6
. Top ow: (le ) physical capi al
Kp
, g een capi al
Kg
and go e nmen cap-
i al G, (middle)
CO2
concen a ion M, ( igh ) esou ce R. Middle ow: (le ) in es men s
ip
and
ig
and ax
e enue
ep
, (middle) empe a u e T, ( igh ) ex ac ion a e u. Bo om ow: (le ) consump ion C and p o-
duc i i y Y, (middle) in as uc u e
𝜈1
and adap a ion
𝜈2
, ( igh ) mi iga ion
𝜈3
466
H.Mau e , W.Semmle
6 Mul i‑objec i e app oach o heop imal con ol p oblem
unde s a e cons ain s
The simul aneous op imiza ion o mul i-objec i e unc ions esul s in a se o ade-o
o Pa e o solu ions. Al hough he e is a weal h o li e a u e in ol ing ini e-dimensional
op imiza ion p oblems (see Eich elde 2008), only a ew pape s a e de o ed o op i-
mal con ol p oblems; see, eg., Kaya and Mau e (2014) and Eich elde e al. (2023).
The p oblem o op imizing o e he Pa e o on o a bi-objec i e con ol p oblem has
ecen ly been add essed by Kaya and Mau e (2023).
In his sec ion, we s udy a bi-objec i e op imal con ol p oblem which a ises om
he ac ha he con ol sys em and he objec i es depend on some pa ame e s which
ha e signi ican impac on he solu ion bu which can no be es ima ed p ecisely. One
such c i ical pa ame e is he elas ici y
𝜖
o he
CO2
concen a ion desc ibing he disu-
ili y in he objec i e (16), esp., (19). The e we ha e chosen he nominal alue
𝜖=1.1
.
Le us deno e he wel a e unc ion by
W(X,U,𝜈,𝜖)
o unde line i s dependence on
𝜖
.
We shall compa e he wel a e
W(X,U,𝜈,𝜖)
o he small pa ame e
𝜖1=0.6
and o
he la ge pa ame e
𝜖2=1.8
. Ou ocus is on he ade-o be ween he solu ions o he
unc ionals
To gene a e he Pa e o on o his bi-objec i e op imal con ol p oblem we apply
he weigh ed sum scala isa ion and hus maximize he ollowing weigh ed sum unc-
ional wi h weigh
w∈[0, 1]
:
S anda d homo opy me hods a e used o compu e he solu ion o weigh s
wi=i∕N,i=0, ..., N,
wi h, eg.,
N=100
. Deno e he s a e and con ol solu ion
depending on he weigh w by
Xw(
⋅
),Uw(
⋅
),𝜈w(
⋅
)
and he objec i es alues by
Fw
1
and
Fw
2
. Then he Pa e o on is de ined he by cu e
PF ={(Fw
1,Fw
2)|w∈[0, 1]}
.
6.1 Pa e o on o s a e cons ain s (37) and
M( )≤3.3
The Pa e o on is conca e as shown in Fig.5a since we a e maximizing. We no ed
ea lie ha he objec i e alues a e nega i e.
Now le us de e mine he poin whe e he Pa e o on has minimal dis ance o he
o igin. To his end we ha e o minimize he so-called mas e unc ion
By inspec ing he nume ical esul s o maximizing he weigh ed objec i e (43) we
see ha he minimum o he unc ion
Fm(w)
is a ained a
w=wop =0.58
, depic ed
in Fig.5b, wi h unc ional alue
F(wop )=28.94
. Sol ing he op imal con ol p ob-
lem (43) wi h weigh
w=wop =0.58
we ob ain
(42)
Fk(X,U,𝜈)=W(X,U,𝜈,𝜖i),k=1, 2.
(43)
Fw(X,U,𝜈)=(1−w)
⋅
F1(X,U,𝜈)+w
⋅
F2(X,U,𝜈).
(44)
Fm(w)=||(Fw
1,Fw
2)||2.
Fw(X,U,𝜈)=−19.64, M( )=3.3, T( )=292.49.
467
Mul i‑objec i e op imal con ol wi hca bon emission and…
The con ol and s a e ajec o ies a e e y close o hose in Fig.2 and a e no shown
he e.
Ins ead o using he weighed-sum scala iza ion (43) we can implemen he Che-
byche scala iza ion desc ibed in Kaya and Mau e (2014, 2023). This amoun s o
maximizing he non-smoo h objec i e
This app oach allows o op imize a mas e unc ion like (44) in a mo e sys ema ic
way using bisec ion o g adien -like me hods.
6.2 Pa e o on o s a e cons ain s (37) and
M( )≤2.6, 50 ≤ ≤200
Again, he Pa e o on is conca e as shown in Fig.6a.
The minimum o he mas e unc ion
Fm(w)=||(Fw
1,Fw
2)||2
is a ained a
w=wop =0.52
. The co esponding con ol and s a e ajec o ies a e a he close o
hose in Fig.4 and can be ega ded as a comp omise solu ion.
(45)
Fw(X,U,𝜈)=max {(1−w)
⋅
F1(X,U,𝜈),w
⋅
F2(X,U,𝜈)},0≤w≤1.
Fig. 5 Pa e o on o unc ionals
F1,F2
unde s a e cons ain s (37) and
M( )≤3.3, 0 ≤ ≤200
. a
Pa e o on
{(Fw
1,Fw
2)|w∈[0, 1]}
, b Mas e unc ion
Fm(w)=||(Fw
1,Fw
2)||2
o
0≤w≤1.
Fig. 6 Pa e o on o unc ionals
F1,F2
unde s a e cons ain s (37) and
M( )≤2.6, 50 ≤ ≤200
. a
Pa e o on
{(Fw
1,Fw
2)|w∈[0, 1]}
, b Mas e unc ion
Fm(w)=||(Fw
1,Fw
2)||2
o
0≤w≤1.
468
H.Mau e , W.Semmle
7 Conclusions
Clima e change and ising clima e isks cu en ly pose g ea challenges o academic
wo k as as policymake s. Those challenges a e ecen ly no only no o su passing
uppe limi s o a mosphe ic
CO2
concen a ion, bu educing he a mosphe ics
concen a ion and excessi e empe a u e ise, gene a ing inc easing wea he
ex emes. We p opose an op imal con ol model o clima e con ol including eigh
con ol a iables and i e s a e a iables ha a e subjec o a a he complex mixed
con ol-s a e cons ain s. Besides he mo e s anda d con ol a iables (consump ion,
in es men s in capi al goods, ex ac ion o non enewable esou ce) we p opose an
ex ensi e dynamic model wi h h ee alloca ions o public capi al (in as uc u e,
adap a ion and mi iga ion) as ime-dependen con ol objec i es and i e economic
s a e a iables.
Such an a emp no only needs o analyze he necessa y op imali y condi ions
and compu ed s eady s a e alues e alua ed by he cu en - alue Hamil onian bu in
pa icula he nume ical e alua ion h ough nume ical p ocedu es. Fo he nume ical
solu ion pa hs we ha e chosen a a he la ge ime ho izon o
=200
yea s, whe e
we ha e compu ed nume ical solu ions in se e al scena ios using disc e iza ion and
nonlinea p og amming me hods. In all scena ios, he solu ions exhibi a u npike
beha io . The main ocus in ou model is on he e olu ion o he
CO2
concen a ion
and a dynamic equa ion o he empe a u e (see Kaya and Mau e 2023) o measu e
he e ec o he changing
CO2
concen a ion on he empe a u e. This way ei he he
ca bon emission and concen a ion o he empe a u e can be used as policy a ge -
since one can be con e ed in o he o he .
We i s explo e o
=200
he pa hs o he a iables wi hou s a e cons ain s,
so he inal ou comes o le he a iables de eloping eely which is no a e y
sa is ac o y pa h o he con ol o he s a e a iables— he
CO2
emissions and
empe a u e go o high le els—gene a es no he con ols, such as in as uc u e,
adap a ion, mi iga ion, shown in Fig.1. Nex we cons ain he
CO2
emission by an
uppe magni ude o 3.3 which gi es he pa h o
CO2
. The
CO2
emission goes o
an uppe bound in a ini e ime, a e 50 s eps,and he ex ac ion o he ossil uel
esou ces decline, Those pa hs co espond o app op ia e con ols, shown in Fig.2.
This is achie ed wi h he alue o he wel a e unc ion
W=−20.96
.
In he nex scena io, Fig.3, we cons ain he
CO2
emission o be no g ease han
he le el 3, which gi es us also he maximum empe a u e ise. Those esul s can
also be achie ed by he app op ia e con ols, in pa icula he ossil uel ex ac ion
a e and wel a e o
W=−24.93
, o a lowe wel a e han in he p e ious case.
The nex scena io is depic ed in Fig.4, wi h he
CO2
cons ain
M( )≤2.6
o all
∈[50, 200]
. This allows also he empe a u e o be cons ained below p e ious
cases, namely o
T( )≤291
o all
∈[50, 200]
. The wel a e is now dec eased as
well:
W=−40.99
.

469
Mul i‑objec i e op imal con ol wi hca bon emission and…
In gene al, we p o ide some dynamic es ima es o how he scaling up o e o s
o mi iga ion and adap a ion can be unded and how he unds should be alloca ed
be ween ( adi ional and clima e ela ed) in as uc u e in es men , mi iga ion and
adap a ion e o s. We ind ha in as uc u e in es men e o s a e in mos cases
high, some imes occu ing wi h a delay e ec . Since in his con ex success ul mi i-
ga ion policy means phasing in o enewable ene gy (see also Mau e and Semmle
2015) we ha e also explo ed wha amoun o adi ional ossil ene gy should be
le insi u in o de o sa is y some
CO2
emission and empe a u e cons ain s.Ye ,
ou con ol ac ions migh no wo k i we a e high abo e he
CO2
a ge , namely i
he e a e—as demons a ed in simple models by G eine e al. (2010) and No d-
haus (2008)— ipping poin s and h esholds beyond which he clima e and whe he
ex emes accele a e. Fo a model p oposing also o he means o inancing clima e
policies, o example clima e bonds, see O lo e al. (2018). Add essing hese issues
equi ed enla ging and sol ing o highe dimensional nonlinea con ol p oblems.
We also showed ha ou nume ical solu ions o ini e ho izon decision model ha e
u npike p ope ies simila o in ini e ho izon models.
We also demons a ed why ou mul i-objec i e con ol app oach should be
complemen ed by he compu a ion o a Pa e o on ha helps us o a ach ce ain
weigh s o he objec i es in he objec i e unc ional. As we demons a ed we can
hen sugges a ce ain comp omise be ween he pe cep ion o highe and lowe isk
o damages.
Acknowledgemen s We hank Ta o Khundadze o aluable assis ance. We also wan o hank Manoj
A olia and P akash Loungani o coope a ion on he issues add essed in his pape .
Funding Open access unding p o ided by In e na ional Ins i u e o Applied Sys ems Analysis (IIASA).
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License,
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Re e ences
A olia M, Loungani P, Mau e H, Semmle W (2023) Op imal con ol o a global model o clima e
change wi h adap a ion and mi iga ion. Ma h Con ol Rela ed Fields 13:583–604. h ps:// doi. o g/ 10.
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Au ho s and A ilia ions
Helmu Mau e 1· WilliSemmle 2,3,4
* Willi Semmle
Semmle [email protected]
Helmu Mau e
helmu .mau e @uni-muens e .de
1 Ins i u ü Analysis und Nume ik, Uni e si ä Müns e , Eins eins . 62, 48149Müns e ,
Ge many
2 Depa men o Economics, The New School, 6E 16 h S , NewYo k, NY10003, USA
3 Uni e si y o Biele eld, Biele eld, Ge many
4 IIASA, Laxenbu g, Aus ia