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

Maurer, Helmut; Semmler, Willi Article — Published Version Multi-objective optimal control with carbon emission and temperature constraints: for achieving a low-fossil-fuel economy Central European Journal of Operations Research Provided in Cooperation with: Springer Nature Suggested Citation: Maurer, Helmut; Semmler, Willi (2025) : Multi-objective optimal control with carbon emission and temperature constraints: for achieving a low-fossil-fuel economy, Central European Journal of Operations Research, ISSN 1613-9178, Springer, Berlin, Heidelberg, Vol. 33, Iss. 2, pp. 449-471, https://doi.org/10.1007/s10100-025-00970-3 This Version is available at: https://hdl.handle.net/10419/323286 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Vol.:(0123456789) Central European Journal of Operations Research (2025) 33:449–471 https://doi.org/10.1007/s10100-025-00970-3 Multi‑objective optimal control withcarbon emission andtemperature constraints: forachieving alow‑fossil‑fuel economy HelmutMaurer1· WilliSemmler2,3,4 Accepted: 26 February 2025 / Published online: 13 April 2025 © The Author(s) 2025 Abstract In this paper we propose multi-objective control to deal with climate change and climate risks and the transition to a low carbon economy. Extending our previous collaborative work as in Atolia etal. (Math Control Related Fields, 13:583–604, 2023), we again build on the Nordhaus type DICE model to include various optimal macroeconomic policies such as mitigation, adaptation and climate-related infrastructure investment studying the dynamics of the decarbonizing of the economy. Based on a finite horizon model that includes the threats of climate disasters arising from CO2 emissions and temperature rise, we deal with preventive measures such as adaptation reducing disaster effects. Our optimal control problem of finite horizon is consisting of a dynamical system with five-dimensional state vector representing stocks of private capital, green capital, public capital, stock of brown energy in the ground, carbon emissions, and temperature. The objective function captures preferences over consumption but is also impacted by atmospheric CO2 , climate risks events and by mitigation and adaptation policies. Given the numerous challenges to climate change policies with multiple objectives the control vector is eight-dimensional including mitigation, adaptation and infrastructure investment. The optimal control problem is studied under various state constraints. In two scenarios we compute the Pareto front for a bi-objective control problem. Optimization over the Pareto front provides us with suitable weights for the two objectives. In particular we explore the role of CO2 constraints, as the Kyoto Protocol has suggested, and temperature constraints, as the Copenhagen–Paris agreements have proposed. Keywords Climate change model· Mitigation· Optimal control· Discretization methods· Turnpike solution· Fiscal policy Mathematics Subject Classification Primary 49N90· 49K15· 49M37; Secondary 91-08 H. Maurer and W. Semmler have contributed equally to this work. Extended author information available on the last page of the article 450 H.Maurer, W.Semmler 1 Introduction The Paris, December 2015, COP 20 agreement on climate change is aiming at reducing the temperature increase to below 2oC relative to pre-industrial level. This implies that effective mitigation policies need to be pursued that not only prevent the CO2 emission from rising further but should reduce the annual emission substantially. The Paris agreement is detailed in the IPCC (2018) report that demonstrates higher probability of limiting global warming to 1.5oC will only be obtained, if a significant reduction of CO2 net emissions from 2020 to 2040 will be achieved. From then on the upper bound of CO2 should not be exceeded anymore. Since those upper limits create great policy challenges we propose here a modeling strategy that, attempts to answer three questions coming up in this context: First, what are the best strategies to keep the CO2 emission bounded by a predefined upper bound, and, correspondingly, how can one steer down the CO2 emission if it already has reached too high a level. Second, how can climate policies be scaled up and what resources should be allocated to mitigation and adaptation efforts, especially for the latter, in particular, when climate risk, due to a lack of emission reduction, is rising and future economic, social, and ecological damages can be expected. A third issue is of how the efforts of mitigation and adaptation are funded and how the funds should dynamically be allocated between traditional infrastructure investment, mitigation and adaptation efforts—and in what sequence. A number of those issues have been studied in Integrated Assessment Models (IAMs) of various kind using scientific modelling to link the economy with the biosphere and the atmosphere. This broad class of IAMs focusing on economy-climate interaction, use various scientific modeling and estimation methods. In our paper we more specifically focus on the seminal work by Nordhaus (2008, 2017) on the economy-climate link. This work specifically introduces the economy-climate interaction in mathematical formulations of an economic growth model, integrating into the model of carbon emission from industrial production, damages from it affecting output, and an optimal mitigation policy. Nordhaus calls his major work a Dynamic Integrated Model of Climate and the Economy, in short a DICE model. For details of the DICE model, see Nordhaus and Boyer (2000) and Nordhaus (2008). We extend the latter Nordhaus type model to include beside mitigation, optimal policies for adaptation and infrastructure investment exploring the dynamics of the transition to a low fossil-fuel economy. Since mitigation policy is mainly aiming at phasing in of renewable energy we also explore what amount of traditional fossil energy is allowed to be extracted when setting some carbon emission and temperature constraints. Whereas Nordhaus employs as objective function preferences over consumption, based on standard growth theory, our objective function captures multiple targets—preferences over consumption, but is also impacted by atmospheric CO2 as well as the mitigation and adaptation policies. Our dynamic model, as it includes the phasing in of renewable energy along with the issues mentioned above, can be considered an extension of the DICE type models. We present a dynamic global model with feedback control, representing an optimal control, that allows us to consider the specific policies of infrastructure 451 Multi‑objective optimal control withcarbon emission and… investment, mitigation and adaptation. The model is micro-founded in the sense that we employ a production technology which uses (private) physical capital and energy as inputs. Labor input is suppressed for simplicity as it is supplied inelastically. There are two sources of energy: non-renewable, brown energy produced by an extractive resource sector and renewable, green energy produced with (private physical) green capital. The emissions from brown energy use are a source of negative externality that directly enters the (instantaneous) felicity function. Note that we do not make damages to households dependent on the temperature, but rather on the stock of emissions. The reason is that the time series data on temperature is very heterogeneous across regions and quite volatile over time. In our model the government levies lump-sum taxes to raise revenues, a portion of which provides direct utility, another portion is invested in public (physical) capital, and remaining part is administrative expense. For models with other sources of capital, such as for example bond financing, see Bonen etal. (2016), and Orlov etal. (2018). The traditional use of public capital is to serve as infrastructure investment that augments the productivity of the production process. This infrastructure investment can be considered representing traditional as well as climate-related infrastructure. In our setup, the government can also use public capital for adaptation and for mitigation and chooses the split between these three competing uses optimally. Formally, the model gives rise to an optimal control problem of finite horizon consisting of a dynamic system with five-dimensional state vector representing the stocks of private capital, green capital, public capital, stock of brown energy in the ground, and emissions. The control vector is eight-dimensional, since it also comprises the split of public capital into mitigation, adaptation and infrastructure as time-dependent control functions excluding the choice of split for public capital mentioned earlier. We characterize the optimal tax and investment policies for the government and examine the resultant paths of important macroeconomic variables, particularly, of those related to energy transition, CO2 emission, and resource extraction. The complexity of the problem, however, necessitates both an analytical approach as well as the use of numerical methods. Solving such a model of finite horizon poses the challenge to show that the turnpike properties are not violated and the trajectories of the finite horizon model can approximate the solution of the infinite horizon case. The turnpike property usually follows from imposing terminal state conditions which represent the stationary solution of the necessary optimality conditions. In our case the turnpike property follows from additional bound constraints of the capital assets. For generic studies of turnpike properties of such models, see also Faulwasser etal. (2020) and Grüne etal. (2021). These numerical solutions allow us to investigate the optimal sequence of climate policy decisions with respect to infrastructure, mitigation and adaptation. In all cases considered in the paper, numerical solutions from our finite-horizon set up recover the turnpike property that is characteristics of the infinite horizon models. In terms of climate policy response, in our model, we find that the optimal policy can keep the CO2 emission bounded for a wide-range of initial conditions of capital 452 H.Maurer, W.Semmler stocks and CO2 levels. Specifically, we consider a scenarios with a high level of capital stocks and high level of CO2 . The remaining part of the paper is organized as follows. Section2 describes the optimal control model of climate change. In Sect.3, we introduce the temperature dynamics which is not contained in the basic model. In Sect.4 we discuss the necessary optimality conditions for the control problem with control and state constraints. Section5 presents the numerical solution method and reports the results from several scenarios concerning, in particular, bounds on the CO2 emission. In Sect.6, we consider two objectives arising from two different elasticity exponents in the welfare functional. We compute the Pareto front in two scenarios. Optimization over the Pareto front provides us with suitable weights for the two objectives. Section7 concludes. 2 Optimal control model ofclimate change We extend the Nordhaus DICE type model to include the adverse effects of climate change with a view to study the optimal policies for mitigation and adaptation to climate change until a transition to fossil-fuel-free green energy is successful. The green energy capital is a perfect substitute for fossil fuel in production. The climate change is modeled as an adverse effect of increase in atmospheric CO2 concentration (M) on utility. The mitigation efforts reduce the proportion of carbon in fossil fuel burned that escapes into the atmosphere as CO2 . In contrast, adaptation alleviates the harmful effects of higher atmospheric CO2 levels. The government raises revenue ( ep ) which is used for direct, utility-enhancing services and provision of public (physical) capital/infrastructure (G), with the possibility of some wastage. To analyze the issue of climate change, besides its traditional use for enhancing productive efficiency in the economy, we allow government to use public capital for mitigation and adaptation. The output of the production process is given by with A,Ag,Au>0 ,𝛼,𝛽,𝜁>0 , and 𝛼+𝛽+𝜁<1 . Kg is the stock of green capital, Kp is the stock of (private) physical capital, and 𝜈1∈(0, 1] , as mentioned above, is the fraction of public capital (G) used for the traditional purpose of enhancing productive efficiency. Finally, u is the amount of fossil fuel resource extracted and used, measured in terms of its carbon ( CO2 ) content. The felicity (utility) function depends on four arguments (i) per-capita consumption C; (ii) the per-capita amount of tax revenue ( 𝛼2eP,𝛼2∈[0, 1] ) used for direct welfare enhancement (e.g., healthcare); (iii) atmospheric concentration of CO2 (M) above the long-run sustainable level, the industrial level; and (iv) the per-capita amount of public capital expenditure ( 𝜈2G,𝜈2∈[0, 1) ) allocated to climate change adaptation. The optimal control model so defined then has five state variables: Kp : private physical capital per capita (1) Y =(𝜈 1 G) 𝛽 A(A g K g +A u u) 𝛼 (K p ) 𝜁 453 Multi‑objective optimal control withcarbon emission and… Kg : private green capital per capita G: public capital per capita M: CO2 (GHG) concentration in the atmosphere R: non-renewable resource (fossil energy) and the five basic control variables are ip : investment in physical capital ig : investment in green capital ep : government’s net tax revenue u: extraction rate from the non-renewable resource C: per capita consumption In addition, we consider the following three allocations of public capital as control functions: • 𝜈1 : standard infrastructure • 𝜈2 : adaptation • 𝜈3 : mitigation We emphasize that these allocations are not just parameters to be optimized but are considered as time-dependent control functions 𝜈k=𝜈k(t),k=1, 2, 3 . The state and control variables are denoted by The time horizon tf>0 (years) is finite. The dynamic system in [0, tf] of the global model of climate change is given by with initial conditions: that will be specified later. The control constraint for the extraction rate u is given 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(t)≤umax ∀t∈[0, tf]. 454 H.Maurer, W.Semmler There are three potential uses of government revenues, as mentioned earlier. The amount 𝛼1ep is invested in public capital, 𝛼2ep provides direct utility, and (1−𝛼1−𝛼2)ep is administrative expense/waste. Of the total public capital, G, a fraction 𝜈1 is the usual/traditional public capital that augments the productivity of the production process. Another fraction 𝜈2 is used for adaptation. The remaining fraction 𝜈3 is used for mitigation. Hence, the infrastructural and climate oriented allocations of public capital satisfy the constraints: Moreover, the system is subject to several control-state and pure state constraints. For defining a resource constraint we introduce the scalar function and impose the mixed control-state equality constraint Note that the mixed constraint does not depend on the allocations 𝜈 . In later computations we shall realize that we also need state constraints on the capital assets Kp,Kg,G and the CO2 concentration M. We prescribe lower bounds for the capital assets as state inequality constraints, and an upper bound for the CO2 concentration To handle these state constraints in a more convenient form we introduce the function and consider the following state inequality constraint Finally, note that our dynamics (5) for the evolution of carbon emission, generating the stock of carbon in the atmosphere, is slightly different from the Nordhaus’ DICE model as proposed in Nordhaus (2017). Nordhaus defines emission dynamics as time varying fraction of net output (after damages) that causes a carbon emission flow, partly absorbed by the ocean, but augmented by land carbon emission, that produces a stock of atmospheric carbon. The latter, called also carbon budget, in turn is the main driver of the global temperature, creating in turn the economic damages as a fraction of output. In our case we see the emission dynamics driven by the (9) 𝜈k(t)≥0, 𝜈1(t)+𝜈2(t)+𝜈3(t)=1∀t∈[0, tf]. (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(t),U(t)) = 0∀t∈[0, tf]. (12) K p(t) ≥ K min p ,Kg(t) ≥ K min g ,G(t) ≥ G min ∀0 ≤ t ≤ tf , (13) M( t )≤ M max ∀ 0 ≤ t ≤ t f. (14) s (X)=(K min p −Kp,K min g −Kg,G min −G,M−M max ) ∗ ∈ℝ 4 (15) s(X(t)) ≤0∀t∈[0, tf]. 455 Multi‑objective optimal control withcarbon emission and… equation (5) with 𝜅>0 a parameter allowing for a stationary stock of atmospheric carbon, 𝜅 M , some equilibrium level of carbon concentration, a parametrization that also other literature has used. If now the quantity of emission is a policy target, as was in the Kyoto Protocol, then equation (5) would imply a stationary solution. On the other hand if temperature is a target, as in the Paris agreement, then our emission dynamics could be translated into a global temperature via equation (25) using that as a target; see below. The two base cases of policy targets can also be found in Nordhaus (2008,Chapter V), giving rise to a stationary behavior of the atmospheric stock of carbon. Let us now introduce the welfare functional. Recall, the felicity (utility) function depends on (i) per-capita consumption C; (ii) the per-capita tax revenue ( 𝜈2eP,𝛼2∈[0, 1] ); (iii) atmospheric concentration of CO2 (M); and (iv) the per-capita expenditure on adaptation ( 𝜈2G,𝜈2∈[0, 1) ). The preferences of the representative household (or the policy maker) are where  M is the preindustrial level of atmospheric CO2 and  M>𝜅� M is a high CO2 level, with 𝜅 M being the level that would not need any adaptation and is the longrun sustainable level. We have chosen the value  M=4.5 in Table1. 𝜌>0 is the time rate of preference, n>0 is the rate of population growth, 𝜎>0 is the inverse of the elasticity of intertemporal substitution and 𝜂∈[0, 1] , 𝜀∈[0, 1] , 𝜉>0 , 𝜔∈[0, 1] , and 𝜅>0 are other parameters. The restrictions on parameters ensure that social expenditures and adaptation are utility enhancing with diminishing marginal utility and carbon emission that increase M reduce utility with increasing marginal disutility. As visible in our objective function we have not taken global temperature to measure the effect on welfare, but rather the stock of atmospheric carbon which appears to be easier to measure as driving variable for damages, see the simulations below. Note that for 𝜎≥1 , we only need 𝜂,𝜀>0 . Parameter values are given in Table1. This approach differs from other models that map emissions to temperature changes and then to reduced productivity-cum-output, see Nordhaus and Boyer (2000). The direct disutility approach better captures the wide ranging impacts of climate change that may include health impacts, ecological loss and heightened uncertainty, in addition to reduced productivity. Finally, note that the discount factor adjusts for the population growth rate n from the pure discount rate 𝜌 as all values are normalized by the population per capita. The optimal control problem (OCP) now consists in maximizing the welfare functional (16) subject to the dynamical constraints (2)-(7), the control constraints (8), (9), the mixed control-state constraint (11) and the pure state constraints (12) and (13). To obtain a more compact form of the optimal control problem we use the vector of state and control variables (X,U,𝜈) introduced above to write the dynamical system (2)–(6) in the form (16) ∫ tf 0 e−(𝜌−n)t1 1−𝜎 {[ C ( 𝛼2ep ) 𝜂 ( 1−exp ( −𝜉 ( 𝜈2G ) 𝜔 ) M−𝜅� M  M−𝜅� M )𝜀]1−𝜎 −1 } dt , 456 H.Maurer, W.Semmler Furthermore, let us denote the integrand of the welfare functional by (17)  X (t)=f(X(t),U(t),𝜈(t)),X(0)=X 0. (18) f0(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 Parameter values Parameter Value Definition 𝜌 0.03 Pure discount rate n0.015 Population Growth Rate 𝜂 0.1 Elasticity of transfers and public spending in utility 𝜖 1.1 Elasticity of CO2 concentration in (dis)utility 𝜔 0.5 Elasticity of public capital used for adaptation in utility 𝜎 2 Intertemporal elasticity of instantaneous utility A1 Total factor productivity Ag 1 Efficiency index of green capital Au 100 Efficiency index of the non-renewable resource 𝛼 0.05 Output elasticity of inputs, (AgKg+Auu)𝛼 𝛽 0.1 Output elasticity of public infrastructure, (𝜈1G ) 𝛽 𝜓 0.1 Scaling factor in marginal cost of resource extraction 𝜏 2 Exponential factor in marginal cost of resource extraction 𝛿p 0.1 Depreciation rate of physical capital 𝛿g 0.05 Depreciation rate of private capital 𝛿G 0.05 Depreciation rate of public capital Ωp ∈[5, 15] q-elasticity of investment spending on private capital Ωg ∈[5, 15] q-elasticity of investment spending on public capital 𝜒p 1 ( 𝛿 p +n)Ω p 𝜒g 1 ( 𝛿 g +n)Ω g 𝛼1 0.3 Proportion of tax revenue allocated to new public capital 𝛼2 0.7 Proportion of tax revenue allocated to transfers and public consumption r 0.07 World interest rate (paid on public debt)  M 2.5 Equilibrium concentration of CO2 𝜅 1.2 Atmospheric concentration stabilization ratio (relative to  M )  M 4.5 Value in disutility term in welfare (16) 𝛾 0.9 Fraction of greenhouse gas emissions not absorbed by the ocean c0.01 Decay rate of greenhouse gases in atmosphere 𝜅 1.2 Atmospheric concentration stabilization ratio (relative to  M ) 𝜃 0.01 Effectiveness of mitigation measures 𝜙 0.9 Exponent in mitigation term ( 𝜈 3 G) 𝜙 463 Multi‑objective optimal control withcarbon emission and… The control and state trajectories are displayed in Fig.2. The capital assets Kp,Kg,G exhibit a turnpike behavior and stay on the boundary of the state constraints(37) for most of the time. Also, the CO2 concentration M has a boundary arc M(t)=3.3 for t≥32 except for a small terminal interval. The temperature settles at T(t)=292.5 for t≥50 . The high temperature motivates us to enforce a more restrictive bound for M(t) at least on the terminal part of the planning period. 5.4 Solution forstate constraint M(t) ≤ 3.0 for t∈[40, 200] We require that the CO2 concentration stay below the value M=3.0 for t≥40 . We get the following numerical results for the welfare and terminal state variables: (38) W(U,𝜈)=−20.96 ∶K p (t f )=2.2, K g (t f )=0.3, G(t f )=0.8, M(t f )=3.3, R(t f )=0.8624, T(t f )= 292.5, Fig. 2 State and control trajectories for terminal time tf=200 , initial states (35), state constraints (37) and state constraint M(t) ≤ 3.3 . Top row: (left) physical capital Kp , green capital Kg and government capital G, (middle) CO2 concentration M, (right) resource R. Middle row: (left) investments ip and ig and tax revenue ep , (middle) temperature T, (right) extraction rate u. Bottom row: (left) consumption C and productivity Y, (middle) infrastructure 𝜈1 and adaptation 𝜈2 , (right) mitigation 𝜈3 464 H.Maurer, W.Semmler Figure3 shows the control and state trajectories. The CO2 concentration M stays on the boundary M(t)=3.0 for t≥40 except on a small terminal interval. The temperature comes down to T(tf)<=292 after a short overshot. To reach the goal of a smaller CO2 concentration the extraction rate u(t) decreases significantly so that the resource R is far from being exhausted. Nevertheless, the welfare W(X,U,𝜈) is not much smaller than in the previous cases. 5.5 Solution forstate constraint M(t) ≤ 2.6, ∀t∈[50, 200] This very restrictive constraint leads to the solution shown in Fig. 4. Numerical results for the welfare and the terminal state variables are (39) W(U,𝜈)=−24.93 ∶K p (t f )=2.2, K g (t f )=0.3, G(t f )=0.8, M(t f )=3.0, R(t f )=0.9112, T(t f )= 291.87. Fig. 3 State and control trajectories for terminal time tf=200 , initial states (35), state constraints (37) and state constraint M(t) ≤ 3.0 . Top row: (left) physical capital Kp , green capital Kg and government capital G, (middle) CO2 concentration M, (right) resource R. Middle row: (left) investments ip and ig and tax revenue ep , (middle) temperature T, (right) extraction rate u. Bottom row: (left) consumption C and productivity Y, (middle) infrastructure 𝜈1 and adaptation 𝜈2 , (right) mitigation 𝜈3 465 Multi‑objective optimal control withcarbon emission and… The CO2 concentration is steered down to M(t)<=2.6 on [50,200]. This has the effect that the temperature remains below T=291 on [50,200] which is only 1 deg. higher then the initial temperature. However, it can not be avoided that the temperature reaches the high value T(t)=292.0 already at t=20 . The exhaustion rate u(t) is nearly zero on [0,50] and remains on a small level which causes the high level R(tf)=0.9236 of the terminal resource. The small exhaustion rate is also responsible for the low production and consumption for t≤50 . Mitigation measures are dominant on [0,50] and push adaptation and infrastructure aside. It is interesting to note that we obtain a similar solution by imposing the following state constraint on the temperature: (40) W(U,𝜈)=−40.99 ∶K p (t f )=2.2, K g (t f )=0.3, G(t f )=0.8, M(t f )=2.6, R(t f )=0.9236, T(t f )= 291.87, (41) T(t)≤Tmax =291 ∀50 ≤t≤200. Fig. 4 State and control trajectories for terminal time tf=200 , initial states (35), state constraints (37) and state constraint M(t) ≤ 2.6 . Top row: (left) physical capital Kp , green capital Kg and government capital G, (middle) CO2 concentration M, (right) resource R. Middle row: (left) investments ip and ig and tax revenue ep , (middle) temperature T, (right) extraction rate u. Bottom row: (left) consumption C and productivity Y, (middle) infrastructure 𝜈1 and adaptation 𝜈2 , (right) mitigation 𝜈3 466 H.Maurer, W.Semmler 6 Multi‑objective approach totheoptimal control problem understate constraints The simultaneous optimization of multi-objective functions results in a set of trade-off or Pareto solutions. Although there is a wealth of literature involving finite-dimensional optimization problems (see Eichfelder 2008), only a few papers are devoted to optimal control problems; see, eg., Kaya and Maurer (2014) and Eichfelder etal. (2023). The problem of optimizing over the Pareto front of a bi-objective control problem has recently been addressed by Kaya and Maurer (2023). In this section, we study a bi-objective optimal control problem which arises from the fact that the control system and the objectives depend on some parameters which have significant impact on the solution but which can not be estimated precisely. One such critical parameter is the elasticity 𝜖 of the CO2 concentration describing the disutility in the objective (16), resp., (19). There we have chosen the nominal value 𝜖=1.1 . Let us denote the welfare function by W(X,U,𝜈,𝜖) to underline its dependence on 𝜖 . We shall compare the welfare W(X,U,𝜈,𝜖) for the small parameter 𝜖1=0.6 and for the large parameter 𝜖2=1.8 . Our focus is on the trade-off between the solutions for the functionals To generate the Pareto front for this bi-objective optimal control problem we apply the weighted sum scalarisation and thus maximize the following weighted sum functional with weight w∈[0, 1] : Standard homotopy methods are used to compute the solution for weights wi=i∕N,i=0, ..., N, with, eg., N=100 . Denote the state and control solution depending on the weight w by Xw( ⋅ ),Uw( ⋅ ),𝜈w( ⋅ ) and the objectives values by Fw 1 and Fw 2 . Then the Pareto front is defined the by curve PF ={(Fw 1,Fw 2)|w∈[0, 1]} . 6.1 Pareto front forstate constraints (37) and M(t)≤3.3 The Pareto front is concave as shown in Fig.5a since we are maximizing. We noted earlier that the objective values are negative. Now let us determine the point where the Pareto front has minimal distance to the origin. To this end we have to minimize the so-called master function By inspecting the numerical results of maximizing the weighted objective (43) we see that the minimum of the function Fm(w) is attained at w=wopt =0.58 , depicted in Fig.5b, with functional value F(wopt)=28.94 . Solving the optimal control problem (43) with weight w=wopt =0.58 we obtain (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(tf)=3.3, T(tf)=292.49. 467 Multi‑objective optimal control withcarbon emission and… The control and state trajectories are very close to those in Fig.2 and are not shown here. Instead of using the weighed-sum scalarization (43) we can implement the Chebychev scalarization described in Kaya and Maurer (2014, 2023). This amounts to maximizing the non-smooth objective This approach allows to optimize a master function like (44) in a more systematic way using bisection of gradient-like methods. 6.2 Pareto front forstate constraints (37) and M(t)≤2.6, 50 ≤t≤200 Again, the Pareto front is concave as shown in Fig.6a. The minimum of the master function Fm(w)=||(Fw 1,Fw 2)||2 is attained at w=wopt =0.52 . The corresponding control and state trajectories are rather close to those in Fig.4 and can be regarded as a compromise solution. (45) Fw(X,U,𝜈)=max {(1−w) ⋅ F1(X,U,𝜈),w ⋅ F2(X,U,𝜈)},0≤w≤1. Fig. 5 Pareto front for functionals F1,F2 under state constraints (37) and M(t)≤3.3, 0 ≤t≤200 . a Pareto front {(Fw 1,Fw 2)|w∈[0, 1]} , b Master function Fm(w)=||(Fw 1,Fw 2)||2 for 0≤w≤1. Fig. 6 Pareto front for functionals F1,F2 under state constraints (37) and M(t)≤2.6, 50 ≤t≤200 . a Pareto front {(Fw 1,Fw 2)|w∈[0, 1]} , b Master function Fm(w)=||(Fw 1,Fw 2)||2 for 0≤w≤1. 468 H.Maurer, W.Semmler 7 Conclusions Climate change and rising climate risks currently pose great challenges for academic work as as policymakers. Those challenges are recently not only not to surpassing upper limits of atmospheric CO2 concentration, but reducing the atmospherics concentration and excessive temperature rise, generating increasing weather extremes. We propose an optimal control model of climate control including eight control variables and five state variables that are subject to a rather complex mixed control-state constraints. Besides the more standard control variables (consumption, investments in capital goods, extraction of nonrenewable resource) we propose an extensive dynamic model with three allocations of public capital (infrastructure, adaptation and mitigation) as time-dependent control objectives and five economic state variables. Such an attempt not only needs to analyze the necessary optimality conditions and computed steady state values evaluated by the current-value Hamiltonian but in particular the numerical evaluation through numerical procedures. For the numerical solution paths we have chosen a rather large time horizon of tf=200 years, where we have computed numerical solutions in several scenarios using discretization and nonlinear programming methods. In all scenarios, the solutions exhibit a turnpike behavior. The main focus in our model is on the evolution of the CO2 concentration and a dynamic equation of the temperature (see Kaya and Maurer 2023) to measure the effect of the changing CO2 concentration on the temperature. This way either the carbon emission and concentration or the temperature can be used as policy target - since one can be converted into the other. We first explore for tf=200 the paths of the variables without state constraints, so the final outcomes to let the variables developing freely which is not a very satisfactory path for the control of the state variables—the CO2 emissions and temperature go to high levels—generates not the controls, such as infrastructure, adaptation, mitigation, shown in Fig.1. Next we constrain the CO2 emission by an upper magnitude of 3.3 which gives the path of CO2 . The CO2 emission goes to an upper bound in a finite time, after 50 steps,and the extraction of the fossil fuel resources decline, Those paths correspond to appropriate controls, shown in Fig.2. This is achieved with the value of the welfare function W=−20.96 . In the next scenario, Fig.3, we constrain the CO2 emission to be not greaser than the level 3, which gives us also the maximum temperature rise. Those results can also be achieved by the appropriate controls, in particular the fossil fuel extraction rate and welfare of W=−24.93 , to a lower welfare than in the previous case. The next scenario is depicted in Fig.4, with the CO2 constraint M(t)≤2.6 for all t∈[50, 200] . This allows also the temperature to be constrained below previous cases, namely to T(t)≤291 for all t∈[50, 200] . The welfare is now decreased as well: W=−40.99 . 469 Multi‑objective optimal control withcarbon emission and… In general, we provide some dynamic estimates of how the scaling up of efforts of mitigation and adaptation can be funded and how the funds should be allocated between (traditional and climate related) infrastructure investment, mitigation and adaptation efforts. We find that infrastructure investment efforts are in most cases high, sometimes occurring with a delay effect. Since in this context successful mitigation policy means phasing in of renewable energy (see also Maurer and Semmler 2015) we have also explored what amount of traditional fossil energy should be left insitu in order to satisfy some CO2 emission and temperature constraints.Yet, our control actions might not work if we are high above the CO2 target, namely if there are—as demonstrated in simpler models by Greiner etal. (2010) and Nordhaus (2008)—tipping points and thresholds beyond which the climate and whether extremes accelerate. For a model proposing also other means of financing climate policies, for example climate bonds, see Orlov etal. (2018). Addressing these issues required enlarging and solving for higher dimensional nonlinear control problems. We also showed that our numerical solutions for finite horizon decision model have turnpike properties similar to infinite horizon models. We also demonstrated why our multi-objective control approach should be complemented by the computation of a Pareto front that helps us to attach certain weights for the objectives in the objective functional. As we demonstrated we can then suggest a certain compromise between the perception of higher and lower risk of damages. Acknowledgements We thank Tato Khundadze for valuable assistance. We also want to thank Manoj Atolia and Prakash Loungani for cooperation on the issues addressed in this paper. Funding Open access funding provided by International Institute for Applied Systems Analysis (IIASA). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/. 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Math Program 106:25–57 Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. 471 Multi‑objective optimal control withcarbon emission and… Authors and Affiliations HelmutMaurer1· WilliSemmler2,3,4 * Willi Semmler [email protected] Helmut Maurer [email protected] 1 Institut für Analysis und Numerik, Universität Münster, Einsteinstr. 62, 48149Münster, Germany 2 Department ofEconomics, The New School, 6E 16th St, NewYork, NY10003, USA 3 University ofBielefeld, Bielefeld, Germany 4 IIASA, Laxenburg, Austria