**3. Results and Discussion**

In the calculations, we took calendar years 2020 and 2100 as the initial *t*<sup>0</sup> and the final (terminal) *tf* time, respectively, which meant that we were examining the climate control problem on the finite time interval 2020–2100. To formulate the boundary conditions and impose a constraint on change in the global mean surface temperature *Tsfc*, we assumed the following:


Then, the permissible increase in the temperature anomaly *T*<sup>2100</sup> by year 2100 relative to 2020 would be *T*<sup>2100</sup> = Δ*T*<sup>2100</sup> − Δ*T*<sup>2020</sup> = 0.4 . This value represents the boundary condition for *T* at *t* = *tf* . The threshold parameter, which defines a path constraint (Equation (23)), is *CT* = 2 − Δ*T*<sup>2020</sup> = 0.9 .

Changes in both global mean surface temperature and deep ocean temperature calculated for different climate change scenarios in the absence of climate engineering interventions are illustrated in Figure 1. The corresponding temperature changes in the year 2100 are shown in Table 2. According to Reference [79], without additional measures to reduce GHG emissions (RCP8.5 scenario), increases in global mean surface temperatures are expected to be between 3.7 and 4.8 ◦C by the year 2100 versus pre-industrial levels (this range is based on median climate response). As seen in Table 2, by year 2100, the model outlined here projects globally averaged surface temperature increases of 4.26, 3.44, and 2.80 ◦C for the RCP8.5, 1pctCO2, and RCP6.0 scenarios, respectively (relative to pre-industrial period). Thus, geoengineering can be regarded as one of supplementary measures needed to achieve the climate targets of the Paris Agreement.

**Figure 1.** Changes in (**a**) global mean surface temperature and (**b**) deep ocean temperature calculated for different climate change scenarios in the absence of climate engineering interventions.

**Table 2.** Calculated temperature changes *T* and *TD* from 2020 to 2100 (changes relative to the pre-industrial level are shown in brackets).


We considered results of calculations for the RCP8.5 (the worst-case) scenario in more detail. Figure 2 shows (a) the optimal albedo of the global stratospheric aerosol layer, (b) the corresponding surface temperature anomaly, (c) the mass of the global aerosol layer, and (d) the optimal emission rate of aerosol particles calculated for RCP8.5 pathway with and without constraint on the global mean surface temperature increase. In the absence of state constraint, the optimal albedo *α*∗ *<sup>A</sup>* and, accordingly, the optimal emission rate of aerosol particles *E*∗ *<sup>S</sup>* would increase exponentially. This optimal aerosols emission rate ensures that the target temperature anomaly *T*<sup>2100</sup> = 0.4 is satisfied. However, within the given time interval 2020–2100, a temperature rise would exceed the set point *CT*, i.e., *T*(*t*) > *CT* (the "overshooting" phenomenon [80]). The maximum increases in global mean surface temperature for different climate change scenarios are presented in Table 3. The use of the constraint (Equation (21)) allows us to avoid overshoot; however, compared to the unconstrained case, keeping the increase in global mean surface temperature below the target constrained level *CT* would require additional amount of aerosols (see Table 4). For example, for the RCP8.5 scenario, the total mass of aerosol particles injected in the stratosphere from the year 2020 to 2100 is about 73.6 Tg S, which is about 2 times larger than *M*∗ *<sup>S</sup>*, *tot*, calculated by solving an unconstrained OCP.

**Figure 2.** Results for the RCP8.5 pathway: (**a**) optimal albedo of aerosol layer *α*∗ *<sup>A</sup>*; (**b**) the corresponding temperature anomaly *T*∗; (**c**) total mass of aerosols *M*∗ *<sup>S</sup>*; and (**d**) the optimal emission rate *E*<sup>∗</sup> *S*.

**Table 3.** Maximum global mean surface temperature anomaly *T* calculated without state constraint.


**Table 4.** The mass of aerosols *MS*, *tot* (Tg S) injected into the stratosphere from 2020 to 2100.


Results obtained for 1pctCO2, RCP6.0, and RCP4.5 scenarios are represented in Figures 3–5, respectively. These figures show that the overshooting phenomenon is also observed for the 1pctCO2 and RCP6.0 scenarios. The only exception is the RCP4.5 scenario.

**Figure 3.** Results for the 1pctCO2 scenario: (**a**) optimal albedo of aerosol layer *α*∗ *<sup>A</sup>*; (**b**) the corresponding temperature anomaly *T*∗; (**c**) total aerosol mass *M*∗ *<sup>S</sup>*; and (**d**) the optimal emission rate *E*<sup>∗</sup> *S*.

**Figure 4.** *Cont.*

**Figure 4.** Results for the RCP6.0 pathway: (**a**) optimal albedo of aerosol layer *α*∗ *<sup>A</sup>*; (**b**) the corresponding temperature anomaly *T*∗; (**c**) total mass of aerosols *M*∗ *<sup>S</sup>*; and (**d**) the optimal emission rate *E*<sup>∗</sup> *S*.

**Figure 5.** Results for the RCP4.5 pathway: (**a**) optimal albedo of aerosol layer *α*∗ *<sup>A</sup>*; (**b**) the corresponding temperature anomaly *T*∗; (**c**) total mass of aerosols *M*∗ *<sup>S</sup>*; and (**d**) the optimal emission rate *E*<sup>∗</sup> *S*.

If the optimal control problem is considered with control variable constraint αA(t) ≤ *U*, then the target value for the temperature anomaly in the year 2100 may not necessarily be achieved (this depends on the value of the constraint *U* and the scenario in question). Assuming, for example, that *U* = 0.02, then the corresponding instant mass of aerosols is estimated to be 4.8 Tg S. In such a case, for the RCP8.5 scenario, the calculated temperature anomaly in the year 2100 relative to 2020 would exceed the target value by 0.3 ◦C, which is equivalent to exceeding the pre-industrial level by 1.8 ◦C. It needs to be recalled that constraint on the control variable is associated with a possible limitation on resources required to implement the project, namely, the amount of aerosols available to the project executors.

We emphasize that the results of calculations discussed above are for illustration purposes only. The primary outcome presented in this paper is the optimal control-based approach that can be used to design projects targeting purposeful manipulation of climate and weather.
