**1. Introduction**

Climate change is among the most significant threats to human civilization in the 21st century and beyond [1]. The Paris Climate Accord proposed to hold average temperature increase "to well below 2 ◦C above pre-industrial levels" and to pursue efforts to keep warming "below 1.5 ◦C above pre-industrial levels" [2]. To reach these goals, eight countries have already presented long-term low-emission strategies, which aims to reduce greenhouse gas emissions; several countries are currently in the process of preparing such strategies [3]. Meanwhile, the World Meteorological Organization's (WMO) "Statement of the State of the Global Climate in 2017" released in January 2018 said, "The global mean temperature in 2017 was approximately 1.1 ◦C above the pre-industrial era" [4]. There is high confidence that planetary warming will continue throughout the 21st century even if we immediately stopped emitting greenhouse gases into the atmosphere (e.g., References [5–9]). Some resent studies (e.g., References [10–13]) suggest that geoengineering technologies can serve as a supplementary measure to stabilize climate as "in the absence of external cooling influence" [14], it is hard to achieve the Paris Agreement climate goals.

Solar radiation management (SRM) by injection of sulfur aerosols into the stratosphere [15,16] is one of the most feasible and promising solutions for inducing negative radiative forcing (RF) from aerosols in order to at least partially compensate the positive RF from atmospheric greenhouse gases. The current state of understanding of climate engineering technologies, including SRM, has been discussed in References [17–26]. Over the years, climate models have played a key role in exploring

geoengineering techniques and predicting and quantifying their potential effects on Earth's climate (e.g., References [27–36]). Due to the uncertainties inherent in climate models that could not be sufficiently reduced over the last decade [37], the resulting range of possible outcomes of hypothetical geoengineering efforts remains quite vague. To handle the climate response uncertainties, some studies (e.g., References [38–45]) have suggested modeling the Earth's climate as a control system with feedbacks, which allows planning scenarios for geoengineering using the so-called "design model". This formulation makes it possible to design the control law and calculate the amount of SRM forcing as a function of time needed to offset the rise in global mean surface temperature due to human-caused positive RF. Meanwhile, exploring Earth's global climate as controlled dynamical system, we can approach geoengineering from the perspective of optimal control theory [46–49]. Within the optimal control framework, the goal of geoengineering can be formulated in terms of extremal problem, which involves finding control functions and the corresponding climate system trajectory that minimize or maximize a certain objective functional (also referred to as performance measure or index) subject to various constraints (e.g., References [50,51]. If **x** is the state vector of climate system and **u** is the vector of control variables, then the abstract extremal problem can be formulated as follows:

$$\mathcal{J}\left(\mathbf{x},\mathbf{u}\right) \to \text{extr}, \; \mathcal{F}\left(\mathbf{x},\mathbf{u}\right) = 0, \; (\mathbf{x},\mathbf{u}) \in \mathcal{M} \subset \mathcal{X} \times \mathcal{U} \tag{1}$$

The statement of this problem includes a set X ×U on which the (real) functional J (**x**, **u**) is defined and constraints imposed on state and control variables given by the model of control object F(**x**, **u**) = 0 (dynamic constraints) and by the subset M in X ×U. The solution to the extremal problem (Equation (1)) is the optimal process (**x**∗, **u**∗). Thus, by solving the optimal control problem (OCP), we can obtain the mathematically rigorous control law and the corresponding system's trajectory that are relevant for the specified performance measure J (**x**, **u**).

This paper deals with a simple mathematical model for controlling the global mean surface temperature *Tsfc* in the 21st century by the injection of sulfur aerosols into the stratosphere to limit the global temperature increase in the year 2100 by 1.5 ◦C above pre-industrial level and keeping global temperature over the period of 2020–2100 within 2 ◦C as required by the Paris climate agreement. The objective is to minimize resources (the total mass of aerosols) required to achieve the desired final state of the climate system. In the model, the positive RF produced by the rise in the atmospheric concentrations of greenhouse gases is specified in accordance with the Representative Concentration Pathways [52] and the 1pctCO2 (1% per year CO2 increase) scenario.

The mathematical statement of OCP is the collection of the following key elements: objective function defined to judge the effectiveness of control process, mathematical model of the controlled object, equality and inequality constraints to be satisfied by state and control variables, and boundary and initial conditions (if any) for state variables. To imitate the behavior of the climate system, we applied a two-component energy balance model [53–55] in which the global mean surface temperature anomaly (perturbation) represents the variable that interests us the most, and the albedo of the global aerosol layer is designated as the control variable. We derived analytical expressions for both the optimal albedo of the global aerosol layer and the corresponding change in the global mean surface temperature.

The results of illustrative calculations are presented for the period 2020–2100. For each climate change scenario, the optimal albedo of the aerosol layer—and therefore the aerosol emission rates—as well as the associated global mean surface temperature changes were found.

We need to emphasize that the main reason for using such a model is that similar two-layer models have been considered and analyzed in a number of papers considering the response to forced climate change. For example, Geoffroy et al. [56,57] obtained and discussed the general analytical solutions of the two-layer model for different hypothetical climate forcing scenarios and suggested the approach of calibrating the model parameters to imitate the time response of coupled general circulation models (CGCMs) from CMIP5 to radiative forcing. Gregory et al. [58] analyzed the two-layer model and discussed the transient climate response, the global mean surface air temperature change under two scenarios: one with a step forcing (the abrupt 4xCO2 experiment) and one with the 1pctCO2 scenario. Despite the fact that the two-layer model is one of the simplest tools to mimic climate dynamics under external radiative forcing, it was able to simulate the evolution of average global surface temperature over time in response to both abrupt and time-dependent forcing with reasonable accuracy (e.g., References [56,59]).

In the two-layer model, climate control is carried out via changing Earth's planetary albedo by injection of sulfur aerosols into the stratosphere ("albedo modification"). Sulfur aerosols increase the amount of sunlight that is scattered back to space, thereby reducing the amount of sunlight absorbed by Earth. Inherently, the planetary albedo is an average of the local albedo, averaged over the entire globe. The local albedo, in turn, is a highly variable dimensionless parameter that depends on a number of local factors, such as the composition of the atmosphere and in particular the presence of aerosols, the cloud amount and properties [60], the sea ice cover [61], the land use [62–64], the snow cover [65], etc. A typical value of Earth's planetary albedo is about 0.3 [66].

As change in the albedo of our planet is a powerful driver of climate (indeed, a 1% change in the Earth's planetary albedo generates the radiative effect of 3.42 Wm<sup>−</sup>2, which is commensurate with radiative forcing due to a doubling of CO2 concentrations in the atmosphere), scientists have proposed "albedo modification" as a powerful tool to deal with global warming (e.g., References [16,20,23,29,31,33,67]).
