# Leveraging control theory tools to enable real-time policy action for sustainable social-ecological-technical systems

John M Anderies, Jean-Denis Mathias

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John M Anderies, Jean-Denis Mathias. Leveraging control theory tools to enable real-time policy action for sustainable social-ecological-technical systems. Communications Earth & Environment, 2025, 6 (1), pp.806. 10.1038/s43247-025-02767-3. hal-05367306

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# Leveraging control theory tools to enable real-time policy action for sustainable social-ecological-technical systems

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John M. Anderies<sup>1,2,4</sup> & Jean-Denis Mathias<sup>3,4</sup>

The idea of planetary boundaries sets limits for a safe operating space for humanity and offers a guide for tackling global sustainability challenges. But staying within these limits requires understanding how globally interconnected social, environmental, and technological systems behave when decisions are made with incomplete information. Here, we draw on the notions of observability and controllability from modern control theory, an engineering approach to steering complex systems, to show how decision-makers can act in real time without perfect models or full knowledge. Our analysis illustrates a natural mapping of design parameters onto real-world policy decisions and highlights how to manage the interplay between past, real-time, and projected information to improve the controllability, and thus the sustainability, of social-ecological-technical systems.

The Planetary Boundaries (PBs) Framework<fnref n="1" /> has become an iconic representation of the sustainability challenge modern globalized societies face: the creation and maintenance a safe operating space (SOS) for humanity. This planetary-scale framing of sustainability is an implicit recognition that we have entered the Anthropocene that defines nine boundaries related to Earth system dynamics that, if crossed, may cause the Earth system to move into a new regime. In the past, feedbacks between nature and society were biased in the direction from nature to society. Earth system dynamics impacted migration, conflicts, innovation, and social organization more than the other way around. In the Anthropocene, human impacts have reached a scale in which feedbacks between society and nature are more balanced and tight, and occur at the global scale². Concisely put, global society may now have the capacity to push the Earth system beyond several PBs.

While the notion of PBs provides a target, a subset of a state space in technical terms, to guide human development³, it opens up many challenging questions about how to *navigate* in that state space to remain in the SOS. The fact that the 2015 update of the original study framed the assessment of the status of the PBs in terms of “control” variables reflects the long and deep connection between mathematical control theory and sustainability⁴⁻⁸ and control theory is still used widely in policy debates, e.g. to calculate the social cost of carbon⁹. Even though modern sustainability scholarship has moved beyond a control-centric view of top-down policy interventions that characterized earlier work to a focus on building adaptive, transformative, and governance capacity and linking knowledge and action¹⁰, sustainability science and control theory still share a core set of

concerns: enabling a system to function in difficult dynamically evolving circumstances using feedback structures to process information (make decisions) and dynamically guide interventions in a system. Capacity to adapt, transform and link knowledge to action all require effective feedback structures that translate information about the system (where are we relative to the SOS?) into actions (investments in technology, institutions and organizations, social capital, education) that recursively *feed back* into the system.

This situation drives our core research question: given fundamental limits on what observations can be made about a system state in real time and limits on our capacity to translate them into actions, how do we design effective, practical real-time policy (feedback structures) to improve our capacity to manage the safe operating space of a social-ecological-technical system (SETS)? To address this question, we leverage the PID (proportional-integral-derivative) control design paradigm as it provides and intuitive mapping between the mathematical structure of feedback controls and real-world policy decision processes. The PID approach is based on the idea that the controller makes corrections to the system state based on the ‘error’ between what is observed of the system state and a goal (e.g. the distance between the present system state and a planetary boundary). Corrections are based on present (P), past (I), and estimates of future (D) information which provides an intuitive analogue for how policy decisions are actually made. This approach allows us to explore how biases in what is observed and the relative weights given to present, past, or expected future outcomes in policy decisions impact our capacity to guide dynamic SETS. More specifically, we note that real-world policy makers behave like PID

<footnote n="1">¹School of Sustainability and School of Human Evolution and Social Change, Arizona State University, Tempe, AZ, USA. ²Beijer Institute of Ecological Economics, The Royal Swedish Academy of Sciences, Stockholm, Sweden. ³Université Clermont Auvergne, INRAE, UR LISC, Centre de Clermont-Ferrand, Clermont-Ferrand, Aubière, France. ⁴These authors contributed equally: John M. Anderies, Jean-Denis Mathias✉ e-mail: m.anderies@asu.edu</footnote>
controllers because they must generate real-time policy based information available to them (present, past, and future estimates). One of our examples focuses on the COVID crisis (see real-time data dynamics in SI) where decisions were based on the current state of the system (P-controller), past actions (I-controller) as a history of the crisis developed, and on mental models used to predict future impacts of policies (D-controller). Despite the fact that control theory has been applied to models of complex social-ecological systems, the implications of applying it to observations in real time have seldom been addressed in the case of social-ecological systems. The key contribution of this paper is to demonstrate how to operationalize the PID control paradigm to manage SETS in real time within the constraints of real-world governance structures where tight design processes and conditions for optimization do not exist.

Within the PID paradigm, we leverage the notions of controllability and observability. Controllability represents the capacity to translate information about the system to steer it from an initial state towards a given target. This, in turn, requires making decisions based on observations of the system state. The notion of observability captures the fact that such observations are not straightforward, may depend on a chain of technical sensors and mental models, and can be costly. Navigating planetary boundaries provides a ready example: global-scale collective action challenges limit controllability regardless of observability of the Earth system¹¹. Likewise, if collective action dilemmas could be overcome, limited observability may generate disagreement about what to do and slow action. Finally, even with agreement about what to do and the will to act, good decisions based on the wrong or limited observations can fundamentally change the dynamics of controlled systems and reduce our capacity to reach and maintain a SOS. In this study, we focus on this last case. We assume there is a will to act and draw attention to the link between observability and our capacity to reach a SOS.

## Results

Given the broad readership of this journal, we first briefly contextualize our approach. We are not aiming to develop new methods or theory. As noted in the introduction, control theory in its various forms is widely used on resource and environmental management problems (see Methods). This is no surprise as control theory is a powerful tool for analyzing any dynamic, uncertain decision problem. Several insights have emerged from specific studies, i.e. the use of control theory to develop a specific policy response to a specific problem in a specific social-ecological system (see Methods for many examples of such work, including our own work). Rather, our claim, given our work at the intersection of control theory, dynamical systems theory, and sustainability, is that important principles from the many applications of control theory have not been translated for practical use by non-specialists, e.g. policy actors. Thus, we intentionally rely on very well developed tools from control theory, especially the PID control paradigm given its intuitive appeal, to translate some of these principles in a broadly understandable way.

In what follows, we explore some aspects of the interplay between observability and controllability. First, we highlight how observability - and the choice of what we observe for making decisions - may affect the controllability of a system given a desired policy target in a natural resource extraction context. This first example considers a stable environment (the system's dynamics remain the same) whereas we consider a changing environment (system's dynamics vary) in the second case study. For this purpose, we explore the robustness of PID-based policies by analyzing how controllability may change because of exogenous drivers in contagion problems. Finally, coping with future uncertainties requires exploration of how different possible policies may impact future reachable states of system. For this purpose, we explore in the third case study how reachable states may change when decisions are based on observations of natural versus social system states and whether that information is based on historical or projected system states in pollution (e.g. atmospheric CO₂ concentration) emission problems. Taken together, the insights from this analysis suggest: 1) policies should integrate how their capacity for action strongly depends

on indicators and observations that are available on the managed system; 2) the use of past, real-time or projected information may impact robustness of decision-making process in different and possibly non-intuitive ways; and 3) stakeholders should analyze policies not only in terms of controllable states but also with regard to reachable states to effectively cope with unexpected events.

## A general framing of SETS problems

Control problems can be conceptualized in very general terms as filling or draining water containers to a desired level. The 'water' may be fuel, voltage, traffic density, fish in the ocean, CO₂ in the atmosphere, trust, knowledge, or social capital. Whatever the 'water' (stock) represents, modern control theory seeks to keep the stock levels in a safe operating space by controlling the flows in and out of the container. An essential feature of sustainability science is that some containers are natural systems that can empty (e.g. carbon cycling) and fill (e.g. net primary production) themselves.

If $x(t)$ is the stock level at time $t$, the general mathematical representation of this processes is

$$ \Delta x(t)/\Delta t = G(x(t), t) + I(x(t), u(t), t) \quad (1) $$

where $G(x, t)$ (growth) and $I(x, u, t)$ (impact) represent the natural and human induced net replenishment rates, respectively. Regardless of the complexity of these functions, whether non-linear, periodic, or uncertain, control remains a basic process of managing inflows and outflows to control one stock ($x \in \mathbb{R}$) or many interacting stocks ($x \in \mathbb{R}^n$).

In natural resource problems, $x$ is the biomass of a target species and $u$ is the harvest effort. $G(x)$ represents natural population dynamics of the target species and $I(x, u)$ describes technology that translates effort $u$ and the available resource stock $x$ into realized harvest. Since harvest removes biomass, Eq. (1) becomes $\Delta x(t)/\Delta t = G(x) - I(x, u)$. In the climate change problem, $G(x)$ again captures the dynamics of a target population, $x$, which, rather than a desirable biological species, is an undesirable chemical species. Because plants remove carbon and human activities add carbon via emissions, Eq. (1) becomes $\Delta x(t)/\Delta t = -G(x) + I(x, u)$. $G(x)$ depends on the atmospheric carbon concentration, $x$, which is a proxy for temperature, ocean acidification, and fertilization effects.

Both of these problems relate to (dis)investment in natural infrastructure that processes materials and energy either as a valued flow (e.g. food) or waste (e.g. atmospheric CO₂). In both cases, investment can be passive - i.e. society can simply allow them to self-regenerate. However, because regeneration may take decades or centuries, these "slow-burn" problems are challenging for humans to address given our information processing biases, i.e. underweighting low probability events and loss aversion¹², and time discounting. The contagion/invasion (infectious disease, innovation, (mis)information) problem also takes the form $\Delta x(t)/\Delta t = -G(x) + I(x, u)$ where $x$ is the number of infected people. $G(x)$ again represents an ecology of interacting species, e.g. hosts, pathogens, or ideas. Human activities, $u$, impact the spread of species though $I(x, u)$. The challenge is to manage the dynamics of 'invasive species', which tend to occur on significantly shorter time scales than the first two problems.

## Observability and controllability in natural resource management

Choosing a relevant measurement variable, hereafter denoted as $y(t)$, can be difficult and expensive for managers in terms of the knowledge and sensing infrastructure required. Control action may be irrelevant or damaging if $y(t)$ does not contain pertinent system information and, even worse, is the only available measurement (i.e. is used because it is all that is available). We illustrate some aspects of this challenge using the standard model of natural resource extraction inspired by Clark et al.⁵,¹³,¹⁴ in which $G(x) = g(x(t) - \alpha)$ ($k - x(t)$) and $I(x, u) = u(t)qx(t)$ where $g$ is the intrinsic resource growth rate, $\alpha$ is the critical depensation threshold, $k$ is the carrying capacity, $u$ is harvesting effort, and $q$ represents technological productivity (see Methods). In
this case, Eq. (1) becomes

$$ \Delta x(t)/\Delta t = g(x(t) - \alpha)(k - x(t)) - u(t)qx(t). \quad (2) $$

For clarity, we assume the simple policy goal of driving the population to a level, $x_{MSY}$, to support maximum sustainable yield (MSY) by driving effort to the corresponding level $u_{MSY}^{15}$. We illustrate three cases in which observability is restricted to a single system output: i) biomass, $x(t)$, ii) effort, $u(t)$, and iii) harvest $u(t)qx(t)$. Effort is adjusted using a PI-controller:

$$ \Delta u(t)/\Delta t = \gamma_p e(t) + \gamma_I \sum_{\tau=0}^{\tau=t} e(\tau)\Delta t \quad (3) $$

with $u(t) \in [0, 1]$ and where $e(t)$ is the error measured by the deviation of the observed outcome from the relevant target $x_{MSY}$, $u_{MSY}$, or MSY, respectively (see Methods for further details), and $\gamma_p$ and $\gamma_I$ are the ‘gains’ or rapidity of the adjustment to present error and past error history, respectively.

In Fig. 1, we only consider a P-controller ($\gamma_p = 0.05$; $\gamma_I = 0$) for the sake of clarity (the PI case is treated in Supplementary Note 1, see Supplementary Fig. 1) and analyze the controllability of the system under different observability conditions by comparing trajectories beginning at points A, B, and C. The controllability set (colored region) is defined as the set of initial states

<figure><img src="image_4.png" /><figcaption>**Fig. 1 | The impact of observability on controllability for a natural resource harvesting system.** The colored regions show the controllability sets for a natural resource extraction system for our 3 observability conditions as labeled, the color representing the time to reach the target equilibrium. The black curves with arrows are system trajectories starting from different initial conditions A-E. Trajectories starting outside the controllability set can never reach the target equilibrium indicated by the red star while those starting inside do. Equilibria are stable when biomass or effort is observable but unstable when the only the harvest is observable in the case of the MSY target Panel (c). In Panels (a-c), the goal state, shown by the red star, is the equilibrium where the biomass isocline (red curve) intersects the effort isocline (green line/curve). When biomass Panel (a) and effort Panel (b) are observable, the equilibrium is stable whereas the equilibrium is degenerate when harvest Panel (c) is observable. Starting at point D, the system very slowly converges</figcaption></figure>

that will reach the target (red star). Figure 1a shows that if only biomass is observable, the system can be controlled from A and B but not from C. Figure 1b shows that if only effort is observable, the system can be controlled from A and C but not from B. These different initial conditions should be understood as potential system states that can result due to exogenous shocks and illustrate how a manager can loose control of a system due to observability constraints. Finally, Fig. 1c shows that the system cannot be controlled from A, B, or C when regulated based on harvest.

The fisheries management literature has observed the limitations of MSY and have proposed many alternatives$^{16,17}$. One would be a stock target above $x_{MSY}$ to build resilience against environmental variability. Our analysis of a policy that aims for a stock level 50% above $x_{MSY}$ is shown in Fig. 1d. This policy is more conservative and the controllability set is significantly enlarged. Notice that there are two equilibria defined by the intersections of the harvest isocline (green curve) with the biomass-effort equilibria (red curve). Whereas decision-makers focus on the (stable) equilibrium at the red star, the second (unstable) equilibrium at the lower right is of importance because its location delimits the controllability of the system. Our results also show that stability of MSY-based policies depends on the observation from which the system is controlled. MSY-based policies using harvest-based observations involve multiple attractors whereas MSY-based policies using biomass- or effort-based observations do not.

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to the target. However, if the system exceeds this target (e.g. an effort a little bit higher than $u_{MSY}$ due to, for example, a measurement error), the system will slowly collapse because the harvest isocline is tangent with the biomass isocline. This means that from initial conditions around the target (point E for instance), even though the system will approach the equilibrium and ‘slow down’ to give the illusion that the system is stabilizing, the system will eventually collapse in the long term. Comparing Panel (d) with Panels (a and b) shows that more conservative targets are required to compensate for instabilities generated by policies based on observing output (harvest) that combine information about system states to achieve the same level of controllability when observing system states (biomass and effort) directly. Comparing Panels (c and d) to Panels (a and b) shows how good decisions made with the wrong observation can fundamentally change the dynamics of controlled systems.
Next we consider a more realistic case where effort is controlled indirectly by economic mechanisms (e.g. landing tax, license/quota fee), modeled by $a(t)$, as follows

$$ \Delta u(t)/\Delta t = \beta u(t)(1 - u(t))(px(t) - c - a(t)) \quad (4) $$

where $p$ represents the price of the harvested biomass, $c$ the fixed costs of harvest and $a(t)$ the level economic regulation ($a(0) = 4.5$). This model captures the idea that when profits are positive ($px > c$) effort enters the fishery and vice versa with $a(t)$ reducing the economic attractiveness of the resource. We implement a full PID controller and for the past errors, we suppose that decision-makers only consider data over the previous 5 years. This PID control takes the form

$$ \frac{\Delta a(t)}{\Delta t} = \gamma_p e(t) + \gamma_I \sum_{i=0}^{5} e(t - i\Delta t)\Delta t + \gamma_D \frac{\Delta e(t)}{\Delta t} \quad (5) $$

The rate of change of effort is constrained by social, technical, and behavioral factors, so we restrict $a(t)$ to $[0; 2a(0)]$.

Figure 2 summarizes the performance of this control when biomass is observable. We tested different P-, I- and D-gains to illustrate the impacts of past, present, and estimated errors on the controllability set and the time to reach the target (red star). Comparing with Fig. 1, as we might expect the controllability set is smaller for economic versus direct regulation. In the reference case (Fig. 2a) with all gains equal to 0.2, only point I is within the controllability set. Points D, H and C have a high initial biomass that induce effort increases making it more difficult to bring down later. The trajectory from D shows how the I-error is not able to correct the trajectory when the biomass is below $x_{MSY}$: despite high biomass and low effort, the control maintains high biomass for too long, slowing the correction of the trajectory

when biomass decreases. Also notable is that the heatmap (time to reach the target) has a spiral shape due to the underlying dynamics (spiral equilibrium). Decreasing the effect of past errors (I-gain, Fig. 2b) limits this effect and enlarges the controllability set, capturing points B and D. Increasing the effect of future errors (D-gain, Fig. 2c), on the other hand, accentuates the spiral dynamics, enabling control of the system from point B but not from D. Finally, increasing the effect of current errors (P-gain, Fig. 2d) also accentuates the spiral (for instance see the difference of dynamics from point I between Fig. 2d and Fig. 2a), and significantly shrinks the controllability set and increases the time to reach the target.

## Adapting control parameters in a changing world? The case of contagion processes

In addition to the strict notion of observability explored in the previous example, all systems suffer from the related issue of measurement bias. For example, Anderies et al.¹³ have analyzed the influence of bias in knowledge infrastructure systems by illustrating how non-linear systems can be very sensitive to different decision-making processes. COVID-19 is a good case in point for how policy-makers frequently must make decisions with limited observability which introduces biases in what information they choose to observe. The French Government's response, for example, prioritized real-time data on cases (a P-like policy) early in the pandemic then focused on the basic reproduction number, $R_0$ which predicts the *expected* number of cases generated by a single person (a D-like policy) combined with past behavioral responses to lockdowns to compensate for policy response delays (an I-like policy). This example demonstrates how decision-makers may use PID design processes whether or not they recognize they are doing so as described in more detail below. In this sense, PID is a natural approach for the analysis of real-world policy problems.

<figure><img src="image_6.png" /><figcaption>a - Reference case ($\gamma_P = \gamma_I = \gamma_D = 0.2$)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>b - Decrease of the past error influence ($\gamma_I = 0.04$)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>c - Increase of the future error influence ($\gamma_D = 1.5$)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>d - Increase of the current error influence ($\gamma_P = 0.6$)</figcaption></figure>

**Fig. 2 | The impact of PID parameters on controllability in the case of economic regulation.** Controllability sets (colored regions) in the case of economic regulation where biomass is the observable variable. Analogous to Fig. 1 each panel shows the target ($x_{MSY}$) as a red star and trajectories (black curves with arrows) for several representative initial conditions for the reference case Panel (a), reduced influence of the past Panel (b), increased influence of the future Panel (c) and current Panel (d)

errors on the controllability sets compared to the reference case. The colors again show the time to reach the target. Note that trajectories depend on the dynamics of $a(t)$ and thus the system is higher than two dimensions. The controllability sets are thus projected onto the biomass-effort space for sake of clarity. As a result, although it may appear that uncontrollable trajectories cross the controllability set (see point D of Panel (d) for instance), they actually do not because $a(t) \neq a(0)$.
We explore these issues further in a general model of contagion used to study the spread of disease¹⁸ and the diffusion of innovation¹⁹. We consider the standard Susceptible-Infected-Recovered (SIR) model (see Methods):

$$ \frac{\Delta S}{\Delta t} = S(t) - (R_0(t)\phi I(t)S(t) - \alpha R(t)) \quad (6) $$

$$ \frac{\Delta I}{\Delta t} = I(t) + (R_0(t)S(t) - 1)\phi I \quad (7) $$

$$ \frac{\Delta R}{\Delta t} = R(t) + (\phi I(t) - \alpha R(t)). \quad (8) $$

We suppose that the government controls $R_0$ between a perfect lockdown ($R_0(t) = 0$) and doing nothing, $R_0(t) = R_{max}$ based on disease (e.g. infectiousness, virulence) and population (social behaviors, customs, immunological health) characteristics. We use a PI controller (use of past and real-time but no future estimation). We consider the French case with 67 million people and a policy target of 3000 intensive care unit (ICU) cases. We consider two variants of contagion ($R_0$) and virulence ($v = %$ of infections ending up in ICU): 1) low-high ($R_0 = 3, v = 2$), 2) high-low ($R_0 = 6, v = 1$). PI parameters influence the speed of approach to the target (see colorbars) in Fig. 3a, b as well as the stability of the system. First, rapid convergence depends on striking a balance between using present and past information (dark blue areas tend up and to the left). Fixing the I-parameter at 1 and increasingly weighting present information, the convergence time initially decreases and then begins to increase again. This illustrates an over-reaction to the present situation. More interestingly, for $R_0 = 6$, PI-values are lower than for the $R_0 = 3$ variant overall. This is counterintuitive in that one would expect a more rapid response would be required to cope with higher contagion. However, the high-contagion variant involves fast dynamics which, when combined with fast controller dynamics (high P and I parameters) decreases overall system stability and can prevent the

stabilization of the system (see the trajectories plotted for $P = 4$ and $I = 1$). In systems with complex supply chains (e.g. health care) stability is desirable, and we thus may have to counterintuitively slow down our response to maintain it. Fig. 3c and d, illustrate the impact of contagiousness on the time required to reach the ICU target as P- and I-gains are varied. It is more efficient to have a high P-gain for lower contagion, consistent with Fig. 3a, b. However, high P-gain for high contagion destabilizes the system which may be compensated with a lower I-gain. Finally, Figure 3e shows the actual number of ICU patients in France and the government's response over time and illustrates the potential impact of changing the weights of PID controller as more information becomes available. The first lockdown (mainly a P-controller) may have extended too long. The second lockdown was shortened as estimates of $R_0$ were used (a PD-controller). The third lockdown response incorporated past data as well (a full PID controller) and the lockdown ended very early (in terms of numbers of ICU patients). This case illustrates the role of learning and new information that can be used to dynamically adjust PID controllers.

## 2024: The hottest of past years or the coolest of future years

If policy makers compare 2024 to past years or projections of future years, 2024 is the hottest or coolest, respectively. How sensitive are climate dynamics to using past data or future projections in decision-making? Recall that the atmosphere is a bucket to which $CO_2$ is added by human activities (I of Eq. (1)) and naturally removed by the carbon cycle (G of Eq. (1)). We use a simplified version of the DICE model with two state variables: atmospheric carbon $C(t)$ and the world capital stock $K(t)$ to define G and I. We assume G is a linear function of the stock, i.e. $G(C(t)) = gC(t)$ where g is a constant. I is the emission rate from the world economy as function of $K(t)$, $f(K(t))$, which can be reduced via our control variable $\mu(t)$ (See Methods). That is,

<figure><img src="image_10.png" /><figcaption>**Fig. 3 | The impact of PI parameters and contagiousness on infectious disease controllability.** Contagiousness and the capacity to reach the target number of ICU cases. Panels a and b show how the time required to reach the ICU target (represented by colors as in previous figures) depends on variant type and PI values. For combinations of P- and I-parameters outside the colored region, the system is unstable (compare insets in Panels a and b). Panel c shows the time required to reach the target for low and high contagiousness and for different values of $\gamma_p$ and $\gamma_i$. A steep increase in time to reach the target indicates when controllability is lost. In</figcaption></figure>

Panel (d), the red volume contains the set of all parameters for which the target can be reached. The points A-E are on the boundary of this parameter set and correspond to those in (c) illustrating loss of controllability. Panel e shows the number of patients in intensive care units over time (data extracted from data.gouv.fr) in blue overlain with the national lockdown policies with the red vertical lines (note that a combination of more or less restrictive measures were carried out throughout the pandemic which are not reported in this Figure).
$I(K(t), \mu) = (1 - \mu(t))f(K(t))$. This leads to our equation for $C(t)$:

$$dC(t)/dt = -g(C(t)) + (1 - \mu(t))f(K(t)). \quad (9)$$

The world capital stock $K(t)$ grows via investment, $h(K(t))$ and decays at rate $\delta$. Investment is reduced by the climate change abatement, $a(t)$, and damage, $d(t)$, costs. This leads to the dynamics of $K(t)$:

$$dK(t)/dt = \left[ \frac{1 - a(C(t))}{1 + d(C(t))} \right] h(K(t)) - \delta K(t). \quad (10)$$

(See Methods for model details). In what follows, we compare two outputs to determine $\mu(t)$: $C(t)$ (normalized error relative to a reference concentration of 350ppm) and economic costs $a(t) + d(t)$, which both depend on $C(t)$ with a goal of zero economic costs. The baseline policy simply increases the emission reduction rate by 10% every 5 years, a so-called 'open-loop' control with no feedback. In Fig. 4a, errors are normalized to 1 (errors divided by the maximum error in the 2010-2100 period) to see when they reach their maximum (the larger the error, the stronger the policy feedback response). The derivative error is higher early on for both outputs (solid = CO₂), dashed = economic) which means that policies based on current and past information will react more slowly. While it may seem obvious that acting on how the system is *changing* rather than how it is now (P) or has been (I) would induce a more rapid response, this is not always the case. This depends on the details of the problem and the initial conditions.

Next, all CO₂-based errors (see stars in Fig. 4a)) reach their maximum earlier than the economic-based errors (see diamonds in Fig. 4a). If policy-makers focus on CO₂ concentration, the system will react sooner because the D-error is significant very early. Focusing on economic indicators makes mitigating climate change difficult because the maximum cost (a trade-off between abatement costs and damages) occurs very late. To illustrate, we tested different values of P-, I- and D-gains for both outputs. Better results are obtained by emphasizing D-gain (see Supplementary Note 2 and

Supplementary Table 1 for more details), illustrating the importance of future estimation of the future in 'slow burn' problems. Best policies for each output are plotted in Fig. 4b. CO₂-based policies enable society to mitigate climate change but involve higher costs early on. The economic-based policy mitigates climate change too late because damages are near zero early on and generate no signal to act. At the same time, abatement costs are high, disincentivizing action. Discount rates are central to this trade-off between bearing costs early or later. Even very low discount rates attach very little weight to future events. Carbon-based policies, on the other hand, generate a clearer signal and force action early with its attendant abatement cost. Note that this result depends critically on the form of the damage cost function in the DICE model which is quite sensitive to parameter value assumptions.

Finally, we address the question of reachability in Fig. 4c, d: from the 2010 initial state of the world, what are the reachable global states over time? We tested one million of combinations of P-, I- and D- gains in order to assess all reachable states in the case of CO₂-based (Fig. 4c) and economic-based (Fig. 4d) error. At a glance, the reachable set (red area) based on CO₂-error allows for a lower value of CO₂ concentration in the long run. Another key observation is the shape of these sets: whereas the CO₂-based set includes a wide range of CO₂ concentration from 2020 on, the economic-based set is very narrow until 2040. This means that whatever the P-, I- and D-gain combination, the CO₂ concentration will be nearly the same. This illustrates the weak effect of economic-based climate policies.

## Discussion

The analyses of these core sustainability problems using the PID paradigm highlights the fact that SETS dynamics can be very sensitive to information processing in feedback systems. While fully acknowledging that information is processed by a diverse set actors embedded in complex, evolving social contexts, our aim was to show in an intuitive way how general PID insights may help inform real-time policy design and implementation processes. Our results emphasize the importance of information by showing that the

<figure><img src="image_11.png" /><figcaption>Fig. 4 | CO₂- vs economic-based incentives for mitigating climate change. Panel a PID errors for the reference policy illustrating the relative importance of errors at different points in time. Panel b Carbon and cost trajectories for different policies compared to reference case. Reachable sets (in red) from the 2010 initial state for</figcaption></figure>

different values of PID-gains (see methods) according to CO₂-based error (Panel c) or economic-based error (Panel d). Black and green curves show the bounds of the PID combinations explored.
same controller, expressed by PID parameters, may lead to different system stability characteristics depending on the observation on which the controller is based. This key principle is suggestive for addressing environmental problems: instead of focusing solely on the controller, focusing on observations that influence the controller may provide significant insights in terms of efficiency and stability of environmental policies.

The resource extraction and climate change cases illustrate the differences between controlling systems based on actual system states (biomass, CO₂) versus process outcome states (harvest, economic damages). The latter is used as proxy and combines system states and creates problems for control. In principle, controlling the system using the same variables as those used to define the goal may lead to better performance. In practice, however, this is often not possible. In the case of a fishery, it may be very difficult to measure the fishery biomass (the goal state) while measuring harvest may be easier. This is the case in many situations where policy actors must work with what they can measure (what is observable) rather than what would be best to measure. Our analysis illustrates that in the case of resource extraction, controlling based on process outcomes (harvest) leads to instability; the goal state cannot be maintained in the face of perturbations and the 'slowness' of the system near the goal sends misleading signals. With climate change, process outcomes entrain values, i.e. subjective measures of the costs and benefits of various states. Controlling based on economic outcomes is very sensitive to assumptions about these values and problems of how they are distributed over time. In the case of climate change where the goal state is observable (atmospheric carbon concentration) our analysis points to emphasizing policy controls that more strongly emphasize carbon targets rather than, e.g., the social cost of carbon in collective decision-making processes that must navigate the tension between cutting emissions and economic performance.

However, recommendations to prioritize biophysical or social indicators depend heavily on the dynamics of the respective systems. Beyond observability, the effectiveness of such indicators also hinges on system inertia and the ways in which biophysical and social dynamics respond and interact with each other. Such interactions can produce unexpected outcomes, even when the indicator is aligned with the objective, as illustrated in Fig. 1. Managing with the goal and control variable both based on harvest (Fig. 1c) yields the smallest controllability set as compared to when the goal and control variable are both based on biomass (Fig. 1a) or effort (Fig. 1b), respectively. This is because the MSY lies on an unstable equilibrium and it is difficult to reach and maintain an unstable state. On the other hand, Fig. 1d demonstrates that a lower stable target expands the controllability set considerably, showing that dynamical properties of the system, such as the presence of stable/unstable states, impacts the performance of management actions much more than the alignment of the goal with the control variable. Finally, Fig. 1 demonstrates that no observable indicator—regardless of how well it aligns with the objective—can enable control of the system from all points A, B, and C. The contagion example further highlights these subtleties by illustrating the importance of how we assign relative weights to past or present information or estimates of the future while managing non-linear dynamical systems. Taken together, these results highlight the importance of considering the complex interplay between observability and the goals set by decision-makers.

In general, it is extremely difficult to apply control theory to real-world SES problems. The uncertainty and complexity are too high for case-specific, customized controller design. However, the PID control paradigm, because it is quite general, relatively simple, and maps very naturally onto real-world policy practice as shown in the COVID case does provide a starting point for the development of solutions. Because PID design is based on trial and error and experience, we cannot say for a given class of SESs what the best balance of P, I, and D control action is as we can't typically run experimental tests on SESs. However, industry experience does provide some guidance. P-controllers provide fast response if steady-state error is acceptable. I-controllers are important when eliminating steady-state error is a priority but can generate overshoot. D-controllers are useful when reducing overshoot and system stability is important. PID controllers are

used to balance requirements for speed, accuracy, and stability. These principles, along with the recognition of the importance of the link between observability and controllability demonstrated here, enable policy actors to consider acceptable conditions for reaching an objective (e.g. is overshoot acceptable?), assess the weights placed on past, present, and future information, and reflect deeply on what they can observe and what must be observed to reach that objective. Thus, a key contribution of this paper is to provide a framing for policy actors to better understand the subtleties of information processing in feedback systems that critically impact the ability to manage systems in real time based on observations.

Finally, in our framing of the SOS in terms of controllability sets, we highlighted the mathematical generality of the problems SETS must navigate. There are, however, critical differences in the flavor of control engineering and sustainability science problems. In control engineering from which the PID paradigm we have applied originates, the main challenge is to control high-dimensional problems that are well understood in terms of materials, dynamics, and sensors and deal with what we might refer to as *observable and controllable complexity*. Sustainability science, on the other hand, must deal with highly non-linear, deeply intertwined designed and living systems, i.e. SETS, for which capacity to observe and control their internal dynamics is much more limited. Nonetheless, ideas from modern control engineering can be applied in to derive general principles regarding how observability impacts the capacity of SETS to generate and maintain a SOS.

## Methods

Our approach builds on an extensive literature on the application of control theory to SETS interactions. These applications come in two broad categories, *optimal control* and *modern control engineering*. The former has a longer history in sustainability problems and is more common in economics. This is partly due to the fact that optimal control is based on the notion of finding a path through state space that maximizes or minimizes some global quantity which has a very natural analogue in economics and sustainability, e.g. finding a consumption/investment path over time that maximizes the social value of a set of capital stocks$^{20,21}$. The stock of interest may be related to the built environment$^{4,22}$, a pollutant$^{23-25}$, atmospheric CO$_2$, a fish, groundwater, forest, or any natural resource stock$^{5,8,26-28}$, or any combination, e.g. refs. 8,29–32. Further, the co-state variables that arise naturally in optimal control capture the cost of violating constraints and have a clear and powerful economic interpretation as the shadow price/ cost of deviating from the optimal stock levels. For instance, the DICE model⁸ characterizes carbon abatement paths using the social cost of carbon.

Despite the appealing analogues between optimal control and societal development pathways, optimal control is limited by stringent observability and controllability demands that cannot be realized in practice and should be seen a useful tool to explore what might be the best thing to do in ideal circumstances. Given our focus on operating in circumstances far from the ideal and on maintaining a system in a SOS rather than system optimization, we turn to tools more well-suited to this task from modern control engineering. A range of tools and techniques from modern control engineering, i.e. robust control$^{33,34}$, PID control methodologies$^{13}$, multi-objective and stochastic optimization$^{35-38}$ have been applied to a range of sustainability-related problems. A classical modern control engineering problem focuses on controlling a system from outputs without relying on a perfect understanding of the underlying dynamics. Rather than build models from the ground up, engineers have developed methods to extract essential information by considering the managed system as a “black box” and using sensors to measure system behavior (outputs) to estimate a model of the internal structure of the system based on the relationship between inputs and outputs. This approach relies on the extent to which outcomes can be observed (*observability*) and the extent to which controllable inputs can impact outcomes (*controllability*)$^{39}$. Work in this area has progressed through the co-evolution of theory and experimental testing$^{39,40}$. Unfortunately, repeated experiments and learning from failures in sustainability contexts is not practical (you can only do one experiment on species
extinction). However, general insights from engineering regarding managing systems with affordable information and practical constraints can be leveraged⁴¹.

## The PID control paradigm

Our contribution relies on taking the complimentary approach of applying the general, intuitive PID control paradigm⁴² at a more conceptual level to explore the core conceptualization of sustainability itself using the notion of controllability sets and to try to understand the capacity and range of real-world policy action. The concept of controllability refers to the capacity to control an initial state towards a given objective. We show how what is observed and how observations are weighted in time (PID parameters) may change the structure of the controlled model, the structure of equilibria, and the capacity of interventions to reach given objectives. We also extend the notion of controllability set which is seldom addressed in the literature as most studies address controllability from a single initial state. The controllability set naturally captures the notion of the safe operating space and allows us to incorporate a rich range of sustainability ideas such as inequality, justice, etc. How would, for example, addressing inequality affect a controllability set? What are the trade-offs in achieving sustainability goals in terms of changes in the controllability set? This work generalizes³³ which explored these questions in terms of robustness-sensitivity trade-offs for a specific case with specific uncertainties. Our main contribution lies in connecting observability to the dynamical properties of the controlled system and, in turn, the controllability properties of the system as a potentially useful way to formally conceptualize sustainability.

Let's consider the general model described in Equation 1: if $x(t)$ is the stock level at time $t$, the general mathematical representation is

$$ \Delta x(t)/\Delta t = G(x(t), t) + I(x(t), u(t), t) \quad (11) $$

where $G(x, t)$ (growth) and $I(x, u, t)$ (impact) represent the natural and human induced net replenishment rates, respectively. Regardless of the complexity of these functions, whether non-linear, periodic, or uncertain, control remains a basic process of managing inflows and outflows to control one stock ($x \in \mathbb{R}$) or many interacting stocks ($x \in \mathbb{R}^n$). We denote $y(t)$ as the output that is observable:

$$ y(t) = H(x(t)) \quad (12) $$

This variable $y(t)$ is then used for defining control $u(t)$ as follows:

$$ u(t) = F(y(t), t) \quad (13) $$

Therefore, control $u(t)$ depends on the observable output $y(t)$ and the F-function. For the F-function, we choose a proportional-integral-derivative (PID). For this purpose, we consider a target $y^*$ to reach. We define the error $e(t)$ as the difference of the actual observable state of the system $y(t)$ with this target $y^*$, meaning that $e(t) = y(t) - y^*$. Then $u(t)$ is defined as follows:

$$ u(t) = \gamma_p e(t) + \gamma_I \int e(t) + \gamma_D \frac{de}{dt} \quad (14) $$

In the following examples, we use a discrete approximation of system dynamics and PID controller dynamics (i.e. $\int$ becomes $\sum$ and $\frac{de}{dt}$ becomes $\frac{\Delta e}{\Delta t}$). We investigate how adaptively changing relative weights given to present, past, and future (estimates) information (i.e. the PID control parameters) impacts three types of models described hereafter: the standard renewable natural resource problem⁵; the management of infectious disease outbreaks using the modified SIR model⁴³ with decaying immunity; and an adapted version of the DICE model⁴⁴ to explore challenges with applying D-type controllers (future estimation of system states) to climate change. In all cases, we run simulations in order to calculate the time required for reaching the objective from different initial states. This time is represented by a colorbar.

## Natural resource extraction model

The most commonly used logistic growth model is given by $G(x) = gx(t)$ ($k - x(t)$). The parameter $k$ is the maximum population biomass that can be supported by the underlying resource. When $x$ is small, the population is not resource limited and grows exponentially at rate $gk$ where $g$ is the intrinsic growth rate of the species normalized in units of $k$. When $x$ approaches $k$, resource constraints drive the growth rate to zero and the population stabilizes at $k$. This is a very robust ecosystem in that any arbitrarily small population can recover to $k$. Many populations exhibit critical depensation; the existence of a minimum population level $\alpha$ below which the population cannot recover due to factors such as the difficulty of finding mates. Incorporating this feature leaves us with $G(x) = g(x(t) - \alpha)(k - x(t))$. Resource harvest is typically modeled as $I(x, u) = u(t)qx(t)$ where $u$ is harvesting effort, e.g. harvest hours per day, $q$ represents technological productivity, e.g. biomass harvested per harvest hour per harvestable biomass $x$. These common assumptions lead to the discrete-time representation of (1):

$$ \Delta x(t)/\Delta t = g(x(t) - \alpha)(k - x(t)) - u(t)qx(t). \quad (15) $$

In our analysis, we set $k = 4$, $g = 0.25$, $\alpha = 0.25$ and $q = 1$. For equation 4, we set $\beta = 0.05$, $p = 4.5$ and $c = 1.5$. $\Delta t$ is here equal to one year. We suppose policy makers can observe either the biomass $x(t)$ (i.e. they adapt their policy according to $e(t) = (x(t) - x_{MSY})/x_{MSY}$ or the effort $u(t)$ (i.e. they adapt their policy according to $e(t) = (u_{MSY} - u(t))/u_{MSY}$) or the total harvest ($e(t) = (MSY - u(t)x(t))/MSY$). PID controls are defined by Equations 3 and 5 in the main text.

## Contagion model

We consider two variants: A, less contagious and more severe ($R_0(0) = 3$; $\phi = 0.1$; $\alpha = 0.1$) and B, more contagious and less severe ($R_0(0) = 6$; $\phi = 0.1$; $\alpha = 0.1$), with $R_{max} = R_0(0)$. The error function $e(t)$ is $e(t) = I_{ICU}/3000 - 1$ and $R_0(t) = R_0(0)/(1 - u(t))$, with $u(t) = \gamma_p e(t) + \gamma_I \sum_{\tau=0}^{\tau=t} e(\tau)$, $u(t) \in [0, 1]$, and $I_{ICU}$ is the number of infected individuals in ICU.

## Climate model

We simplified the DICE-2010 model by: 1) having a constant contribution of oceans to atmospheric CO2 (70 Gt of carbon/5 years, i.e. the mean contribution in DICE 2010); and 2) having economic damages based on CO2 concentration rather than temperature. Temperature dynamics with radiative forces are then removed. We consider that +100ppm-damages are equal to +1degree-damages. This enables us to capture the main dynamics with only two state variables, the atmospheric carbon concentration $C(t)$ as well as the world capital stock $K(t)$, as follows:

$$ K(t + \Delta t) = 1.25 \frac{1 - a(t)}{1 + d(t)} p(t) K(t)^{0.3} l(t)^{0.7} + 0.95 K(t) \quad (16) $$

$$ C(t) = [1 - \mu(t)] p(t) K(t)^{0.3} l(t)^{0.7} + 0.9120 C(t - \Delta t) + 70 \quad (17) $$

$\Delta t = 5$ years and parameters are set according to the DICE-2010 model. The abatement cost $a(t)$ is $a(t) = bp(t)^*s(t)/2800\mu(t)^{2.8}$. $bp(t)$ is the backstop price given by $bp(t) = 0.975bp(t - \Delta t)$ with $bp(2010) = 344$. $s(t)$ is the baseline carbon intensity $s(t) = s(t - \Delta t)\exp^{5gs(t)}$ with $gs(t + \Delta t) = 0.9995gs(t)$ ($s(2010) = 0.489$; $gs(2015) = -0.01$). The damage cost $d(t)$ is $d(t) = 0.0027(C(t - \Delta t)/213 - 3)^2$ with 300ppm being the concentration reference for damages. $l(t)$ is the world population dynamics (UN medium scenario). $p(t)$ is the total productivity factor given by $p(t) = (p(t - \Delta t))/(1 - gp(t))$ with $gp(t) = 0.079\exp^{-0.030(t-2010)}$ and $p(2010) = 3.79762$. The error is defined either by: $e(t) = (C(t) - 350)$ in ppm for CO2-based incentives or $e(t) = a(t) + d(t)$ for economic-based incentives. Then, the control law becomes $\mu(t) = \gamma_p e(t) + \gamma_I \sum_{\tau=0}^{\tau=t} e(\tau) + \gamma_D(e(t) - e(t - \Delta t))/\Delta t$.
## Data availability

All model output generated or analyzed during this study is included in this published article and its Supplementary Information file and the code to generate and analyze the model output is available as indicated below.

## Code availability

The code is available at https://github.com/jdmathias/PID. Running this code will require Matlab. The code is also provided at codeocean (https://doi.org/10.24433/CO.9393130.v1) to allow all results of the paper to be reproduced without installing Matlab software.

Received: 11 April 2025; Accepted: 4 September 2025;

Published online: 10 October 2025

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## Acknowledgements

J.D.M. thanks the French National Research Agency (project VIRGO, ANR-16-CE03-0003-01 and project MATRES, ANR-24-CE03-1092) for their financial support.
**Author contributions**

J.M.A. and J.D.M. conceived of the study and contributed to the theoretical conceptualization and theoretical model development. J.D.M. led the implementation of the numerical model, performed the simulations, and produced the summary figures. J.M.A. and J.D.M. interpreted the results of the model analysis. J.M.A. and J.D.M. wrote the manuscript.

**Competing interests**

The authors declare no competing interests.

**Additional information**

**Supplementary information** The online version contains supplementary material available at

https://doi.org/10.1038/s43247-025-02767-3.

**Correspondence** and requests for materials should be addressed to
John M. Anderies.

**Peer review information** *Communications Earth & Environment* thanks the anonymous reviewers for their contribution to the peer review of this work.

Primary Handling Editors: Yann Benetreau. A peer review file is available.

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