One of the most difficult problems in forecasting the effects of the COVID-19 pandemic is to generate reliable signals for local governments in the presence of highly uncertain data and model parameters. To resolve this, in the present work a modelfree approach is taken, which is able to generate reliable signals for social distancing measures (or against them) based only on the available data of COVID-19 patients showing moderate to severe symptoms (and independent of possibly asymptomatic infected or patients with mild symptoms). The number of those symptomatic infected, who require medical attention, can typically be measured accurately.
It is well documented [6, 10] that social distancing measures help to reduce infection rates and have an effect on the containment of the spread of SARS-CoV-2. On the other hand, social distancing has negative effects on both the economy and the mental and emotional health of the population. Therefore, governments face the hard decision of when to enact social distancing and when to relax the measures. The present paper may serve to provide a decision making mechanism based on a model-free feedback control design.
An advantage of a model-free approach (compared to model-based techniques such as MPC (model-predictive control) used in [7, 11] for instance) is that it is inherently robust and arbitrarily scalable. The latter means that, because it is independent of quantities such as the infection, recovery or death rates, the control approach can be applied to any country or region or city, without the need to identify the specific parameters for this region (which is data that is mostly not available).
The above described question of when to enact social distancing measures or even a lockdown is a typical control theoretic question. Modelling this question utilizing a binary control input (i.e., with values in {0, 1}) a suitable feedback controller is able to generate the required signals. To this end, we combine a widely used model for the description of the COVID-19 pandemic -a version of an SEIR (Susceptible-Exposed asymptomatic-Infected symptomatic-Recovered) model from [12] -with a control component proposed in [11] . The latter adds additional dynamics to the model which account for the effects of social distancing policies represented by the value of the control input and the response of the population to them (paying heed to possible delays). The control objective is to keep the number of infected with moderate to severe symptoms (which typically require hospitalization) below a threshold defined by the number of available ICU (intensive care unit) beds. To achieve this we exploit the bang-bang funnel controller developed in [9] , which is able to guarantee error margins in tracking problems and the control input switches between only two values.
Funnel control proved an appropriate tool in several applications such as temperature control of chemical reactor models [5] , control of industrial servo-systems [4] and underactuated multibody systems [1] , voltage and current control of electrical circuits [3] , DC-link power flow control [13] and adaptive cruise control [2] .
2. Epidemiological model for the COVID-19 pandemic. As suggested in [12] we use a version of an SEIR model for the dynamics of the COVID-19 pandemic. This so called SIRASD (Susceptible-Infected-Recovered-Asymptomatic-Symptomatic-Deceased) model has been modified in [11] in order to account for possible social distancing policies. The resulting epidemiological model has the following dynamics:
with the gain function
.
In (2.1) the total population of a considered region is split into the following classes:
• susceptible individuals S(t),
• infected but asymptomatic individuals I A (t),
• infected and symptomatic individuals I S (t),
• recovered individuals R(t),
• deceased individuals D(t) (due to the disease). It is easily seen that the derivative of the sum of the above quantities is zero,
for all t ≥ 0, thus it stays constant over time and we may define the initial population (assuming D(0) = 0) by
The other parameters used in (2.1) are summarized in Table 2 .1 and we emphasize that α A , α S , β A , β S , ρ, p, γ 0 , γ 1 ,ψ ∈ [0, 1]. Note that, as in [11] , we assume that the class of asymptomatic infected also includes those with only mild symptoms, which typically do not seek medical attention and are hence often not registered as infected. The class of symptomatic infected includes those with moderate to severe symptoms which may require ICU beds and are registered by local authorities -thus, I S (t) is typically accurately measured.
The last equation in (2.1) models the dynamics of social distancing policies and contains additional parameters to be explained in due course. As introduced in [11] , the function ψ can be seen as a time-varying population response which decreases the transmission coefficients β A and β S in the case that social distancing measures are in place. Possible delays in the response are modelled by the parameters γ 0 , γ 1 . Note that we have ψ(t) ∈ [0, 1] for all Parameter Epidemiological meaning β A , β S transmission coefficients (contact or infection rates) for an asymptomatic or symptomatic individual to transmit the disease to a susceptible individual, resp. α A , α S recovery rates for asymptomatic and symptomatic infected, resp. p proportion of individuals who develop symptoms ρ probability of a symptomatic infected individual to die from the disease before recovering γ 0 , γ 1 settling-time parameters used to determine the average time of the population responsē ψ parameter which determines the strictest possible isolation γ K positive gain coefficient t ≥ 0, where ψ(t) = 1 stands for the case of no distancing at all and ψ(t) = 0 would mean that any contacts between people are suppressed. Of course, the latter case is unachievable in practice, which is accounted for in the model. The control input u is assumed to take only binary values, i.e., u(t) ∈ {0, 1} for all t ≥ 0. This control signal models the policy enacted by the government, where u(t) = 0 means that no isolation measures are in place, and u(t) = 1 means that the government has determined social distancing. The dynamics with which these measures influence the response of the population are modelled by the last equation in (2.1). Here, the value of the parameterψ ∈ (0, 1), see Table 2 .1, is additionally influenced by a gain function K ψ defined in (2.2); the value of K ψ (t) decreases as the proportion of asymptomatic infected individuals increases. Assuming that initially ψ(0) = 1 (no isolation) we may infer that
For further details on the model (2.1) we refer to [11, 12] . 3. Control objective and controller design. The objective is to determine a control input signal u : R ≥0 → {0, 1} which guarantees that the number of available ICU beds is not exceeded. This control signal may serve as an orientation for local governments whether and when to enact social distancing measures. Since the class of symptomatic infected individuals encompasses those which may require ICU beds we seek to keep I S (t) below the number n ICU of available ICU beds, with tolerance ξ ∈ [0, 1] accounting for symptomatic infected which do not require intensive care. In other words, the aim is to achieve
In order to obtain a feedback control law which is able to guarantee the above transient behavior, we exploit the idea of bang-bang funnel control from [9] . "Bang-bang" means that the control input switches between only two different values, hence this technique is suitable for our purposes. We stress that system (2.1) does not belong to the class of systems investigated in [9] and hence feasibility of bang-bang funnel control needs to be investigated separately. However, due to the special structure of (2.1) a simpler control law is possible here. To be precise, we consider the controller
where ϕ + , ϕ − , ε + , ε − are non-negative parameters satisfying ϕ − + ε − < ϕ + − ε + and by u(t−) we denote the left limit u(t−) = lim h 0 u(t − h) of the piecewise constant function u at t. The controller is initialized by u(0−) = 0.
In [9] ϕ + and ϕ − are time-varying functions which determine a performance funnel for a certain signal to evolve in and ε + , ε − are "safety distances" required to guarantee that the signal evolves within this funnel. In the present paper, the aim is to achieve I S (t) ∈ [ϕ − , ϕ + ], thus, in view of the above control objective, we may choose
We emphasize that the feedback control strategy (3.2) only requires the measurement of the number I S (t) of symptomatic infected individuals at time t, which, as mentioned in Section 2, can be measured accurately.
In the following we identify conditions on parameters in Table 2 .1 and the "safety distances" ε + , ε − so that, for any admissible initial conditions the closed-loop system (2.1), (3.2) has a global and bounded solution which satisfies (3.1). Moreover, the control input u should have only a finite number of jumps in each compact set. Solutions are considered in the sense of Carathéodory, i.e., they are assumed to be locally absolutely continuous and satisfy the differential equation almost everywhere. A solution is said to be maximal, if it has no right extension that is also a solution.
4. Main result. In order to show feasibility of the controller (3.2) we require a set of assumptions on the parameters of (2.1). In essence, these assumptions mean that by maintaining a strict a lockdown, i.e., u(t) = 1 for t ≥ 0, it is possible to guarantee (3.1). If this is not possible, then no switching strategy can be successful.
We like to note that (A1) and (A2) are typically satisfied in real epidemiological scenarios, see also Section 5. The values S 0 and R 0 in (A3) are initial values for S(0) = S 0 and R(0) = R 0 , and both are assumed to be positive to avoid technicalities and keep the proof simple. In fact, the assumption R(0) > 0 is true for the COVID-19 pandemic in practically every country in the world and nearly every region by the date this article is written (and taken as t = 0). Apart from that, assumptions (A3) and (A4) are quite conservative since they are designed for worst-case scenarios, and hence they may impose hard restrictions in practice. However, the controller (3.2) may even be feasible if these assumptions are not satisfied and appropriate values for ε + and ε − can be identified by simulations. We may also observe that (A1)-(A4) do not impose any obvious restrictions on ε − , but ε + needs to be large enough by (A4). The bounds on ε − in (A5) only repeat the assumptions made in Section 3. In the following main result of the present paper we prove that, under assumptions (A1)-(A5), the application of the bang-bang control law (3.2) to the epidemiological model (2.1) guarantees the bound (3.1).
Theorem 4.1. Consider system (2.1) and assume that (A1)-(A5) hold for some ε + , ε − , S 0 and R 0 . Let the initial values be given such that
and, additionally,
Then the controller (3.2) with (3.3) applied to (2.1) leads to a closed-loop system which has a global and bounded solution (S, I A , I S , R, D, ψ) : R ≥0 → (R ≥0 ) 6 such thatİ S is bounded, (3.1) holds and u defined by (3.2) has locally finitely many jumps.
Proof. We divide the proof into several steps.
Step 1 : We show the existence of a local solution. In fact, repeating the arguments of [8, Cor. 3.3] Step 2 : We show thatİ S is bounded on [0, ω). This is a consequence of (2.3) and
Step 1, by whicḣ
Step 3 : We show that u has only finitely many jumps in each interval [a, b) ⊆ [0, ω). Seeking a contradiction assume that there exists an interval containing infinitely many jumps. Then there exist sequences (t n ), (s n ) ⊆ [a, b) with t n < s n < t n+1 for all n ∈ N and, without loss of generality, t n → b for n → ∞ (otherwise we choose a smaller interval) such that u(t) = 1 for all t ∈ [t n , s n ) and u(t) = 0 for all t ∈ [s n , t n+1 ) and all n ∈ N. Then (3.2) implies that I S (t n ) = ϕ + − ε + > ε − = I s (s n ). Therefore, by the mean value theorem there exists λ n ∈ (t n , s n ) such thaṫ
which contradicts boundedness ofİ S from Step 2.
Step 4 : We show that ω = ∞. Since (S, I A , I S , R, D, ψ) is bounded as S(t) + I A (t) + I S (t) + R(t) + D(t) = N and ψ(t) ≤ 1 for all t ∈ [0, ω), the case ω < ∞ is only possible when the jumps in u accumulate for t → ω, but this is excluded by Step 3.
Step 5 : We show that I A (t) ≥ 1−p p I S (t) for all t ≥ 0. To this end, set z(t)
for all t ≥ 0, and hence Grönwall's lemma implies z(t) ≥ e −α A t z(0). Therefore, we have
where the last inequality follows from the assumption (4.1).
Step 6 : We show that S(t) ≥ S min (S 0 , R 0 ) for all t ≥ 0, where S min (S 0 , R 0 ) is as defined in (A3). We find that, invoking N − D(t) ≥ R(t) ≥ R 0 ,
for all t ≥ 0. Then Grönwall's lemma implies that, for all t ≥ 0,
Step 7 : It remains to show (3.1). To this end, let t 0 ≥ 0 be such that I S (t 0 ) = ϕ + − ε + and
where t 1 = ∞ is possible, if the above set is empty. We show that I S (t) ≤ ϕ + for all t ∈ [t 0 , t 1 ), which in view of I S (0) ≤ ϕ + − ε + proves the claim.
First, we show thatÏ S (t) ≤ −M ε − for all t ∈ [t 0 , t 1 ) for M > 0 as defined in (A3). Observe that by (3.2) we have u(t) = 1 for all t ∈ [t 0 , t 1 ) and henceψ(t) = γ 1 K ψ (t)ψ − ψ(t) < 0. Furthermore, we calculatë
for all t ∈ [t 0 , t 1 ). Next we aim to estimate the last term in the brackets above, that is
Step 5, (A1)
Invoking (A2) and the fact that I S (t) > ε − for all t ∈ [t 0 , t 1 ) we may further estimate the above term by
where ζ was defined in (A3). With this we find that, invoking ψ(t) > K ψ (t)ψ,
Step 5, (A1), (A3)
Step 2
Define the polynomial q(t) := −ε + + pβ A N (t − t 0 ) − 1 2 M ε − (t − t 0 ) 2 and observe that it attains its maximum at τ = t 0 + pβ A N M ε − , hence
for all t ∈ [t 0 , t 1 ). This finishes the proof of the theorem.
We like to note that in Theorem 4.1 it is assumed that the initial value I S (0) lies within a certain interval. If the upper bound I S (0) ≤ ϕ + − ε + does not hold, then even a hard lockdown with u(t) = 1 for t ≥ 0 may not be sufficient to guarantee (3.1). Furthermore, in (4.1) it is assumed that I A (0) ≥ 1−p p I S (0) which is not a restrictive assumption since the number of asymptomatic infected is typically much larger than the number of symptomatic infected individuals; (4.1) is always satisfied, if I S (0) = 0, which is typically the case at the beginning of a (local) outbreak.
Simulations. In this section we illustrate our findings by a simulation of the epidemiological model (2.1) under the feedback control law (3.2) . For the simulation we use the parameters that have been identified in [11] for the example of Brazil. Since it is not the purpose of the present paper to model the COVID-19 pandemic for a specific region, but to demonstrate that a model-free approach works independent of the model parameters, we simply use the parameters from [11] which are summarized in Table 5 .1. i.e., we have ten asymptomatic infected, who, for instance, caught the disease during a stay in another region. Furthermore, we assume that 15% of the population already developed immunity due to a prior disease and hence belong to the class of recovered individuals. This number may be reasonable for the COVID-19 pandemic (at least in some regions in Germany) as e.g. the COVID-19 Case-Cluster-Study [14] (Heinsberg study) suggests.
The number of available ICU beds is based on the capacity in Germany, which are 32637 beds (based on data from de.statista.com, July 21, 2020) for a population of about 8.3 · 10 7 . Preserving this ratio, we obtain for our example of N inhabitants a number of n ICU = 40 (rounded to the next integer) ICU beds. As tolerance we choose ξ = 0.1 and we recall that ϕ + = (1 + ξ)n ICU is the threshold that the number I S of symptomatic infected should not exceed, cf. (3.1) . For purposes of comparison, Fig. 5.1 shows the simulation results in the case that the government takes no action and hence u(t) = 0 for all t ≥ 0. It can clearly be seen that the number I S (t) quickly increases above the available ICU capacity and the number of deaths rises dramatically; note that the effect of the exceeded ICU capacity is not considered in the model and hence the number of deaths might even be much higher.
In contrast to this, Fig. 5 .2 shows the same scenario but with social distancing measures enacted according to the control (3.2) . For this simulation we have chosen ε + = 2ξ · n ICU , which is able to guarantee (3.1). Furthermore, we have chosen two different values for ε − , namely ε − 1 = 1.75ξ · n ICU and ε − 2 = 3ξ · n ICU . Similar to various studies before, the simulations depicted in Fig. 5.2 show that social distancing measures are capable of reducing the total number of infected individuals and, as a consequence, the total number of disease induced deaths. The feedback controller (3.2) is able to guarantee (3.1) as shown in Fig. 5.2 (a) . It can be seen that, while periods with u(t) = 1 ensure that the ICU capacity is not exceeded, periods with u(t) = 0 (no social distancing) result in quick increases of infection numbers and the switch in the controller (3.2) is triggered again after only a short period of time (at most a few weeks). This shows that lifting social distancing measures over a longer period of time will not be feasible until the end of the pandemic. As shown in Fig. 5 .2 (d) the control input u 2 corresponding to ε − 2 exhibits a faster switching pattern than the input u 1 corresponding to ε − 1 . Several other simulations show that larger values of ε − lead to a faster switching with shorter periods between the switches, but, on the other hand, the pandemic is defeated at an earlier time point (i.e., the time T > 0 for which u(t) = 0 for all t ≥ T can be made smaller the larger ε − is chosen). These are two conflicting objectives (few switches vs. shorter pandemic) and the government has to decide which should be favored; the controller design parameters may be adjusted accordingly.
Another observation reveals that the total number of deaths, i.e., D max = lim t→∞ D(t) depends on the choice of ε − . Since minimizing D max seems a reasonable goal we have performed a set of simulations so that this is achieved, which led to the above value of ε − = ε − 1 . As shown in Fig. 5 .2 (c) the number of deaths is indeed higher for ε − 2 and, as shown in Fig. 5.2 (b) , the total number of infected (represented by N − R(0) − S(t)) is larger for ε − 2 .
Finally, by way of comparison, we like to note that effective control methods for the COVID-19 pandemic based on model-predictive control (MPC) have been developed in [7, 11] . However, MPC requires accurate model data and measurements of all state variables for feasibility. As shown in [7] , uncertainty in the data and measurements can be compensated to a certain extent by using e.g. interval predictions, however it is not possible to prove a priori that MPC does not exceed the available ICU capacity. This is one of the advantages of the approach presented here.
We have presented a novel feedback controller which may serve as a decision making mechanism for governments during the COVID-19 pandemic. The controller is based on the bang-bang funnel controller from [9] and hence model-free, robust and arbitrarily scalable. The controller does not require any parameters of the underlying epidemiological model, but only the measurement of the number of individuals with moderate to severe symptoms, and it is able to keep this number below a threshold determined by the ICU capacity at any time.
Simulations illustrate that the proposed controller (3.2) is able to achieve the control objective and that a relaxation of social distancing policies may quickly lead to increasing infection numbers. Although a relaxation of the distancing measures over a period of only 1-2 weeks may seem pointless, the simulations suggest that a temporary increase of infection numbers, while preventing that the ICU capacity is exceeded, ensures a less extended time course of the COVID-19 pandemic compared to a strict lockdown. At the same time this allows a resumption of social and economic activities during these periods. By a continuous improvement of the available data and the simulations it should be possible to obtain better forecasts of when the input signal switches, which would allow the people to prepare for possible measures or relaxation.
Although the results presented in this work are quite promising regarding automated signals for social distancing measures, this is only a first step towards a universal technique. Future research needs to focus on methods which incorporate different levels of social distancing measures compared to only the two levels (full measures or no measures at all) considered in the present paper. A balanced use of such regulations is more practicable and will be accepted by a wider public. Typical examples are cancelation of big events, carrying face masks in supermarkets and public transport or working from home when possible; these measures may be included in the model by additional control values u i ∈ (0, 1) as suggested e.g. in [7] .
Another topic of future research is the combination of different models for different countries or regions, where different control values (due to government policies) are active. In particular, it needs to be investigated how the interactions between different regions, based on migration flows, influence the spread of the disease. Such an approach will possibly reveal which social distancing measures must be taken in neighboring regions with different outbreak levels.
Last but not least, we like to note that the approach presented here is not restricted to the model (2.1), but the model-free feedback controller (3.2) can be applied, mutatis mutandis, to any epidemiological model available in the literature, appended by the dynamics for the population response.