Odille Bellenguez

Nadia Brauner

Alexis Tsoukiàs

Is there an ethical Operational Research?

The ethical dimension of Operational Research and Decision Aiding, although not a new subject, turns to be a concern, both for the large public and the OR community, because of the wide spread of autonomous artefacts endowed with decision capacity thanks to the use of models, methods and tools developed within our field. The paper addresses the question of whether there exists an "Ethical Operational Research", identifies the ethical questions which are specific to our professional community and suggests research topics which, although independently developed, are relevant for handling such questions.

1

Introduction

There is an increasing interest and discussion about "Ethical Operational Research" and more generally about "Ethical" or "Responsible" Decision Support (see [START_REF] Tsoukiàs | Social responsibility of algorithms: an overview[END_REF]). The topic is not really a new one: there is a EURO Working Group on Ethics 1 since 2002 and there is a "Prometheus Oath" written by J.P. Brans, founder of this Working Group (see [START_REF] Brans | OR ethics and decision: the OATH of PROMETHEUS[END_REF]). Interested readers can see two excellent surveys in [START_REF] Brans | Ethics in OR/MS: past, present and future[END_REF] and [START_REF] Ormerod | Operational research and ethics: A literature review[END_REF] of the literature on this topic.

Under such a perspective this contribution just continues an ongoing discussion already started in the 60s and early 70s (see [START_REF] Ackoff | The social responsibility of operational research[END_REF], [START_REF] Churchman | Challenge to Reason[END_REF]) and continued since then (see [START_REF] Gallo | Operations research and ethics: Responsibility, sharing and cooperation[END_REF], [START_REF] Gass | Ethical guidelines and codes in operations research[END_REF], [START_REF] Gass | On ethics in operational research[END_REF], [START_REF] Menestrel | Ethics in operations research and management sciences: a never-ending effort to combine rigor and passion[END_REF], [START_REF] Wenstøp | Operations research and ethics: development trends 1996-2009[END_REF]). The reason for which these topics turn to be discussed is related to the increasing diffusion of "autonomous artefacts" with augmented decision capacity. Both the wide public, but also scientists and policy makers are concerned by the wide spread of devices and processes which decide or recommend decisions using "algorithms" or "methods" which are felt to be non-controllable, dangerous, biased, unfair, inexplicable with unknown long-term impacts ( [START_REF] Casteluccia | Understanding algorithmic decision making: opportunities and challenges[END_REF]).

We share such concerns with a near discipline, Artificial Intelligence 2 and more precisely with Computational Social Choice. Most of these concerns are not really new (see the discussion in [START_REF] Tsoukiàs | Social responsibility of algorithms: an overview[END_REF]), but it pays to continue the discussion as we do with this paper.

The paper is organised as follows: in section 2 we introduce a general framework for the paper discussion. In section 3 we identify the type of "ethical questions" that could be of interest for this paper. In section 4 we discuss the questions which can be raised within our professional practice as decision analysts (or designers of decision support devices). In section 5 we briefly detail some research topics which although stand alone are at the same time useful in order to improve how we handle the topics discussed in section 4. We summarise the discussion in section 6 where we ultimately provide a reply to the question in the title.

Setting

In order to frame our discussion we are going to use a precise setting introduced and discussed in [START_REF] Meinard | On the rationality of decision aiding processes[END_REF], [START_REF] Meinard | What is legitimate decision support? Cahier du LAMSADE[END_REF] and [START_REF] Tsoukiàs | On the concept of decision aiding process[END_REF]. 1 See https://www.euro-online.org/websites/ethicsandor/ 2 See the topics discussed at the conference: https://facctconference.org/ or the Mechanism Design for Social Good Working Group: https://www.md4sg.com/. See also the High-Level Expert Group for Trustworthy Artificial Intelligence recommendation to the European Commission: https://digital-strategy.ec.europa.eu/en/library/ ethics-guidelines-trustworthy-ai.

We consider a situation where a "client" (an entity implied in a decision process), asks an advice to a "decision analyst" about how to improve his/her conduct with respect to that decision process. We also consider situations in which the "client" does not seek an advice for a precise decision process, but for a class of decision processes, the advice coming under the form of a device/system/software which is supposed to elaborate recommendations; the decision analyst in such cases being the designer of such system.s. Under such a perspective the decision support activities can be seen as: -either the direct interactions between client and analyst in order to elaborate a recommendation; -or the design of an appropriate system which on its turn will compute or help to compute a recommendation. Clearly many combinations between these two extreme cases are possible in the real world.

As already partially mentioned in [START_REF] Tsoukiàs | From decision theory to decision aiding methodology[END_REF], the use of a formal decision aiding methodology implies considering three different dimensions.

1. An axiomatic dimension, establishing the conditions under which it is possible to use protocols/algorithms/models in a meaningful and useful way.

2. An algorithmic dimension, considering the size of the solution space, the necessary data (availability, accessibility, storage, quality) as well as the necessary computing resources.

3.

A pragmatic dimension related to the conditions under which a decision aiding process is valid and legitimate (see [START_REF] Landry | Model legitimisation in operational research[END_REF], [START_REF] Landry | Model validation in operations research[END_REF], [START_REF] Meinard | What is legitimate decision support? Cahier du LAMSADE[END_REF]).

3 Ethics for whom?

First of all we need to identify different categories of concerned individuals. To be more precise: as Operational Researchers or Decision Analysts we may raise ethical questions for different purposes and under different perspectives. Not all of them are necessarily of interest for a scientific investigation.

We maybe raise ethical questions just because we are conscious citizens. These are the typical questions which all of us some day need to consider, but generally are related to our own individual values. These cannot be matter of study, analysis or guidelines and principles of conduct within our discipline if not respecting the very general values our societies consider relevant. But there is nothing specific to the fact that we are decision analysts.

Just to be more precise: accepting to provide support and models for military operations can raise ethical and/or moral questions to any among us, but the positive or negative reply is a matter of personal choices. Some of us will be happy to do it, others not and others might be indifferent. We cannot establish any ethical guideline on how to handle such issues. Some among us, besides being citizens are also teachers or researchers (in OR). Independently from our specific discipline and research field, there exist ethical questions related to our precise role of scientists. Such ethical questions concern our conduct as teachers (with respect to our students and colleagues) and researchers (with respect to our near scientific communities and the science in large).

Such issues are generally handled through deontology charts (how to behave appropriately with the students, how to conduct experiments, how to write papers, how to quote the existing literature etc.), or specific debates in philosophy of sciences (see e.g. [START_REF] Coutellec | Penser l'indissociabilité de l'éthique de la recherche, de l'intégrité scientifique et de la responsabilité sociale des sciences[END_REF]), but are not specific to our discipline and research areas (see e.g. the Practical guide "Integrity and responsibility in research practises" from the CNRS ethics committee [START_REF]Practical guide "integrity and responsibility in research practises[END_REF], the Singapore Statement on Research Integrity [START_REF] Wcrif | Singapore statement on research integrity[END_REF], the OECD Best Practices for Ensuring Scientific Integrity and Preventing Misconduct [START_REF]Best practices for ensuring scientific integrity and preventing misconduct[END_REF], The European Code of Conduct for Research Integrity by the European Academies [START_REF]The european code of conduct for research integrity[END_REF]).

There are instead two areas of ethical concerns which are specific to our domain and role of Decision Analysts. These are related to our profession (providing advice to decision makers or designing tools to be used by decision makers) and to our research in the broad area of Operational Research and Decision Analysis.

Professional Ethics

It is interesting to note that large part of the debate and the literature about the ethical dimension of our discipline and the practice related to it, originate from discussions about professionalism and deontology in our profession started at the late 60s. At that time the idea of creating a professional body of "chartered" operational researchers or decision analysts (later on called "OR Fellows" by the ORS in UK) ignited a debate which lasted several decades (and still persists) before reaching any practical conclusion ( [START_REF] Ackoff | The future of operational research is past[END_REF], [START_REF] Ackoff | Resurrecting the future of operational research[END_REF], [START_REF] Churchman | Operations research as a profession[END_REF], [START_REF] Kirby | A festering sore: the issue of professionalism in the history of the operational research society[END_REF], [START_REF] Orsa | Guidelines for the practice of operations research[END_REF], [START_REF] Rosenhead | Report of the commission on the future practice of operational research[END_REF]). This discussion moved beyond the UK and USA professional bodies (see [START_REF] Rauschmayer | Why good practice is not enough -ethical challenges for the or practitioner[END_REF]) and we can try to summarise under a specific perspective (the one of conducting rationally decision aiding processes).

As decision analysts we provide support to clients. We are not the only professionals who provide decision support: lawyers, accountants, physicians, psychotherapists, engineers, just to give some examples, help their clients to handle their problems and they do so using some scientific knowledge and approach, thus distinguishing their profession from just informal intuitive advice to friends and relatives. There are two topics we need to consider here: -What does make our decision support different from other equally scientifically based decision support activities? In other terms why decision analysts are not psychotherapists? -Since we nevertheless share some features with other professions, who already considered the problem of deontology, compliance, unsatisfied clients, young professional training etc., why our profession should not establish similar protocols, practices, training modules etc.?

We are not going to pursue further this discussion because it is out of the scope of this paper, but we can identify the ethical problems to handle within our profession: under which conditions we can claim that our professional advice to a client satisfies appropriate ethical standards and who establishes such standards?

We need to separate two distinct cases raising potentially different types of "ethical questions". The first one is the case where analysts directly provide at a client some advice on how to handle a problem within a decision process, a typical case being organising the shifts of the personnel at the emergency department of a hospital or managing a large call for tenders for software COTS for an IT industry.

The second case consists in designing generic methods, protocols or software aimed at being used for a precise class of decision problems, possibly customising such products for and with specific clients. Typical examples here include, supply management packages, flow-shop scheduling procedures, generic recommender systems etc.

There are certainly mixed cases as well as cases where specific applications become generic ones (a well known example being the yield management procedures originally designed for a specific client and then developed as stand alone customisable methods; see [START_REF] Smith | Yield management at American Airlines[END_REF]).

Despite the apparent differences between these two cases we will develop a unique argumentation based on two aspects: -the use of a decision aiding methodology; -the unveiling of hidden or implicit hypotheses and assumptions within our models and/or methods.

Decision Aiding Methodology

We start considering our profession as being characterised by the use of a formal language, the pretention of using/introducing a rationality model within the client's decision process and the use of algorithms (see [START_REF] Tsoukiàs | From decision theory to decision aiding methodology[END_REF]). As introduced in [START_REF] Tsoukiàs | On the concept of decision aiding process[END_REF], a decision aiding process can be defined as the interactions between a "client" (asking for advice) and an "analyst" (providing the advice) and can be represented through a set of "deliverables" such as: -a description of the problem situation; -a problem formulation; -an evaluation model; -a final recommendation.

In constructing such deliverables we make choices (as analysts). For instance:

-in order to represent the likelihood of an event we may adopt a probability measure (while other measures are possible); -in order to compute a majority for a voting procedure we may adopt the "Borda" rule (but others are possible); -in order to model the impact of the combined realisation of some decision variables we assume this impact being linearly defined with respect to the variables (but other choices are possible). Such "technical" choices are most of the time uncontrollable by the client and we (the analysts) are the only able to measure the consequences and to guarantee the meaningfulness and usefulness of their use.

We also do further hypotheses which are less technical, but nonetheless important. For instance: -in calculating the economic impact of a given infrastructure we consider a "territory", but how this territory is chosen/defined? -in designing a supply chain we consider the client's costs and time constraints, but are these the only constraints we should take into account? -in order to set up a vendor rating procedure should we compare the suppliers between them or only with respect to quality standards? Despite such choices being agreed with the client(s), it is unlikely the client(s) really realise the extension to which modifying any of these hypotheses can modify the outcome of the decision aiding process. In other terms, beyond any generic deontological constraint due to the fact that we have a professional relationship, we need to consider the specific constraints our condition sets. We can try to summarise these through the following points.

• Are we sufficiently critical? The fact we work for a client does not mean we cannot or we should not have a personal and independent perspective about what the client claims to be the problem to consider. We need to be able to show to the client aspects of the problem or other problems she does not see. At the same time we should be ready to modify our perspective and learn from the client's claims, values and beliefs.

• Where does rationality come from? This topic is extensively discussed in [START_REF] Meinard | On the rationality of decision aiding processes[END_REF]. The point to raise in this discussion is that there are several sources of "rationality", from external norms and standards to subjective behaviours and argued beliefs. What we need is understanding which among such different sources we use with that precise client for that precise decision process we are involved to. The pretention of introducing one or more dimensions of rationality to the client's reasoning for her problem, is what, most of the times, legitimates our action as analysts. Having a clear idea about where such dimensions of rationality come from is essential in establishing an appropriate professional relationship with the client.

• Can we explain, justify and easily revise or update? Most of the times we deliver both a model and the results of applying a set of methods to the constructed model. This implies choosing among what our technical knowledge and skills provide, following what the client claims being her problem. However, we do many technical choices which not always are "obvious", at least for the client and/or the other involved stakeholders. It might not be always necessary, but we need to ask ourselves: should I be asked, am I able to explain why we did such a technical choice, to completely justify such a choice against an appeal to a court and to defend the choice against an "adversary" analyst? Moreover, since modelling for decision support purposes is always a learning process, we also need to ask ourselves: how easy is it to revise and update the model and the methods in case the data change, the values and the opinions change, the problem setting (and formulation) can change.

• Is the result convincing for us, for the clients and for the involved stakeholders? Providing decision support means constructing convincing arguments for some potential action to undertake. Such convictions concern three different categories of stakeholders. We first need to convince ourselves that our advice is sound with respect to our technical knowledge and our methodology. We then need to convince the client that our advice is appropriate with respect to the problem the client has, the decision process for which the advice has been asked: the client needs to feel owner of the advice received. Finally we need to convince the rest of the stakeholders that the advice to the client was legitimately designed, that we have been critical and that the impacts of our advice being adopted have been understood.

Hidden hypotheses

Although in a professional setting we deploy a formal decision aiding methodology, we are always induced in assuming a number of hypotheses as granted or given. Some of these can become explicit through an appropriate use of our methodological knowledge. The fact that using a linear (additive, separable) utility function in order to aggregate the impacts of decisions along different attributes implies assuming that such impacts are commensurable and can be traded among them, is part of our methodological knowledge which will consistently impede us to use such a method in case this hypothesis does not hold (i.e. the client does not accept it). However, there are potential misuses and errors which can occur under certain implicit hypotheses, for which we are not really trained and prepared to handle. Such implicit assumptions (often ignored exactly because implicit) need to be explicitly identified and handled if we want to build a trusted relationship with our clients or in case we want a trustworthy use of any autonomous artefact using our models and methods.

• Cognitive biases. Decision Analysts are subject to the same cognitive limitations as other humans. They have personal values, personal cognitive limits, personal habits, culture and feelings. How much these influence the way through which the client's problem and the information provided are modelled? Other professional bodies impose specific training in order to handle such questions or establish specific protocols and external assessors to be used by those clients who may doubt about the analysts' biases. It is certainly true that our profession does not consider any specific training or appealing procedure, although these problems exist and should at least be discussed with our clients.

• Exceptional circumstances. Consider a classic risk management model and a situation characterised by extreme risks (extremely unlikely to occur events, but with extremely high impacts in case they actually occur; see [START_REF] Nott | Extreme Events: A Physical Reconstruction and Risk Assessment[END_REF]). Would our model still work appropriately? Consider a model of extreme risk theory: would this still apply for emerging risks management (see [START_REF] Bier | The Gower handbook of extreme risk : assessment, perception and management of extreme events[END_REF])? Consider a supply chain model. Would this work and be robust under any possible circumstances? And would that model still apply if the supply chain problem concerns humanitarian logistics (see [START_REF] Tomasini | Humanitarian Logistics[END_REF])?

The above examples are just cases where mainstream methods and models have been proven to be inappropriate when exceptional circumstances occur. This leads to a general question: given any method or model we suggest using for advising a client, will this advice still hold if such exceptional circumstances occur and if not what do we suggest the client to do? Which raises the question of whether we know the application limits of generalisation of whatever we suggest as recommendation to our clients.

• Data. All our methods and algorithms require data. Not only the ones provided by the client, but also data about the "territory", the "landscape", the "culture", the "economical and social context", the "organisation" where the decision process for which our help is requested is going to be used. Data are collected, stored, transferred, transformed, manipulated, along "pipelines" which are far from being with no im-pact upon the final outcome. Moreover, data, although they belong to nobody, can be protected by "rights", private or collective, social, economic or cultural. In designing methods and models we need to consider both the data pipeline quality as well as the rights protection issue (see [START_REF] Christophides | An overview of end-to-end entity resolution for big data[END_REF]).

• Algorithms. Most of the times our methods require efficient algorithms. Most of the times we need to trade-off between efficiency and accuracy or even optimality. Most of the times we also need to take into account other features of the algorithms such as manipulability, strategic proofness, security, robustness to adversarial attacks, blackbox effect etc. Most of the times our clients are not aware of what is the impact of choosing an algorithm instead of another. Clients are also usually unaware of the software differences when algorithms are coded and of the computing resources necessary to run them. It is unlikely our clients will ever be tempted to learn all such topics, but is our "ethical" obligation to know them and let the client understand which are the stakes at play when choosing a precise algorithm and a precise software implementation.

• Impossibilities. Not all methods fit to any type of decision problem. Generally speaking we know that most of the times, given a set of properties to be satisfied by the outcome of a potential algorithm, these are inconsistent. In other terms there is no algorithm able to satisfy at the same time all the desired properties (see [START_REF] Arrow | Social choice and individual values[END_REF], [START_REF] Brandt | Handbook of Computational Social Choice[END_REF], [START_REF] Ph | Exploitation of a crisp relation in a ranking problem[END_REF]). This is not really a problem in terms of computing solutions, but we need to know which properties are satisfied by which algorithms and we need to be able to explain that to our clients. In other terms we need to be able to explain to our clients what an impossibility theorem means for her problem and which are the different partial solutions we can adopt (and at which "price" in terms of satisfied properties).

• Long term consequences. When American Airlines started studying yield management methods in order to manage the ticket pricing (see [START_REF] Smith | Yield management at American Airlines[END_REF]), nobody (within the company and in the broad Operational Research community) could ever imagine the impact these methods will have on the travelling industry and the travelling habits within our societies. Today potentially any operator running a travelling business (including trains and buses) uses a yield management method in order to price dynamically tickets. The whole industry in this field changed its business model and each single consumer modified its willingness to pay for a travelling ticket (independently from business or leisure travelling). We are not going to discuss here whether this had a long term positive or negative impact, although some may discuss the consequences on the tourism industry, the house renting industry, the en-vironmental impact etc. The "ethical" question is that all such impacts have never been discussed neither within the company nor within the society. Nobody anticipated, discussed or even questioned the new business model underestimating the impact of a simple optimal pricing method. Providing models, methods, tools, aiding to improve decision making has impacts which can go far beyond the client and the other involved stakeholders. Most of the ethical questions we rose in the previous paragraphs are related to a precise decision aiding process, the actors involved and the immediate impacts. But we need to also consider impacts which will occur on a long, very long term and for stakeholders, citizens, territories and biomes who never ever thought that changing the optimal production policy of a company could change their lives half a century later.

Ethics in OR research

We have seen that most of the "ethical questions" about our discipline concern its use in the real life and the way through which we handle the relations with our clients, the relevant stakeholders and the use of our decision aiding methodology. The reader may note that many of such questions are related to topics addressed in our research (mostly indirectly) already since the 70s such as the axiomatic analysis of voting procedures, the analysis of behavioural biases or the development of problem structuring methods. The question we raise at this point is: "are there specific research topics in Operational Research related to our ethical questions"? Being more precise: probably any research topic in our discipline could be relevant for our ethical questions, but are there some new or more relevant ones?

1. Are we aware? The first class of research questions concerns awareness. In section 4 we raised several ethical questions concerning the use of models, methods, algorithms, protocols, etc. The fundamental remark is that none among such tools (which we use in order to advise our clients in their decision processes) are "neutral". Using one instead of another can have short or long term consequences which are independent from what the client asks or the situation requires. The question is: are we aware of such consequences? And the consequent research question is: Do we know how to choose appropriately our tools? In other terms, do we know what each tool can do, cannot do, the conditions under which they can be used and provide meaningful results? Axiomatic characterisations, representation theorems, simulations, are certainly research fields in our discipline which provide results usable in order to reply to the above demands. Under such a perspective we may emphasise that using numerical simulations and experimental settings (testing protocols, behavioural biases and modelling hypotheses) result in very useful tools helping to undercover hidden behaviours and tacit assumptions which could be concealed during the use of any among our tools.

We may also emphasise that the more we use the "open science" paradigm, sharing data, software codes, results and findings, greatly improves our capacity to increase awareness about what, when and how works (or does not work).

2. Do we help others becoming aware? Being ourselves (as analysts) aware of what our tools and our methodology can (and cannot) do is necessary, but not sufficient. Both our clients and the involved stakeholders (possibly the society as a whole?) need to develop awareness of what our tools can (and cannot) do. This of course raises a far larger topic about how scientists can and should communicate their findings to a "non-scientific" audience in a way that increases and strengths awareness and autonomy (see [START_REF] Bauer | What can we learn from 25 years of pus survey research? liberating and expanding the agenda[END_REF] or [START_REF] Bucchi | Handbook of Public Communication of Science and Technology[END_REF]), but remains relevant for our discipline. Large part of the general public gets used in misusing statistics and other quantitative information, in using inappropriately averages, indexes, protocols and codes, producing totally meaningless results and conclusions (see [START_REF] Gigerenzer | Calculated risks: How to know when numbers deceive Simon and Schuster[END_REF] or for more fun [START_REF] Barker | Lying Numbers: How Maths and Statistics Are Twisted and Abused[END_REF]). Even large and prestigious institutions, not to talk about public agencies and governments misuse such information in order to justify regulations and policies (for a famous example see the incredible diffusion of a meaningless index such as the Shanghai ranking of the Universities: [START_REF] Billaut | Should you believe in the shanghai ranking?[END_REF]). Under such a perspective it pays learning to use simple heuristics facilitating communicating quantitative information in a meaningful way (see [START_REF] Gigerenzer | Simple heuristics that make us smart[END_REF]). At the same time developing general frameworks which allow to unify a field of models and provide a unique frame within which interpret, explain and justify methods and protocols helps increasing awareness for any stakeholder (and the general public) involved in a decision process. The reader can see the impact of measurement theory, [START_REF] Roberts | Measurement theory, with applications to Decision Making, Utility and the Social Sciences[END_REF], in establishing a rigorous notion of meaningfulness: [START_REF] Narens | Meaningfulness and invariance[END_REF] [48], or the impact of conjoint measurement theory in unifying the field of multiple criteria decision analysis, [START_REF] Bouyssou | Conjoint measurement models for preference relations[END_REF]. Last, but not least, assuming a problem driven decision support attitude, instead of a method driven one, generally allows to improve communication with the client and enhance awareness about why certain methods will not fit in that precise problem situation, while others might be more appropriate. The result is adopting an "horizontal" or "methodological" view of our discipline and not seeing Operational Research as just a collection of methods (see [START_REF] Tsoukiàs | From decision theory to decision aiding methodology[END_REF]).

3. Are we critical? A decision aiding process is certainly a set of activities involving the client (who asks for advice) and the analyst (who provides the advice). However, we already observed that the choices we do, while conducting the decision aiding process, have impacts far beyond these two stakeholders. Moreover, remaining confined within the client's demand and/or the analyst's perception can result in missing other opportunities the decision aiding process offers. Most problem structuring methods (see [START_REF] Keeney | Value-Focused Thinking. A Path to Creative Decision Making[END_REF], [START_REF] Rosenhead | Rational analysis of a problematic world[END_REF]) emphasise that decision problems are constructed (and not identified) and that solutions critically depend on how problems are formulated. More recently the problem of constructing the set of alternatives (which is at the centre of the decision model) has turned to be a research topic (see [START_REF] Colorni | Designing alternatives for decision problems[END_REF], [START_REF] Ferretti | Studying the generation of alternatives in public policy making processes[END_REF]). Under such a perspective, research in the following three areas is extremely important in order to improve and expand our capacity to interact with the clients and the other stakeholders developing our critical view of the decision aiding process:

problem structuring methods in general, since they provide a general framework for supporting the whole decision aiding process ( [START_REF] Rosenhead | Past, present and future of problem structuring methods[END_REF]); -design theory as a formal tool for developing "out-of-the-box" alternatives beyond the dominant designs usually suggested as solutions ( [START_REF] Alexander | Design in the decision making process[END_REF], [START_REF] Le Masson | Design theory: History, state of the art and advancements[END_REF]); -preference learning because whatever we use in order to elaborate an advice is learned through interaction with the clients and/or accessing data and information ( [START_REF] Fürnkranz | Preference Learning[END_REF]).

4. Do we help others becoming critical? Keeping a critical perspective with respect to the problem situation as it appears to be (or as the client makes it appearing) is certainly important for our ethical questions. However, it is not sufficient. The client and the involved stakeholders also need to develop a critical perspective about what happens both within the decision process they are involved in and the decision aiding process. Such a process is very much a matter of "convincing": first of all ourselves (we remain within standards of meaningfulness), then our clients (they get something they feel it helps them) and then the rest of the world (the whole process was legitimate; see [START_REF] Meinard | On the rationality of decision aiding processes[END_REF], [START_REF] Meinard | What is legitimate decision support? Cahier du LAMSADE[END_REF]). Under such a perspective our clients and the other stakeholders need to be able to reply positively to the question: is this advice going to resist to any arguing against it, arguing grounded on data, procedures, protocols, and authority? A formal framework where such problems are discussed is formal argumentation theory (see [START_REF] Amgoud | Using arguments for making and explaining decisions[END_REF], [START_REF] Ph | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF], [START_REF] Ouerdane | Argumentation theory and decision aiding[END_REF], [START_REF] Trevor | Persuasion in practical argument using value-based argumentation frameworks[END_REF]).

Conclusions

Let us summarise the discussion and the claims we introduced. Ethical questions arise in our everyday life as well as in our professional life independently from what our profession is. The questions we are interested in this paper derive from our specific profession as decision analysts. As a side effect we need to consider which research topics, while independently developed, can help us in handling such ethical questions.

Under such a perspective we need to remember that decisions (what we are supposed to help making) are value driven and not data driven: data are necessary, but not sufficient for making decisions. It is part of our profession to make understand these values, for us, our clients and the involved stakeholders. At the same time we need to remember that aiding to decide is problem driven and not method driven, which means we first need to understand the problem and then we need to think about solving it.

As we show in the paper, ethical questions have been introduced in our discipline since the very beginning. Our discussion emphasises two parallel issues we need to consider when we try to handle such questions. The first concerns awareness of what methods are, can do, cannot do and how choosing any among them is not neutral with respect to the solution computed and the recommendation provided. The second is the development of a critical attitude about the consequences of our modelling choices which goes beyond the usual relation analyst/client.

A first point to make is that there are no universal procedures, protocols, algorithms or methods. Any of them will unfit for some reason and for some kind of problem situation and for some type of client and we need to know how to handle this. Keeping a critical attitude with respect to the clients' demand and to our profession helps on a long run both our clients and our profession. There are many hidden hypotheses and implicit choices in modelling and solving a decision problem and these need to be clear to us (as analysts), to our clients and to the stakeholders who might be involved in the problem situation.

Is there an ethical Operational Research? If ethics consists in applying to our profession standards of morality (in whatever way these have been established) then our reply is negative. But if ethics consists in assuming our responsibility for what we offer to our clients then yes. Although we may not be liable for what our clients decide using our advice and/or our tools, we are responsible for many (avoidable) consequences which can occur. We have a power and we need to use it with responsibility.

Acknowledgements

The paper reflects discussions hold within the Special Interest Group on Ethics in Operational Research of the CNRS funded National Research Network on Operational Research (GDR-RO, http://gdrro.lip6.fr/). We specially thank the participants to the workshop hold at the LAMSADE, Université Paris Dauphine the 26/11/2011, http://gdrro.lip6.fr/ ?q=node/254. Christine Solnon read an early version of this document and her comments greatly improved it.


Dominique Monnet

Alexandre Goldsztejn

Franck Plestan

Generalization of the twisting convergence conditions to non-affine systems

Keywords: Twisting control, nonlinear system, non-affine system

The twisting algorithm is a second order sliding mode technique for which convergence conditions have been established. However, these conditions are limited to systems that are affine in the control. In this paper, new convergence conditions that are not limited to such systems, are proposed. Furthermore, it is shown that in the affine case, state-of-the-art convergence conditions are more conservative than the new conditions. Two examples illustrate the difference of conservatism and the way that the new conditions ensure the convergence of systems non-affine in the control input.

Introduction

Sliding-mode control is now a well-established control theory [START_REF] Edwards | Sliding mode control: theory and applications[END_REF][START_REF] Shtessel | Sliding mode control and observation[END_REF][START_REF] Vadim | Sliding modes in control and optimization[END_REF]. The principle is to drive and maintain the system trajectory on a manifold, called "sliding surface" with a discontinuous control signal, in spite of perturbations and uncertainties. While evolving on this surface, the system has the desired dynamic, and is not sensitive to these perturbations and uncertainties. An undesirable effect of the discontinuous control signal is the so-called chattering effect [START_REF] Boiko | Analysis of sliding modes in the frequency domain[END_REF][START_REF] Levant | Chattering analysis[END_REF][START_REF] Fridman | Chattering analysis in sliding mode systems with inertial sensors[END_REF], that is a highfrequency oscillations phenomenon that can damage actuators. In order to attenuate the chattering, higher order sliding mode techniques have been developed, in contrast with the standard first order sliding mode. Among the higher order sliding mode techniques (see for example [START_REF] Shtessel | Sliding mode control and observation[END_REF][START_REF] Fridman | Recent Trands in Sliding Mode Control[END_REF]), second order sliding-mode (2-SM) control has been extensively studied and successfully applied to real case problems. Recent works propose advanced 2-SM methods based on discontinuous sliding surfaces depending on quantization levels of the uncertainties to further limit control amplitude [START_REF] Paolo Incremona | Second order sliding mode control for nonlinear affine systems with quantized uncertainty[END_REF][START_REF] Ferrara | Advanced and optimization based sliding mode control: Theory and applications[END_REF] and chattering. 2-SM can be achieved through different controller struc-Email addresses: dominique.monnet@ec-nantes.fr (Dominique Monnet), alexandre.goldsztejn@ec-nantes.fr (Alexandre Goldsztejn), franck.plestan@ec-nantes.fr (Franck Plestan).

tures. The approach based on the contractivity property of differential inclusion has provided 2-SM establishment conditions for several 2-SM controllers [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF][START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF][START_REF] Levant | Principles of 2-sliding mode design[END_REF] (among them, the twisting algorithm and the supertwisting controllers). Lyapunov-based conditions for 2-SM establishment have been proposed for twisting [START_REF] Vadim | Sliding mode control in electro-mechanical systems[END_REF][START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF] or super-twisting [START_REF] Moreno | A lyapunov approach to second-order sliding mode controllers and observers[END_REF][START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF] controllers, and for their adaptative extensions [START_REF] Shtessel | Twisting sliding mode control with adaptation: Lyapunov design, methodology and application[END_REF][START_REF] Shtessel | A novel adaptive-gain supertwisting sliding mode controller: Methodology and application[END_REF][START_REF] Shtessel | Super-twisting adaptive sliding mode control: A lyapunov design[END_REF]. These convergence conditions, derived either from the contractivity property or from Lyapunov theory, are sufficient but not necessary. Weaker convergence conditions are of great benefit: they allow more freedom in choosing controller parameters when designing a controller, generally leading to improved designs. In this paper, we focus on the twisting trajectories typically obtained using a twisting controller. By investigating the finite time convergence of these twisting trajectories independently of any controller structure, we obtain new finite time convergence conditions that apply to systems with non-affine control input and no assumption on the controller structure. In addition to the novelty of handling non-affine control inputs and non-twisting controller structures, these new sufficient conditions turn out to be sensibly weaker than the previously proposed conditions [START_REF] Vadim | Sliding mode control in electro-mechanical systems[END_REF][START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF][START_REF] Levant | Principles of 2-sliding mode design[END_REF] when restricted to affine input systems with twisting controller. This paper is organized as follows. Section 2 explains the proposed approach to generalizing convergence conditions to non-affine systems. Section 3 provides a theorem that states the convergence conditions of the twisting trajectory. This theorem leads to two corollaries that give sufficient conditions on the twisting controller's parameters to ensure finite time 2-SM establishment. Through two academic examples, Section 4 illustrates that the new convergence conditions are less conservative than the state-of-the-art ones and can be applied to non-affine systems. Finally, Section 5 summarizes the results and provides directions for future works.

Proposed approach

Consider single input non-linear system given by the differential equation,

ẋ = f (t, x, u), (1)
where x ∈ R n is the state vector and u ∈ R is the control input. The sliding variable is denoted σ(t, x), and it is supposed that the system has a relative degree of 2, i.e., σ does not depend on u while u explicitly appears in the expression of σ, σ = σ(t, x, u).

(2) The function σ(t, x) is differentiable, σ(t, x) is continuous and σ(t, x, u) is typically discontinuous because u is discontinuous. The control objective consists in forcing σ to zero, and we assume that u is a function of σ and σ,

u = ϕ(σ, σ). (3)
In the context of 2-SM, ϕ is discontinuous. We assume that a trajectory x(t) exists ∀t ≥ 0 in the sense of Filippov [START_REF] Aleksej | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF], and that x(t) ∈ X ⊆ R n , with X a compact.

Usually, the time dependency of σ(t), σ(t) and σ(t) refers to some dependency to perturbations p(t). In that case, we assume known bounds p(t) ∈ [p min , p max ] that hold ∀t ≥ 0.

The twisting controller

In the particular case where f is affine in u, the second time derivative of the sliding variable is also affine in u, that is, (2) becomes,

σ = h(t, x) + g(t, x)u(t). (4)
Consider the following well-known twisting controller [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] ϕ(σ, σ) = -r 1 sign(σ) -r 2 sign( σ),

where r 1 > r 2 > 0. In [17, pp.148], the principle of the finite-time convergence proof relies on the following global bounds on h(t, x) and g(t, x),

h(t, x) ∈ [-C, C], g(t, x) ∈ [k, K], ∀x ∈ X, ∀t ≥ 0, (6)
with k > 0, which further ensures g(t, x) > 0. Those bounds allow enclosing the dynamic of σ(t) in the following differential inclusion in the phase plan (σ, σ):

σ ∈ [-C, C] + [k, K]ϕ(σ, σ). (7)
This problem being two-dimensional, a majorant curve in the phase plan (σ, σ) can be used to capture all possible trajectories, and the finite-time convergence of this majorant curve to the origin holds true for all the possible trajectories. The resulting sufficient convergence conditions depending on the parameter r 1 and r 2 are given in the first line of Table 1.

Generalized twisting conditions

The finite time convergence of the trajectories generated by the twisting controller is the consequence of behavior of the differential inclusion [START_REF] Fridman | Recent Trands in Sliding Mode Control[END_REF] that differs in each of the four orthants Σ i

Σ 1 = {(σ, σ) | σ > 0 and σ > 0}, Σ 2 = {(σ, σ) | σ > 0 and σ < 0}, Σ 3 = {(σ, σ) | σ < 0 and σ < 0}, Σ 4 = {(σ, σ) | σ < 0 and σ > 0}. (8)
While in the standard approach the orthant dependent dynamic enforced by the twisting controller is analyzed, we propose here another approach: We directly analyze the conditions on the orthant dependent dynamic to produce a finite time convergence, independently of the system structure (that may be non-affine in the input) and of the controller structure. Such conditions naturally involve bounds m i and M i on the dynamic that are defined orthant-wise:

σ ∈ [m i , M i ], ∀(σ, σ) ∈ Σ i , (9)
which are foreseen to be less conservative than the bounds (6) since the latter hold in all orthants. In practice, orthant dependent bounds are evaluated on each

X i (t) = {x ∈ X | (σ(t, x), σ(t, x)) ∈ Σ i } (10)
instead of being evaluated on the whole space X:

σ(t, x) ∈ [m i , M i ], ∀t ≥ 0, ∀x ∈ X i (t). ( 11
)
Computing such bounds in each of the four orthants roughly requires four times more work, but this is advisable in a design phase aiming to obtain a less conservative design. Those general sufficient conditions are stated in Theorem 1 and summarized in the last line of Table 1. In Table 1 and in the rest of the paper, bounds that need to hold ∀t ≥ 0 are usually enforced to hold ∀p ∈ [p min , p max ].

This general approach is finally instantiated to systems with an affine input and the twisting controller. The resulting sufficient conditions are stated in Corollary 1.1. They are summarized in the second line of Table 1: A comparison with the first line of the table shows that they are similar to the usual sufficient conditions associated to the twisting controller, but defined orthant-wise and are therefore less conservative.

3 Conditions of convergence

General case

Consider differential inclusion [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. Sufficient conditions on the bounds m i , M i are given in the next theorem (Theorem 1), to ensure the finite-time convergence of σ and σ to zero in finite time with a twisting trajectory. First, in the half plan σ ≥ 0, having σ < 0 ensures that the trajectory (σ(t), σ(t)) crosses the semi-axis σ ≤ 0 in finite time in the phase diagram, and vice-versa in the half plan σ ≤ 0. This corresponds to Conditions ( 13) and ( 14) of Theorem 1.

Secondly, every time the trajectory crosses the same semi-axis, it must get closer to the origin. It is shown in the proof that the ratio between the two consecutive intersection points along the same semi-axis is bounded by

|m 2 M 4 | |M 1 m 3 | , (12)
and having this ratio strictly lower than one, corresponding to Condition (15), ensures the finite-time convergence to the origin.

Theorem 1 If differential inclusion (9) holds, M 1 , M 2 < 0, (13)
m 3 , m 4 > 0, ( 14
) and |m 2 M 4 | |M 1 m 3 | < 1, (15)
then σ and σ converges to zero in finite time. Moreover, with the initial conditions σ 0 = 0 and σ0 > 0, the convergence time is lower than

d σ0 1 -q , ( 16
)
where d and q < 1 are defined by,

d = - 1 M 1 + 1 √ M 1 M 2 + 1 m 3 m 2 M 1 + m 2 M 1 m 3 m 4 q = m 2 M 4 M 1 m 3 . (17)
PROOF.

Consider the initial condition σ = 0, σ = σ0 > 0 at time t = 0. Due to Condition (13), σ is strictly decreasing over time and reaches 0 in a finite time denoted t 1 , σ(t 1 ) = 0.

In the sub-space Σ 1 , since σ is upper bounded by M 1 , the trajectory (σ(t), σ(t)) is externally bounded by the trajectory obtained by integrating M 1 , which is the the so-called majorant curve [17, ch.4], [START_REF] Davila | Secondorder sliding-mode observer for mechanical systems[END_REF], as represented in Figure 1. Integrating the upper bound M 1 provides the trajectory (σ 1 (t), σ1 (t)), with,

σ1 (t) = M 1 t + σ0 , σ 1 (t) = 1 2 M 1 t 2 + σ0 t. (18)
This bounding trajectory reaches the semi-axis σ ≥ 0 at time t + 1 ,

σ1 (t + 1 ) = 0 ⇐⇒ t + 1 = - σ0 M 1 , σ 1 (t + 1 ) = - σ0 2 2M 1 . (19)
It follows that σ(t 1 ) ≤ σ 1 (t + 1 ). Moreover, t 1 is upper bounded by t + 1 . Indeed, since σ ≥ 0 and is strictly de-

creasing in Σ 1 , σ(t) ≤ M 1 =⇒ 0 ≤ σ(t) ≤ M 1 t =⇒ t 1 ≤ t + 1 . (20)
Consider now the sub-space Σ 2 , with the initial condition (σ(t 1 ), 0). Since σ is strictly negative in this sub-space, σ is negative strictly decreasing and the trajectory reaches the semi axis σ ≤ 0 in finite time t 2 . In this sub-space the trajectory is externally bounded by the trajectory obtained by integrating m 2 represented by the dotted curve in Figure 1. Let t = t-t 1 be a fictive time variable which is such that t = t 1 ⇐⇒ t = 0 to simplify the following calculus. The trajectory (σ 2 (t ), σ2 (t )) is given by,

σ2 (t ) = m 2 t , σ 2 (t ) = 1 2 m 2 t + σ(t 1 ), (21)
and reaches the semi-axis σ ≤ 0 for

t = t - 2 , σ 2 (t - 2 ) = 0 ⇐⇒ t - 2 = - 2σ(t 1 ) m 2 , σ2 (t - 2 ) = --2σ(t 1 )m 2 (22)
It follows that σ(t 2 ) ≥ σ2 (t - 2 ). Substituting σ(t 1 ) by its upper bound σ 1 (t + 1 ) computed previously yields,

σ(t 2 ) ≥ -σ0 m 2 M 1 . ( 23
)
Bounds to be computed

Sufficient condition for finite time convergence

Levant [17, pp.148] (Corollary 1.2)

C, k, K s. t. h(t, x) ∈ [-C, C] , g(t, x) ∈ [k, K], ∀t ≥ 0, ∀x ∈ X 0 < k , 0 < r2 < r1 C < k(r1 -r2) (r1 -r2)K + C < (r1 + r2)k -C Corollary 1.1 ci, Ci, ki, Ki, i ∈ {1, . . . , 4} s. t. h(t, x) ∈ [ci, Ci] , g(t, x) ∈ [ki, Ki], ∀t ≥ 0, ∀x ∈ Xi(t) 0 < ki , 0 < r2 < r1 C1 < k1(r1 + r2) , C2 < k2(r1 -r2) -c3 < k3(r1 + r2) , -c4 < k4(r1 -r2) C1 + k1(-r1 -r2) c3 + k3(r1 + r2) < c2 + K2(-r1 + r2) C4 + K4(r1 -r2) Theorem 1 mi, Mi, i ∈ {1, . . . , 4} s. t. σ(t, x) ∈ [mi, Mi] ∀t ≥ 0, ∀x ∈ Xi(t) M1 < 0 , M2 < 0 , 0 < m3 , 0 < m4 M1m3 < m2M4
Table 1 Summary on bound definitions and convergence conditions presented in the paper, from the more conservative to the less conservative. Corollary 1.2 and Corollary 1.1 are restricted to affine control inputs and to the twisting controller, i.e., σ = h(t, x) + g(t, x)(-r1 sign(σ) -r2 sign( σ)). On the contrary, Theorem 1 makes no hypothesis on the system or the controller, therefore the design parameters of the controller do not appear explicitly in the constraints.

This substitution can be viewed as considering the initial condition (σ 1 (t + 1 ), 0) from which integrating m 2 provides the majorant curve represented by the solid curve in Σ 2 . Let t = t + 2 be the time taken by the trajectory, obtained by integrating M 2 from (σ(t 1 ), 0) at t = 0, to reach the semi-axis σ ≤ 0,

t + 2 = - 2σ(t 1 ) M 2 . ( 24
)
Since σ is positive decreasing in Σ 2 , it follows that,

σ(t) ≤ M 2 ⇐⇒ σ(t) ≤ M 2 t, ⇐⇒ 0 ≤ σ(t) ≤ 1 2 M 2 t 2 , (25)
which implies that σ(t) reaches zero before the trajectory obtained by integrating M 2 . This further implies that,

t 2 ≤ t 1 + t + 2 ⇐⇒ t 2 -t 1 ≤ t + 2 . (26)
Substituting σ(t 1 ) by its upper bound σ 1 (t + 1 ) in the expression of t + 2 yields the upper bound,

t + 2 ≤ σ0 √ M 1 M 2 . ( 27
)
Following the same reasoning as previously in the half plan σ ≤ 0 for t ≥ t 2 , with t 3 and t 4 the times when the trajectory reaches the semi-axis σ ≤ 0 and σ ≥ 0, respectively, we obtain,

t 3 -t 2 ≤ t + 3 , σ(t 3 ) ≤ σ + 3 , t 4 -t 3 ≤ t + 4 , σ(t 4 ) ≤ σ+ 4 . ( 28
)
The majorant curve is given by integrating m 3 in Σ 3 and Σ 4 in Σ 4 , and the bounds are expressed as

t + 3 = - σ(t 2 ) m 3 , (29)
σ - 3 = - σ(t 2 ) 2 2m 3 , (30)
t + 4 = -2 σ(t 3 ) m 4 , (31)
σ+ 4 = -2σ(t 3 )M 4 .
(32) Substituting σ(t 2 ) by its lower bound and t 2 by its upper bounds in (29) and (30) yields,

t + 3 ≤ σ0 m 3 m 2 M 1 , (33)
- σ0 2 m 2 2M 1 m 3 ≤ σ - 3 (34)
Substituting σ - 3 by (34) in ( 31) and (32) finally yields,

σ(t 4 ) ≤ σ0 m 2 M 4 M 1 m 3 , (35)
and

t 4 ≤ t + 1 + t + 2 + t + 3 + t + 4 = d σ0 , (36)
where d is given by [START_REF] Shtessel | Sliding mode control and observation[END_REF]. Let σi = σ(t i ) be the value of σ the i th time the trajectory crosses the semi-axis σ ≥ 0 at time t i . From equation (35) and Condition (15), we have,

σi+1 σi ≤ q = m 2 M 4 M 1 m 3 < 1. ( 37
) σ σ Σ 1 Σ 2 Σ 3 Σ 4 σ0 σ 1 (t + 1 ) σ(t 1 ) σ2 (t - 2 ) σ(t 2 ) -σ0 m 2 M 1 σ(t 3 ) σ(t 4 )
Figure 1. Illustration of Theorem 1. The arrowed curve represents a possible trajectory and the solid curve the majorant curve. The dotted curve bounds any trajectory from the initial condition (σ(t1), 0). The cones represent the set vector field obtained with the differential inclusion [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], and the black arrows the vector field that provides the majorant curve.

As a consequence, σ(t) tends to zero. By Equation (36), and since q < 1, the convergence time

T = ∞ i=1 t i is bounded by, ∞ i=1 t i ≤ d ∞ i=1 σi-1 ≤ d ∞ i=1 q i σ0 = d σ0 1 -q . ( 38
)
This completes the proof of Theorem 1.

Theorem 1 is illustrated in Figure 1. The trajectory (σ(t), σ(t)) is represented by the arrowed curve, the solid curve represents majorant curve that provides the bounds on σ(t 1 ), σ(t 2 ), σ(t 3 ) and σ(t 4 ).

The convergence Condition (15) of Theorem 1 does not impose that the majorant curve, and by extension any possible trajectory, have to cross the opposite semi-axes closer from the origin. For example, the ratio m 2 M 1 can be greater than one, implying that a trajectory starting from (0, σ0 ) with σ0 > 0 might cross the semi-axis σ ≤ 0 at σ1 with | σ1 | > σ0 ; the trajectory will still converge to the origin in finite time provided that Condition (15) is respected. This case is illustrated in the second example of Section 4.

Remark 1

The finite-time convergence conditions based on Lyapunov approaches generally do not allow the trajectories to get away from the origin along the axes. This is due to the fact that the proposed Lyapunov candidate functions V are symmetric with respect to the origin (V (σ, σ) = V (-σ, -σ)), which implies that V (0, σ) = V (0, -σ) and V (σ, 0) = V (-σ, 0) [START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF][START_REF] Shtessel | Twisting sliding mode control with adaptation: Lyapunov design, methodology and application[END_REF]. Therefore, any trajectory starting on a semi-axis will necessarily cross the corresponding level curve of the Lyapunov candidate before crossing the opposite semi-axis further away from the origin. In that respect, any convergence conditions derived from symmetric Lyapunov functions are more conservative than the conditions of Theorem 1.

Theorem 1 states sufficient convergence conditions on the bounds m i and M i , but does not specify how these bounds are defined. So far, any bounds such that the differential inclusion (9) holds, can be used with Theorem 1. However, the tighter these bounds are, the greater the chances to satisfy the conditions of Theorem 1 are.

The tightest bounds such that the differential inclusion (9) holds globally are the maxima and minima of σ over Σ i ,

m * i = inf t≥0 x∈Xi(t) σ(t, x), M * i = sup t≥0 x∈Xi(t) σ(t, x). ( 39
)
In practice, the structure of the controller is chosen to control system (1) that depends on tunable parameters, held in a vector denoted r. It follows that the bounds m i , M i depend on r. The problem becomes twofold: the bounds m i , M i , ideally the optimal ones given by (39), have to be computed and a relevant structure for the controller such that there exists r satisfying the conditions of Theorem 1 must be chosen. In some particular cases, as illustrated in Section 4, it is possible to compute explicit expressions for the optimal bounds as functions of r and derive conditions on r from Theorem 1. The problem of choosing a relevant controller structure is devised in Section 4, that proposes a more general version of the twisting controller. The affine case (4) with the twisting controller ( 5) is easier to deal with, since it is always possible to derive explicit expressions of m i , M i as a function of r and to derive explicit conditions on r.

Affine case

Consider the affine case (4) with the twisting controller [START_REF] Aleksej | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF]. In this case, it is possible to derive bounds on σ that are expressed as linear functions of r 1 and r 2 . As a consequence, the conditions of Theorem 1 directly provide conditions on r 1 and r 2 that enforce finite-time convergence as stated by Corollaries 1.1 and 1.2.

Corollary 1.1 Consider the system (4) with the twisting controller [START_REF] Aleksej | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF]. Let c i , C i be bounds on h over the four sub-domains X i ,

h(t, x) ∈ [c i , C i ], ∀x ∈ X i (t) ∀t ≥ 0, ( 40
)
and k i , K i be bounds on g,

g(t, x) ∈ [k i , K i ], ∀x ∈ X i (t) ∀t ≥ 0. ( 41
)
If (r 1 , r 2 ) satisfies 0 ≤ k i , i = 1, 2, 3, 4, 0 < r 2 < r 1 , (42)
C 1 < k 1 (r 1 + r 2 ), C 2 < k 2 (r 1 -r 2 ), ( 43
) -c 3 < k 3 (r 1 + r 2 ), -c 4 < k 4 (r 1 -r 2 ), (44)
and

C 1 + k 1 (-r 1 -r 2 ) c 3 + k 3 (r 1 + r 2 ) < c 2 + K 2 (-r 1 + r 2 ) C 4 + K 4 (r 1 -r 2 ) ( 45
)
then σ and σ converge to zero in finite time.

PROOF. Condition (42) implies that r 1 + r 2 > 0, and the following bounds on σ can be derived in Σ 1 ,

σ(t, x, σ, σ) ∈ [c 1 , C 1 ] -[k 1 , K 1 ](r 1 + r 2 ) =⇒ σ(t, x, σ, σ) ∈ [c 1 -K 1 (r 1 + r 2 ), C 1 -k 1 (r 1 + r 2 )].
(46) Doing the same computation in the other domains Σ i yields the bounds,

m 1 1 = c 1 -K 1 (r 1 + r 2 ), M 1 1 = C 1 -k 1 (r 1 + r 2 ), m 1 2 = c 2 -K 2 (r 1 -r 2 ), M 1 2 = C 2 -k 2 (r 1 -r 2 ), m 1 3 = c 3 + k 3 (r 1 + r 2 ), M 1 3 = C 3 + K 3 (r 1 -r 2 ), m 1 4 = c 4 + k 4 (r 1 -r 2 ), M 1 4 = C 4 + K 4 (r 1 -r 2 ). (47)
By injecting these bounds into Conditions ( 13), ( 14), [START_REF] Polyakov | Lyapunov function design for finite-time convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF], one gets Conditions (43), ( 44), (45). It follows that if the conditions of Corollary 1.1 are satisfied, then the conditions of Theorem 1 are also satisfied.

Corollary 1.1 states directly conditions on (r 1 , r 2 ) that ensure finite-time convergence, provided that bounds k i , K i , c i , C i are known, whereas Theorem 1 states conditions only on the bounds m i , M i but not on the tunable parameters. This is due to the fact that the bounds (47) considered in Corollary 1.1 are explicit functions of r 1 , r 2 and injecting them into the conditions of Theorem 1 makes them into conditions on r 1 , r 2 .

Finally, Corollary 1.2 proposes more conservative conditions than Corollary 1.1 but requires only bounds on h and g over X instead of the four sub-spaces X i . Corollary 1.2 provides the well-known convergence conditions of the twisting algorithm [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], and is stated as in [START_REF] Shtessel | Sliding mode control and observation[END_REF]. The proof is nevertheless provided to make the connection with Theorem 1.

Corollary 1.2 [17, pp.148] Consider the system (4) with the twisting controller [START_REF] Aleksej | Differential Equations with Discontinuous Righthand Sides: Control Systems[END_REF]. Let C be an upper bound on |h| over the state space,

|h(t, x)| ≤ C, ∀x ∈ X, ∀t ≥ 0, (48)
and k, K be bounds on g,

g(t, x) ∈ [k, K], ∀x ∈ X, ∀t ≥ 0. ( 49
)
If (r 1 , r 2 ) satisfies k > 0, 0 < r 2 < r 1 , (50)
C < k(r 1 -r 2 ), (51) and
(r 1 -r 2 )K + C < (r 1 + r 2 )k -C, (52)
then σ and σ converge to zero in finite time.

PROOF. In a similar way as in the proof of Corollary 1.1, and due to Condition (50), one gets the bounds on σ,

m 2 1 = -C -K(r 1 + r 2 ), M 2 1 = C -k(r 1 + r 2 ), m 2 2 = -C -K(r 1 -r 2 ), M 2 2 = C -k(r 1 -r 2 ), m 2 3 = -C + k(r 1 + r 2 ), M 2 3 = C + K(r 1 + r 2 ), m 2 4 = -C + k(r 1 -r 2 ), M 2 4 = C + K(r 1 -r 2 ). (53)
With these bounds, Condition (51) implies Conditions ( 13), ( 14) of Theorem 1 and Condition (52) implies Condition (15)

Discussion

The convergence conditions of Theorem 1 and Corollaries 1.1 and 1.2 reveal three levels of conservatism. Indeed, it can be noticed that if the conditions of Corollary 1.2 are satified, then the conditions of Corollary 1.1 are satisfied by considering

C i = -c i = C, k i = k, K i = K. (54)
Furthermore, if the conditions of Corollary 1.1 are satisfied, then the conditions of Theorem 1 are satisfied as shown in the proof of Corollary 1.1. As a consequence, there is an implication relationship between the conditions of Theorem 1 and Corollaries 1.1 and 1.2.

The conservatism can be further explained by the quality of the bounds on σ that can be used by the different convergence conditions. The conditions of Theorem 1 allow to consider tighter bounds on σ than the conditions of Corollary 1.1, which themselves can be used with tighter bounds than the conditions of Corollary 1.2 due to the expression of these bounds. The tightest enclosure [m 1 * i , M 1 * i ] that can be derived from bounds (47) is obtained with

c * i = inf t≥0 x∈Xi(t) h(t, x), C * i = sup t≥0 x∈Xi(t) h(t, x), k * i = inf t≥0 x∈Xi(t) g(t, x), K * i = sup t≥0 x∈Xi(t) g(t, x), (55)
and the tightest enclosure [m 2 * i , M 2 * i ] that can be derived from bounds (53) is obtained with (56)

C * = sup
These bounds are such that,

[m * i , M * i ] ⊆ [m 2 * i , M 2 * i ] ⊆ [m 3 * i , M 3 * i ], (57)
and are equals only in particular cases. As a consequence, using the conditions of Theorem 1 with m 2 * i , M 2 * i is more conservative than using them with m 1 * i , M 1 * i , which is more conservative than using m * i , M * i .

Remark 2 Being less conservative means that the new conditions describe a larger set of convergent tuning. The most typical objective being to use small gains, the new conditions will lead to smaller gains using such a performance objective. This can be in contradiction with other objectives like the converging time.

The twisting controller is appropriate if σ has the affine expression [START_REF] Fridman | Recent Trands in Sliding Mode Control[END_REF] and g is positive. In this case, choosing r 1 and r 2 sufficiently large ensures the convergence conditions. However, Theorem 1 can be applied in the general case where σ in non-linear in u, and the twisting controller may not always be a relevant control structure to ensure the convergent conditions. This is emphasized by the second example in Section 4.

Academic Examples

This section proposes to study two examples. The first one illustrates the different levels of conservatism of the three corollaries. The second example shows that Theorem 1 can be employed in the case where σ is non-affine in u. The software Mathematica was used to generate Figures 2 and3.

Example 1: Conservatism illustration

Consider the system ẋ1 = x 2 , ẋ2 = x 1 + (1 + |x 1 |)u + p(t), (58)
with p an unknown perturbation bounded in [p min , p max ]. Notice that this system has a form similar to (4) with h(t, x) = x 1 + p(t) and g(t, x) = 1 + |x 1 |. The sliding variable is defined as σ = x 1 . Supposing that the system is controlled by a twisting controller, the second time derivative of the sliding variable reads as σ = x 1 + (1 + |x 1 |)(-r 1 sign(x 1 ) -r 2 sign(x 2 )) + p(t).

(59) It follows that h and g are expressed as,

h(x, t) = x 1 + p(t), g(x) = 1 + |x 1 |. (60)
For this example, the expressions of m i and M i can be determined by studying the sign of

∂ σ ∂x 1 = 1 + sign(x 1 )(-r 1 sign(x 1 ) -r 2 sign(x 2 )), (61)
that depends only on r 1 and r 2 in the four sub-spaces X i .

In the following, state r 1 > r 2 > 0 and limit the study to the bounded subset of the state space

X = [-1, 1] × [-1, 1]
. Computations yield the following bounds. conditions are represented by the dark grey set in Figure 3. Figure 4 displays the simulation of the system with the initial condition (σ 0 , σ0 ) = (0, 0.8) and the parameters (u 3 , u 4 ) = (2.1, 0.9). As one can remark, the system converges by a twisting convergence even if σ moves away from the origin in the half space σ > 0.

• In X 1 =]0, 1[×]0, 1[, one has C * 1 = 1, k * 1 = 1 and M * 1 = 1 -2(r 1 + r 2 ) + p max if r 1 + r 2 < 1, -(r 1 + r 2 ) + p max otherwise. ( 62

Conclusion

In this paper, new convergence conditions that are not limited to systems that are affine in the control nor to the twisting controller structure are proposed in order to guarantee the finite time convergence. The approach relies on considering a piece-wise set vector field over several domain of the phase plan, that enables to ignore the underlying system and interconnected controller. It was shown over two academic examples that the new conditions

(1) are less conservative than state-of-the-art conditions based on majorant curve for affine systems, (2) can enforce a twisting convergence for non-affine systems.

In particular, the second example illustrated that the twisting controller might not ensure twisting-like convergence of non-affine systems. A more flexible control structure was proposed that enabled to meet the convergence conditions. Although more flexible, this structure might not work for any non-affine plants. Future work includes finding control strategies for large classes of non-affine plants.

The examples featured in Section 4 could be solved analytically, but that might not be possible for more complex real case problems. Indeed, it is not always possible to derive explicit expressions of bounds on σ in the non-affine case. However, σ has an explicit expression depending on r, which makes it possible employ a global non-convex numerical solver to compute these bounds [START_REF] Nikolaos | Baron: A general purpose global optimization software package[END_REF]. A numerical approach would spare the search of analytic solutions, and enable to consider more complex and flexible control structure to ensure the convergence of non-affine plants. Future work includes the formulation of the problems as programs that can be solved with numerical solvers.

Another direction for future research would be to extend the generalized approach based on piece-wise differential inclusion proposed in this paper to other existing 2-SM algorithms, such as the super-twisting.

2 (

 2 b) [pmin, pmax] = [-1, 1]

2 (Figure 2 .

 22 Figure 2. Sets of (r1, r2) values that ensures establishment of 2-SM resulting from the convergence conditions. Dark gray: Theorem 1, gray: Corollary 1.1, light gray: Corollary 1.2.

4 Figure 3 .

 43 Figure 3. Sets of (u3, u4) values that ensure establishment of 2-SM resulting from the conditions of Theorem 1. The white crosses represent the values used for simulation.

Figure 4 .

 4 Figure 4. Phase diagram of Example 2 simulated with (u3, u4) = (2.1, 0.9) or equivalently (2.1, -0.9), (-2.1, 0.9) or (-2.1, -0.9).

Acknowledgements

This work was partially supported by Région Pays de la Loire, Centrale Nantes, West Atlantic Marine Energy Community, LS2N and CNRS as part of the O2GRACE project. This work was partially supported by the French Agence Nationale de la Recherche (ANR) [grant number ANR-16-CE33-0024]. The authors would like to thank the reviewers for their helpful comments.

1 + 2(-r 1 + r 2 ) + p min otherwise, It can be seen in Figure 2 that the light gray set defined by the conditions of Corollary 1.2 is contained in the gray set, defined by the conditions of Corollary 1.1, which is itself a subset of the dark gray set defined by the conditions of Theorem 1. This illustrates the different levels of conservatism of the three Corollaries yields by the inclusion (57). (67) However, Theorem 1 does not require σ to be affine in u and can be employed to find a control u that ensures finite-time convergence. To do so, compute m i and M i as functions of u.

Condition (13) imposes -1 + u 2 < 0 in X 1 whereas Condition ( 14) imposes -3 + u 2 > 0 in X 3 . With the twisting controller ( 5), these conditions become

As a consequence, there exists no (r 1 , r 2 ) such that the twisting controller enforces these two conditions.

Let us consider a more general controller structure than the twisting controller, given by,

with u i ∈ R. With controller (69), Condition (13) becomes,

and Condition ( 14) becomes,

Choosing arbitrarily u 1 = u 2 = 0, Condition (15) rewrites, (-1)(-3 + 4u 2 3 ) < (-3)(-1 + 4u 2 4 ) (72)

The set of values for (u 3 , u 4 ) satisfying the convergence


Maria Nerantzaki

Jean-François Lutz
email: jflutz@unistra.fr

Multistep Growth "Polymerizations"

Abstract.

Chain-growth and step-growth polymerizations are the two main mechanisms that are classically used to synthesize polymers. In this article, we describe a third approach that is usually not listed in polymer textbooks, namely multistep growth "polymerization". This synthetic strategy does not proceed in one pot as the aforementioned ones but relies on a series of distinct chemical reactions. It includes different types of stepwise syntheses such as solidphase iterative chemistry, exponential iterative growth and dendrimer synthesis. Although these methods are more time-consuming and experimentally-demanding than chain-growth and stepgrowth polymerizations, they allow a very high level of control over macromolecular structure.

In particular, they enable a perfect control over molecular parameters such as chain-length, molecular weight, molecular weight distribution, tacticity and comonomer sequence. Here, the common "multistep growth" mechanism shared by all these approaches is described and critically analyzed. Furthermore, the structure, properties and applications of the formed polymers are discussed. This manuscript can serve as a reference guide to understand and rationalize the emerging domain of precision polymer synthesis.

Introduction.

Classical polymer textbooks describe two main mechanisms to synthesize polymers, namely chain-growth polymerizations and step-growth polymerizations. [START_REF] Fontanille | Chimie et physico-chimie des polymères[END_REF][START_REF] Elias | Macromolecules, 4 volumes set[END_REF][START_REF] Odian | Principles of Polymerization[END_REF] In the former approach, a growing polymer chain carrying a reactive center reacts with monomers and incorporates them one-by-one; while in the latter one, multifunctional monomers reacts initially with another and lead progressively to the formation of reactive oligomers of increasing length. In both mechanisms, all successive reaction steps proceed in one pot, which is easy and practical.

Consequently, these methods are predominantly used in academia and industry to synthesize polymers. However, although widespread, these one pot processes are based on statistical factors, which result in the formation of non-uniform macromolecules. Even highly-controlled versions of these methods, such as living anionic polymerization, [START_REF] Szwarc | [END_REF] reversible activationdeactivation polymerization, [5] chain-growth polycondensation [6] and polymerizations regulated by external events [7] lead to polydisperse samples. Consequently, polymer scientists are trained during their education to handle the concepts of average chain-length, average molecular weight and polydispersity; some notions which are less familiar to organic chemists who are accustomed to describe pure compounds.

However, over the last years, there is growing research about the synthesis of molecularly-defined macromolecules. This trend is in large parts inspired by biological polymers such as nucleic acids and proteins, which exhibit advanced structures, properties and functions. [8] Thus, the search for molecular "precision" [9] is today one of the main objectives of polymer synthesis because it opens up the possibility to control in an absolute manner important parameters such as chain-length, polydispersity and microstructure (i.e. monomer sequence [10] and tacticity [11] ). Consequently, the number of publications dealing with the synthesis of uniform oligomers and polymers has drastically improved during the last decade. [12] Uniform polymers are predominantly synthesized by stepwise chemistry using strategies that are at the interface between classical polymer chemistry and organic synthesis, [13] even though some of them can also be achieved with bio-engineering methods. [14] Stepwise chemical approaches include divergent and convergent dendrimer synthesis, [15] solid-phase iterative synthesis, [16] liquid phase iterative synthesis [17] and iterative exponential growth. [18] In these approaches, the successive steps do not occur in one-pot as in chain-growth and step-growth polymerizations.

They involve distinct reactions steps separated by purification procedures. As a consequence, these methods enable the attachment of discrete building-blocks (e.g. a single monomer unit or a molecularly-defined macromolecular synthon) and ultimately the synthesis of uniform macromolecules. In 2016, it was proposed to unify all these approaches under the generic term multistep growth polymerizations. [19] Indeed, since these tools are increasingly used in synthetic polymer chemistry, they shall be from now on described and categorized in polymer textbooks, even though the use of the term "polymerization" to describe these processes may be debated and shall probably be placed between quotation marks. In this context, the aim of the present article is to give a rapid overview of these tools for polymer synthesis. Their advantages and limitations are critically discussed within the next pages.

Fundamentals and mechanisms

As defined in the introduction, a multistep growth polymerization is a process involving successive reactions and purification procedures. Figure 1 shows the mechanisms of different classical multistep growth approaches. The oldest strategy is the synthesis of linear polymer chains by stepwise iterative synthesis (Figure 1a). [10] The roots of this approach go back to the first half of the 20 th century and early attempts to synthesize artificial peptides in solution. [20][21] Stepwise peptide synthesis was afterwards simplified by the use of solid-or soluble-supports as introduced by Merrifield [22] and Shemyakin, [23] respectively. In all cases (solution-, liquidphase [24] or solid-phase [22] synthesis), the mechanism is the same, as shown in Figure 1a. A bifunctional monomer AB containing a protected, thus inactive, function A and a reactive function B is selectively reacted with an initiating molecule or support containing the reactive function A. In these conditions, the uncontrolled polymerization of the AB monomer is avoided and instead the controlled insertion of a single building-block is obtained. The formed adduct is the purified (i.e. column chromatography for solution phase synthesis, precipitation for liquid phase synthesis or filtration for solid-phase synthesis) and function A is deprotected to react with the next monomer. These steps are repeated a certain number of times until a polymer of desired chain-length is obtained. The process also enables synthesis of sequence-defined polymers since monomers with different substituents can be used at each step. [10] However, such strategies are, of course, not limited to protected AB-type monomers that are only discussed here as a typical example. Many other strategies exist, including several protectinggroup-free approaches. For additional information on those aspects, the reader is referred to prior reviews. [25][26] Another interesting strategy to achieve uniform linear polymer chains is iterative exponential growth (Figure 1b) that was initially referred to as molecular multiplication. [27] This approach was pioneered by Whitig and coworkers [28] and also involves bifunctional AB building blocks. The main difference between it and stepwise iterative growth described in the previous paragraph is that it involves two different derivatives of the AB building block instead of one. As shown in figure 1b, one building block has a protected A group and a reactive B group; whereas the other building block display the opposite configuration. Consequently, the coupling of these two molecules leads to a defined dimer terminated by protected functions A and B. The formed adduct is then divided in two batches. In one of them, function A is selectively deprotected, whereas function B is selectively deprotected in the other one. The coupling of these two dimers leads therefore to a defined tetramer. The process is then repeated a desired number of times to obtain uniform 8-mer, 16-mer, 32-mer and so on.

Another widespread multistep strategy is dendrimer synthesis. The first example of such a synthesis was reported by Vögtle and coworkers. [29] Afterwards, the field has been extensively studied in the 1980s and 1990s. [30] Dendrimers are branched macromolecules with a precisely controlled topology and molar mass. The strategies to prepare them are relatively comparable to those described in the two previous paragraphs but rely on multifunctional monomers (i.e. at least trifunctional) instead of bifunctional ones. Figure 1c shows, for example, a divergent strategy involving a trifunctional AAA reactive core and a building block ABA, in which B is reactive and both A groups are protected. Repeating coupling and deprotection steps n times leads to a uniform dendrimer of defined generation Gn. It shall be remarked that the example displayed in Figure 1c was selected to emphasize analogies with stepwise iterative synthesis (Figure 1a) and iterative exponential growth (Figure 1b) but is certainly not the most applied strategy to synthesize dendrimers. Indeed, many other methods exist, in particular protectinggroup free approaches. Furthermore, both divergent and convergent strategies can be applied.

Additional details and information about dendrimers and their preparation can be found in excellent classical reviews. [15,[31][32] Molecular structure of multistep growth polymers A major advantage of multistep growth approaches over conventional one-pot polymerizations is the possibility to control accurately the molecular structure of the formed polymers. As described in the previous section, an important feature that is shared by all multistep growth processes is molecular-uniformity. Indeed, polymers prepared by these methods exhibit in most cases perfectly-controlled chain-length, molecular weight and molecular weight distribution.

However, purifications are mandatory to isolate such uniform polymers. Indeed, these methods are not exempt of defects and the purity of the formed macromolecules depends on the yields of the successive steps. For instance, in stepwise iterative synthesis, near-quantitative coupling yields are typically required. [13,25] With average yields of 99%, after 20 successive coupling steps, the fraction of chains having the correct chain-length is still 81% but after 100 steps, it goes down to only 35%. Therefore, these methods cannot be repeated indefinitely. For stepwise iterative synthesis, the maximum attainable chain-length is generally assumed to be about 200 residues, even though the longest synthetic polymer reported so far had 107 monomer units. [33] For iterative exponential growth, the synthesis of 128-mer is to our knowledge the highest chain-length ever reached (i.e. 7 amplification steps). [34] For dendrimers, giant macromolecules with molar mass in the megadalton range have been reported (i.e. G9). [35] Although not completely impossible, it seems currently very difficult to go beyond these limits. The synthesis of large molecularly-defined macromolecules is therefore an open challenge in current polymer chemistry. One strategy to achieve such a goal is the controlled ligation (either covalent or noncovalent) of molecularly-defined macromolecular synthons prepared by multi-step growth approaches. [36] For example, our group has recently reported the synthesis of giant sequencedefined macromolecules by DNA-guided self-assembly of bio-hybrid precursors containing synthetic polymer segments connected to single-stranded DNA extremities (Figure 2). [37] Using this strategy, it was possible to prepare linear supra-macromolecules containing a precise sequence of 352 non-natural monomers. Yet, the self-assembly process is not perfect and only a fraction of the self-assembled macromolecules exhibits the expected primary structure. Nonetheless, this first example emphasizes that there are valid strategies to go beyond the current limitations of multi-step growth polymerizations.

Another important advantage of multistep growth approaches is the possibility to perfectly control the microstructure (i.e. monomer sequence and tacticity) of the formed polymers. [11] This is a major difference with chain-growth and step-growth approaches, in which this aspect is, in most cases, uncontrolled. For instance, in stepwise iterative strategies, the possibility to incorporate building blocks at given stages of the synthesis enables a perfect degree of control over monomer sequence. [10] Hence, the precision of sequence regulation in these syntheses is comparable to what is found in biological systems. Sequence control is, of course, less precise in iterative exponential strategies. Because of the multiplication mechanism of these syntheses, the formed sequences are in general periodical. [13] However, some related strategies, such as the cross-convergent method, [38] allows synthesis of aperiodic sequences. [39] Tacticity can also be controlled in multistep growth strategies if enantiomerically pure building blocks are used (e.g. as in peptide or in DNA synthesis). This aspect also be controlled in some chain-growth polymerizations such as living anionic polymerizations. [START_REF] Szwarc | [END_REF] Yet, stepwise iterative growth allows synthesis of precisely-controlled stereo-sequences (i.e. eutactic sequences [40] ), which are beyond the classical situations (i.e. isotactic, syndiotactic, heterotactic) that can be achieved in chain-growth copolymerizations. However, these possibilities are currently barely explored, even though some interesting examples of precision polymers with controlled stereosequences have already been reported. [41][42] Multistep-growth approaches also allow precise control over macromolecular topology, even though this aspect has not been as widely investigated as in chain-growth polymerizations. [43] Besides linear and dendritic topologies, branched [44] and cyclic architectures [45][46] can also be attained in stepwise synthesis. One last important aspect that shall also be discussed is the chemo-diversity of the macromolecules that can be produced by multistep growth approaches. Although iterative strategies were first developed for the synthesis of biological macromolecules such as oligopeptides and oligonucleotides, they have now been extended to a very broad variety of synthetic polymers. [13] For example, polyesters can be synthesized by stepwise iterative chemistry, [47] iterative exponential growth [18] and dendrimer strategies. [48][49] In fact, as in chain-growth and step-growth polymerizations, the spectrum of chemical reactions and building blocks that can be used in multi-step growth syntheses is extremely diversified, not to say almost infinite.

Properties and applicability

As described in the previous section, the main advantage of multistep growth "polymerizations" is the control over macromolecular structure that they provide. In that regard, these approaches are undoubtedly superior to conventional polymerization methods. Consequently, they open up a brand new range of properties and applications, which are closer to those of functional biopolymers rather than commodity plastics. These applications are discussed in details in comprehensive reviews [15,[START_REF] Swisher | Sequence-Controlled Polymers[END_REF][START_REF] Meier | [END_REF] and therefore only a few of them will be briefly listed herein.

For example, both linear sequence-defined polymers and dendrimers have been proven to be interesting and efficient platforms for catalysis. [52][53] Sequenced-defined polymers exhibit interesting biomedical properties such as antibacterial activity [46] or the capacity to selectively bind poorly-soluble drugs. [54] For all these applications, it has been proposed that controlled monomer sequences allow highly-specific molecular recognition and binding. [55] It has also been evidenced that the information-storage properties of DNA can be mimicked in synthetic polymers. [56] For example, digital sequences can be efficiently stored in synthetic polymers prepared by solid-phase chemistry, [57][58][59] iterative solution synthesis, [60] dendrimer synthesis [61] or the cross-convergent method. [38] This topic has progressed very rapidly in recent years and it is now possible to store complete data files such as short text or pixelated images in a single macromolecule. [62] As shown in Figure 3, the current record in the area is a storage capacity of 440 bits/chain. [63] Furthermore, sequence-coded polymers can be used in anti-counterfeiting and traceability applications. [64] Commercial products such as the polymer taggant Poltag ® are already on the market.

Yet, multistep syntheses are much more technically-demanding, time-consuming and expensive than classical one-pot methods. Thus, there are some legitimate concerns about the scalability and industrial applicability of this new generation of precision polymers. [65] It shall be clarified that the applications listed above are not classical commodity or specialty applications of polymers. For example, anti-counterfeiting applications require typically ppm or ppb of informational polymers to label commercial products. Thus, small scale (typically gram-scale or multi-gram scale) synthesis is sufficient for many potential applications. Yet, if needed, higher scales (i.e. multi-kilos) can be implemented in large reactors. For instance, the solid-phase synthesis of peptides is applied industrially up to the 100-kilo scale. [66] Methods for the kilogram production of dendrimers have also been reported. [67][68] Thus, the industrial scalability of multistep growth polymers is not an by itself. In fact, the cost of synthesis is probably more the limiting factor. Hence, precision polymers, synthesized by multi-step growth strategies, are probably best suitable for high-end applications, which combine a low/medium-scale production with a high-profit margin.

Outlook

It is now obvious that conventional chain-growth and step-growth polymerization methods, which have been discovered during the early days of polymer science [69] and have been extensively investigated, have probably reached their limits in terms of structural control. [9] Consequently, to attain further perfection in polymer design, new polymerization concepts have to be identified and studied. The multistep growth strategies described in this short article already go in this direction. Indeed, the last three decades of polymer research have evidenced that linear and dendritic macromolecules with perfectly defined molecular structures can be prepared. Still, one may question if all these methods are really "polymerizations" and if they really belong to the field of polymer chemistry? As emphasized in the previous pages, the answer to all these questions is probably yes. Even though multi-step growth methods employ time-consuming repeating steps that are inspired by organic chemistry and total synthesis, the construction and characterization of molecularly-defined macromolecules requires undoubtedly the expertise of polymer scientists. Nevertheless, these methods still have very important drawbacks, for example regarding attainable chain-length, cost and applicability, as emphasized in this manuscript. However, these hurdles can be overcome with the development of engineering and robotic solutions. [33,[70][71] Hence, the search for macromolecular structural "perfection" will most probably remain a dominating theme in polymer synthesis in the coming decades. regions that probably correspond to double-stranded DNA. [37] a. Figure 3. Data storage application of polymers prepared by stepwise iterative synthesis. [63] (i)

Numerous variations of these mechanisms exist.

+ (i) (ii) (iia) (iib)  Ð ~ 1.0  Defined Ø (i) A A A B A A A A A A (ii) A A A A (i) (ii)  Ð ~ 1.0  Exact chain-length (multiple of 2)  Periodic monomer sequence (iii)
The portrait of Antoine de Lavoisier (Author: François Séraphin Delpech, public domain,

Figure 1 .

 1 Figure 1. Classical Multistep growth mechanisms: (a) stepwise iterative synthesis, (b) iterative exponential growth, (c) divergent dendrimer synthesis. In all displayed examples, function A reacts with function B. Step (i) is therefore a coupling step. The grey box symbolizes a protecting group for function A. Step (ii) is therefore a deprotection step. In all cases, the repetition of steps (i) and (ii) leads to the synthesis of uniform macromolecules. In strategy a, a final cleavage step (iii) is required in some cases (e.g. when using solid or soluble supports) to liberate the final macromolecule. The strategies displayed in this figure are only examples.

  support, soluble support, small molecule = cleavable site (optional)

Figure 2 .

 2 Figure 2. (a) DNA-guided assembly of four uniform poly(phosphodiester)s prepared by stepwise iterative phosphoramidite synthesis into large molecularly-defined supramacromolecules. The uniform precursors contain synthetic segments (red) connected to DNA single-strands (blue). The sequence-specific hybridization of the DNA strands leads to a programmed superstructure with a precisely-controlled chain-length. (b) Height AFM characterization of the formed superstructure on a dried substrate. (c, d) Height AFM characterization of the formed superstructure in aqueous solution. The arrows indicate thicker

  into a 20x22 image. (ii, iii) The pixels were transformed into a 440-bits string, which was compressed into 264-bits. (iv) The sequence was translated into a chemical monomer sequence. (v-vi) The polymer is then analyzed by mass spectrometry and subjected to stepwise CID fragmentation. (vii) The spectra are decoded by a software. (viii) Ultimately, information is decompressed and the pixelated image can be visualized.

Acknowledgements

M.N. thanks the Bodossaki Foundation (Athens, Greece) for research funding for the period 2018-2021. J.F-L. thanks the CNRS and the Institut Thématique Interdisciplinaire "Chimie des Systèmes Complexes" (ITI-CSC) for financial support.