On The Subject of Thinking Machines

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

person and you are faced with the task of determining who the person is (The New Problem), a activity (The Process) begins in your brain that goes through the search space of all the people whose face you can remember (The Experience), you then begin to consider the nose, eyes, skin color, dressing, height, speech and any other observable treats (The Features), the process then attempts to match these features to a particular person based on people we have seen before, if no satisfactory match is found, the brain concludes that this is a stranger (The Decision).

Consider a computer vision system on the other hand, trying to perform the same task using Convolutional Neural Networks, when the image of a person is imputed, the 3 Dimensional Tensor of pixels are observed, (The Features), the network then searches (The Process) for the presence of previously learned features called kernels or filters (The Experience), it then compute which of these features are present in the new image and returns a set of class scores (The Decision).

The process is remarkably similar, except for the process by which it is done, for example, Convolutional Neural Networks do not put the position of features into consideration, a nose at the position of the ear makes no difference to a CNN, the process in humans puts this into consideration, however, Capsule Networks, recently proposed by Geoffrey Hinton et al. [START_REF] Sabour | Dynamic Routing Between Capsules[END_REF] seeks to address this weakness.

Thinking is not an act of magic that happens illogically, it is a well defined sequence of discrete actions that include observing problems, comparing observations to learned features and making decisions based on the results of our comparisons. An interesting problem that may be considered not to follow these sequence include emotional decisions pertaining to love, anger, sadness etc. Such decisions are often taken stochastically and cannot always be defined within the framework of thought that we formulated, decisions in such circumstances are often exploratory rather than exploitative. Does this disprove our earlier assertion? No, intelligent machines are fully capable of stochastic actions, the concepts of exploration and exploitation are central to the field of Reinforcement learning, a key enabler of autonomous systems. We shall emphasize further on this soon.

In view of this, we argue that machines do think, just differently from the way humans think. This begs the question, if the framework of thought is the same for both machines and humans, why do they think differently? The answer to this lies in the realm of the components of intelligent systems. Artificial neural networks were inspired by neuroscience but their mechanisms are fundamentally different, we have largely given up on searching for systems that work like the brain, rather we keep searching for systems that work well, without minding how far we are deviating from the way the brain functions.

The human brain and the intelligent systems we create are very different, but the framework of thought is the same. Machines can think and it's only a matter of time before they truly become capable of thinking without using much energy.

Can Machines Imagine

Accepting that machines can think is a little challenging for critics of Artificial Intelligence, it is even far greater a challenge to consider machines capable of "Imagination" To answer the question of "Can Machines Imagine" we must first deeply consider the framework of imagination.

Thinking is often based on past experience as we earlier explained, however, imagination is the formulation of ideas which we have not learned "explicitly" from past experience. We added the term "explicitly" to indicate the fact that while imagination results in new untested ideas, it often draws inspiration from previous experiences. Hardly is any imagination completely new without being rooted in an existing idea, such might be a possibility, however, the connection with past experience might be highly implicit so much that we might not realize how much past experience has influenced our imagination. In simple terms, imagination is often an exploratory action that is guided partly by learned features from past experiences. These exploration is stochastic but still partly motivated due to its connection to past experience, such connections can be weak, but very vital. This framework of imagination changes the question, "Can Machines Imagine" to "Can Machines Develop New Ideas" The later question more appropriately describes the first question. Supervised Deep Learning, which is the most popular method for building Artificial Intelligent systems is solely based on learned features and by nature is at present not capable of producing machines that can develop new ideas of its own. Reinforcement learning however is based on machines that can exploit current knowledge just like supervised deep learning, but can also explore its environment by trying new actions to discover better ideas. This is very much like imagination. The ability of an agent to discover new policies or new sub policies in a hierarchical setting [START_REF] Frans | Meta Learning Shared Hierarchies[END_REF], gives such agent the ability to formulate new ideas and even do new things that we never expected it to do in a particular environment. There are limitations in the form of such discovery being limited by the finite set of actions available to the agent at a given time step or in a state, however, humans are subject to very similar limitations, for example, you cannot just imagine that you want to fly without some jet pack or related equipment.

It can be argued that exploratory moves only mimics imagination, however, no one can tell precisely the process by which imagination occurs in the brain, what we are sure of is the effects of intuition, hence the effects of exploratory moves can be characterized as a form of imagination. A lot still needs to be done in infusing machines with great imagination capabilities. To answer the question "Can Machines Imagine" or "Can Machines Develop New Ideas" the answer is Yes. With time, the ability of machines to imagine would improve as new techniques are discovered.

Can Machines Feel

"Not until a machine can write a sonnet or compose a concerto because of thoughts and emotions felt, and not by the chance fall of symbols, could we agree that machine equals brain-that is, not only write it but know that it had written it. No mechanism could feel (and not merely artificially signal, an easy contrivance) pleasure at its successes, grief when its valves fuse, be warmed by flattery, be made miserable by its mistakes, be charmed by sex, be angry or depressed when it cannot get what it wants."

The above statement was made by Professor Jefferson in 1949. It sums up the view of critics of Artificial Intelligence. The central core of the above assertion is the part, "No mechanism could feel (and not merely artificially signal, an easy contrivance)"

The idea that reaction based on artificial signals cannot be described as a feeling is rather contrary to how the human system functions. All forms of feelings in the body, including pain, anger, love, fear etc. are not a resultant of some inexplicable processes in the human body, rather they are actions initiated by the brain based on signals received from the sensory organs. For instance, pain is not actually generated on the skin, instead the brain receives signals from the skin, encoding the degree of touch and it then initiates a set of reactions that makes us sense pleasure or pain.

There exists such people that due to brain malfunction, they cannot feel any pain even if injured.

A feeling is the response the brain gives to a signal.

In the context of this definition, machines can feel. Take a humanoid robot, we can build sensors into the body such that when it is touched, signals would be sent to the processing system of the robot, which in this case is acting as a brain, the cumulative signals over a period of time would form a sequence that can then be fed into a type of supervised deep learning system called Recurrent Neural Networks. The system can then interpret this signal sequence as pleasure or pain, and could initiate responses such as smile, laugh or frown. This entails that feelings are not limited to biological organisms, Machines can Feel. They can compose music in response to signals from the environment, they can feel happy or sad and can enjoy the pleasures of life if trained on sequence of signals and corresponding feelings. There already exist practical demonstrations of these, "Solo" is a home appliance by a company named Uniform. It can play songs based on your facial expressions, if you are happy, it can play songs to help you enjoy your time and if it detects that you are down, it could play songs that would lift your spirit. A lot more is still to be done. Machines can feel and act based on feelings.

Can Machines Be Conscious

This is perhaps the most daunting question about machines. Consciousness is highly regarded as a trait that is unique to living things. It is generally a state of being aware of your existence.

The question about machines can be more appropriately asked as "Can machines become the subject of their own thoughts" Having established that machines can think, it is most appropriate to consider a conscious machine as any machine that can think about itself. This has a lot of implications. A machine that can do so would begin to care about its existence and can make efforts to prevent its creator from performing any action that can cause it to stop functioning. Such an ability is often a scary prospect, many believe machines possessing such abilities would become terminators, however the question of "Are conscious machines desirable " is not the same as "Are conscious machines possible" Machines can indeed become the subject of their own thoughts. Here is an example. Consider a robot that is programmed to learn not just from how its actions influence its environment but also from how the actions of other agents in its environment influence its internal state. Such other agents maybe humans or other robots. In such a scenario, robot A might learn that some actions of an adversarial agent G is causing it to loose power faster than normal, possibly by making it overwork, robot A may then decide not to obey commands giving to it by Agent G, since data from past experience indicate a negative effect from commands initiated by Agent G. It might also learn that the actions of another Agent D improves its internal state, hence it might develop more affinity for Agent D by obeying D more than G. Such a machine is conscious and is possible within the framework of Deep Reinforcement Learning.

The potentials of these can be scary but can be beneficial too. Robots that are conscious could learn to become more effective at work by learning which conditions would improve their working ability, such robots could learn to even work in collaboration with other agents, all on their own without reprogramming.

The big difference between humans and conscious machines is the role of the "subconscious" in humans. The subconscious part of humans remains largely mysterious, many actions are initiated by it, hence, we can at least have a sense of superiority over robots based on the fact that there is at present no proof that we can infuse machines with sub consciousness within the framework of Deep Learning and Reinforcement learning. Time would tell if someday, breakthrough in neuroscience and Artificial Intelligence can unlock the secrets of the brain and make it possible to build super-intelligent systems.

Conclusions

All around us, the applications of artificial intelligence are present. Google Assistant, Microsoft Cortana, Hound Assistant and Siri have greatly changed the way we interact with our smartphones. Alexa, Amazon Echo, Google Home and Apple's Home Pod are making home automation available to everyone. The effects on industries are even more pronounced. Every aspect of business, from Stock Trading, Recruitment, Manufacturing, Marketing, Decision making etc. are becoming automated. Autonomous cars are gradually becoming common sight in advanced cities. The applications of artificial intelligence would fundamentally transform our society in the same way that digital computers did. This is the fourth Industrial Revolution. However, despite the promises of a more advanced civilization, a lot of people are rightly scared of the consequences of widespread automation and military uses of artificial intelligence. There are fears of job displacement due to automation. These fears are genuine and need to be addressed. Disruption is inevitable but all stakeholders including scientists, captains of industries, investors, directors, shareholders and the government must work towards applying artificial intelligence in such ways that it would augment the roles of humans at work rather than replace them. There exist such jobs in which machines need to absolutely replace humans, this include very dangerous works that lead to the death of many workers yearly. For example, jobs that expose humans to harmful radiation should be done by machines. Rescue operations in the event of disasters can be greatly improved with robots, by these we can save many lives.

Military uses of Artificial Intelligence is also a dangerous concept which should be avoided. Great scientists including Yoshua Bengio and Elon Musk has spoken against weaponizing artificial intelligence.

We need a future where our activities are assisted by Safe Artificial General Intelligence (AGI). A great effort to achieve this is the OpenAI initiative, a nonprofit research organization that is dedicated to creating intelligent systems that would help and not destroy humans.

Let us all work together to create a better future for the human race.

AUTHORS

JOHN OLAFENWA


domain: sdv shs

text:
Ilya Klabukov
email: ilya.klabukov@gmail.com

The BioDARPA Concept

teaching and research institutions in France or abroad, or from public or private research centers.

INTRODUCTION

Defense Advanced Research Projects Agency (DARPA)the U.S. federal agency which mission is to maintain the technological superiority of the U.S. military and prevent technological surprise from harming the American national security [START_REF] Dugan | Special Forces" Innovation: How DARPA attacks problems[END_REF][START_REF] Munos | The DARPA Innovation Model, Creativity[END_REF]. The DARPA was created in response to Soviet Rocket Science Challenge in 1958, and up to now reconciles technological ambitions and hard reality [START_REF] Klabukov | Review of DARPA FY2015 Research Programs[END_REF].

The ideas of realization of a darpa-like agency are formulated for microelectronics [START_REF] Alic | Designing a civilian DARPA[END_REF], energy [START_REF] Bonvillian | ARPA-E and DARPA: Applying the DARPA model to energy innovation[END_REF] and since 1990 th also for biomedicine [START_REF] Cook-Deegan | Does NIH Need a DARPA[END_REF]. This was especially critical for projects that were too radical for traditional institutions like the National Institutes of Health (NIH) and the National Science Foundation (NSF) [START_REF] Marshall | Too radical for NIH? Try DARPA[END_REF].

Since early 2000 th biological projects have become the place in Agency programs [START_REF] Eisenstadt | What Is DARPA Doing in Biology?[END_REF] and the biomedical DARPA challenge was formulated [START_REF] Greenberg | A biomedical DARPA? Yes, but not at NIH: a cutting-edge research center would suffocate in the NIH's institutional quagmire[END_REF].

In 2014 the Agency created a new Biological Technology Office (BTO) to emphasized the new importance of biomedical technology [START_REF] Laursen | DARPA redesign[END_REF] and was allowed experts to talk about the phenomenon of "military-bioscience complex" [START_REF] Reardon | The military-bioscience complex[END_REF]. The office focuses on basic and applied research in the areas of gene editing, biotechnologies, neurosciences and synthetic biologyfrom powered exoskeletons for soldiers to brain implants that can control mental disorders [START_REF] Reardon | The military-bioscience complex[END_REF]. The BTO is responsible for all neurotechnology, humanmachine interface, human performance, infectious disease, and synthetic biology programs within the Agency.

CHALLENGES

In the 20 th century there were many challenges that stimulated the intensification of scientific and technical thought: the Manhattan Project, the Sputnik Challenge, the Apollo program. Among today's global challenges it's especially possible to distinguish the phenomena "colonization of space" and "immortality of the human". And if the first problem is still far from the full-scale embodiment, then the second one can be already formulated.

The Immortality of the Human project represents a complex problem which solution will demand permission of the whole group of technological tasks, many of which are listed in the collection of tasks in engineering biology [START_REF] Klabukov | Engineering Biology Problems Book[END_REF]. Among the necessary for realization of concepts of technological decisions it is possible to note "life-like test-stand", "test-range for bioengineered devices", "pharmaceutical foundry in the Gut", "molecular scalpel for fetal and postnatal surgery in vivo" [START_REF] Klabukov | Foundations for Molecular and Enzymatic Functional Surgery[END_REF], "biological Internet" [START_REF] Klabukov | DARPA-2016: On the rise of Biological Internet[END_REF], "humanized animal models", "organ-on-chip for clinical trials", "bio-decks for human enhancement", etc. Good example of transhumanistic challenges are presented in the Regenerative Medicine Roadmap 2.0 [START_REF] Batin | Regenerative Medicine Roadmap[END_REF] and the Roadmap to Immortality (Fig. 1) [START_REF] Batin | Roadmap to Immortality[END_REF]. 2) supporting social science research that advances the field of engineering biology and contributes to the adoption of new products, processes, and technologies; 3) expanding the number of researchers, educators, and students with engineering biology training; 4) accelerating the translation and commercialization of engineering biology research and development by the private sector; 5) and improving the interagency planning and coordination of federal government activities related to engineering biology [START_REF] Johnson | Engineering Biology Research and Development Act of[END_REF].

Professional communities and committees

Professional communities and committees are presented of the Standards Coordinating Body For Cellular/Gene and Regenerative Therapies and Cell-Based Drug Discovery [START_REF] Hayakawa | Report of the International Regulatory Forum on Human Cell Therapy and Gene Therapy Products[END_REF], Synthetic Biology Open Language Developer's Group [START_REF] Contreras | Intellectual property issues and synthetic biology standards[END_REF], iGEM Foundation [START_REF] Shetty | Engineering BioBrick vectors from BioBrick parts[END_REF], the NIST's public activity, Department of Defense workgroups, etc.

Public activities

The International movement for Life Extension has a bright and long-term history [START_REF] Stambler | A History of Life-Extensionism in the Twentieth Century[END_REF]. Now the public movements are represented by various activities like the "Transhumanist party" by Zoltan Istvan [START_REF] Lee | The Transhumanist Platform and Interview with 2016 U.S. Presidential Candidate Zoltan Istvan[END_REF], Michael Batin's transhumanists community [START_REF] Batin | The Immortality Doctrine[END_REF], Dmitry Itskov's "2045 Initiative" [START_REF] Petersen | Hope of Immortality[END_REF], Craig Venter Institute's startups [START_REF] Shimasaki | Biotechnology Entrepreneurship: Starting, Managing, and Leading Biotech Companies[END_REF], KrioRus company [START_REF] Bernstein | Freeze, die, come to life: The many paths to immortality in post-Soviet Russia[END_REF], etc.

ECONOMICS

The development of advanced technology sectors has always required the robust economic base like as long-term Pentagon contracts for microelectronics and radar technologies in Silicon Valley [START_REF] Steinbock | The Challenges for America's Defense Innovation[END_REF] or the Internet economy boom for the "web 2.0" applications investments.

The economic base of biomedical technologies is only at the beginning of its development. Combination of programs of Department of and Human Services (DHHS), Military Health System (MHS), DTRA, BARDA (Schoch-Spana, 2012) with civil activities are not enough for sustainable development for biotechnological superiority. This poses a serious barrier to the development of the biomedical industry, but in the different plans and road maps is expected rapid growth in this market (HealthNet, Russia) to more than $1 trillion by 2035 [START_REF] Schubert | Russia's "National Technology Initiative[END_REF]. In the field of advanced biomedical technologies can be used a government-supported leverage to research and market stimulation both the key directions and related fields (Fig. 2).

ORGANIZATION MODEL

The previous attempts of realization of darpa-like models were implemented in the form of the government agencies (ARPA-E, i.e. Energy), the interrelated projects collaboration (HS-ARPA, i.e. homeland security), or even agency community (IARPA, i.e. intelligence).

Existing an experimental models of R&D activities like the CIA's venture funds "In-Q-Tel" [START_REF] Lerner | [END_REF], "OnPoint Technologies" and other venture investment tools [START_REF] Klabukov | Venture Funds and Other Advanced Technologies for National Intelligence Services[END_REF]. In many ways, these tools are made possible by M&A activity in the defense and aerospace sector in the mid-1990s.

In various discussions it was noted that technological development incentives could be:

1) funding mechanisms to support the early development phase of countermeasures (the "valley of death"); 2) creation of a "BioDARPA" that would invest in transformational bioresearch. Such research would be "project driven" and linked to identified national needs ("Roundtable discussion : when terror strikes--preparing an effective and immediate public health response,"

2005).

▪ Development of shining-new Health and Wellness markets for personalized products ("markets for things that do not exist") [START_REF] Connell | Creating markets for things that don't exist[END_REF]; ▪ Establishment of a very early stage technology companies operates to creating future markets for things that do not exist [START_REF] Connell | Creating markets for things that don't exist[END_REF]. For examples of this "companies of The hypothetical formation and location of "BioDARPA Concept" in the U.S.

Governance Innovation System are presented on Fig. 3. Supposed that new business models (pharmaceutical industry framework) such as precompetitive research, crowdsourcing, networked innovation, virtual companies, drug repurposing, and forced disruption (e.g., DARPA) are spreading and enriching the biomedical innovation ecosystem [START_REF] Munos | Disruptive Innovation and Transformation of the Drug Discovery and Development Enterprise[END_REF].

The problem is noted that 90% of medical research stands to benefit only 10% of the population. While others dispute the numbers, there is little doubt that disparities exist in the global agenda for medical research [START_REF] Mccarthy | Research and Reality: Towards Responsible Medical Research for Catholic Universities[END_REF]. This bring up a point to accelerate the translation into clinical practice that can be achieved by experimental clinical sites. BioDARPA's Super-system (system-of-systems) complexity can be realized as mosaic exposome ("cloud model") around core organization system (Fig. 4).

ETHICS

It's obviously that needing for liberation of biology to biotech revolution [START_REF] Bailey | Liberation biology: The scientific and moral case for the biotech revolution[END_REF] but influence of traditional conservative forces is for the present quite big. The studies in human enhancement and synthetic biology ethics are need to addition of new Responsible Research and Innovation (RRI) concepts [START_REF] Nerlich | The dilemma of raising awareness "responsibly[END_REF]. Further development of technologies for mass market production of organs in humanized animals edited using CRISPR/Cas9 system is associated with common models of "life" perception. It is necessary to overcome many more of ethical barriers (Fig. 5).

CONCLUSION

The BioDARPA Concept provides the advancing development and broadcasting in clinical practice of the advanced biomedical technologies. The problems of creation of this "firm" (or a cloud forms) are associated with the complexity of their constitution (system-ofsystems model), clear statement of the program goals (the White Book), source selection (both performers and consulting firms), staff recruitment, and location of the BioDARPA in the National Innovation System (both defense and civilian parts).

The ResearchGate Project Group "BioDARPA: Biomedical Breakthroughs and Superiority" † is founded for data collect discussing on actual themes ‡ . † "BioDARPA : Biomedical Breakthroughs and Superiority". ResearchGate Project Group's Link: https://www.researchgate.net/project/BioDARPA-Biomedical-Breakthroughs-and-Superiority ‡ Unfortunately that's some activities like popular twitter account (@BioDARPA) was banned by the trademark holder in 2014.

Figure 1 .

 1 Figure 1. Roadmap to Immortality © Mikhail Batin, Maria Konovalenko, Alexey Turchin, 2013. *

Figure 2 .

 2 Figure2. Government-supported leverage for the rise of national bio-economy © Ilya[START_REF] Klabukov | Foundations for Molecular and Enzymatic Functional Surgery[END_REF]

  hope" are the Lung Biotechnology Inc. (the United Therapeutics startup for produce a humanized organs), Synthetic Genomics, Inc. (Craig Venter's startup of attenuated viruses technologies for the vaccines production), Organovo Holdings, Inc. (3D-Bioprinting equipment and products), ViThera Pharmaceuticals, Inc. (geneticallyengineered probiotic bacteria and yeast for the delivery of therapeutic proteins in the human gastrointestinal tract), Cornell (genetically-engineered synanthropic bacteria to prevent of the cholera-associated pathologies), Synthorx (systems of synthesis of biologics and biofeedstocks based on a 6-letter genetic code), Amyris (synthetic biology bio-products), etc. ▪ Government-supported public programs for stimulation of investment activities and foster consumption; ▪ Increase a confidence in the market sector of new biomedical technology products (government regulation, industry standards, quality control). Today the U.S. Governance Innovation System are presented in the think-tank corporations (RAND Corp., MITRE's Workgroups, Woodrow Wilson International Center), the National Laboratories (Lincoln Laboratory, Lawrence Livermore Laboratory, etc), the Defense Command and Wides (DARPA, Medical Command, etc.) and civil agencies (BARDA, NIST), the New age Industry firms like SpaceX and bio-startups, various Venture and M&A funds (OnPoint Inc.), Non-profit organizations (100-Years Starship, IGEM Foundation), the Governance infrastructure networks (The Protein Capture Reagents Program, National Network for Manufacturing Innovation) and the Crowd networks (ARPA-E, IARPA Community).

Figure 3 .

 3 Figure 3. Hypothetical model of "BioDARPA Concept" in the U.S. Governance

Figure 4 .

 4 Figure 4. BioDARPA's Mosaic Exposome. © Ilya Klabukov, 2017.

Figure 5 .

 5 Figure 5. Conceptual fields of new biomedical breakthroughs: between Revolutionary Wealth and


Bélair Joanny

LAND POLICIES QUESTIONING DISCREPANCIES BETWEEN DISCOURSE AND PRACTICES IN TANZANIA

Introduction

This working paper analyzes practices of consultation associated to two specific recent land policies in Tanzania. It aims at understanding why there are important discrepancies between official policies and practices associated to their implementation. It argues that the role played by international institutions and actors in shaping African politics -neoliberal development policies and discourses -should not be overstated. Instead, I contend that the international influence explains only partly the practices associated to the implementation of Tanzania land's policies: its hegemonic impact is mostly superficial.

Understanding practices associated to land policies requires considering the lasting impact of Tanzania's culturally and historically embedded normative structure of governance and relations of power. Despite its adoption of more liberal policies since the early 1990s, Tanzania displays some enduring ideational features associated to her socialist past such as the conceptualization of the role played by the state in development and the prevalence of top-down governance practices. Thus, practices associated to the participative dimension of development are resulting from actors' navigation between an international neoliberal normative structure, and Tanzania's own normative structure that is shaped by institutional and ideational legacies. First, I show how the official state rhetoric is convergent with neoliberal ideas associated to agricultural development. Then, I explore the discrepancies between official policies and discourses, arguing that practices resulting from these policies are the result of actors' navigating divergent normative systems. I exemplify my argument with two examples. First, I focus on the consultations associated to the review of the New Land Policy. Second, I analyze village mapping practices in selected rural areas of Tanzania. MKUKUTA II provides guidelines for the development of the agricultural sector which are based on several assumptions highly influenced by neoliberal principles. It insists on the fact that surveying and securing land rights for both Tanzanians and investors are crucial to reduce land conflicts. It aims at giving smallholders access to credit to promote investment of their assets through an expected increased productivity, and it wishes to provide a conducive business environment which will foster much needed agricultural foreign-direct investments (FDI). 2 Nationally, the dominant discourse on agricultural development and reform of land policy is aligned with the key principles highlighted in MKUKUTA II. The Ministry for Lands, Housing and Human Settlement Development (Ministry of land) often explains the government's commitment to ensure agricultural and economic development and reduce land conflicts through its ambitious land titling and land use-map program. According to the official rhetoric, planning, surveying and mapping land in Tanzania is both promoting security of land tenure for citizens, and fostering land investments.

CONVERGENCE OF INTERNATIONAL AND NATIONAL RHETORIC ON LAND INVESTMENTS AND DEVELOPMENT

REVIEW OF THE NATIONAL LAND POLICY

The process of reviewing the 1995 Land Policy started in August 2015, but the rationale for the timing of this change remains unclear. The document states that it was triggered by "a changing national and global context, which have induced .1. Also, CSOs felt powerless during the process. Although the consultations trigger collaboration and a fruitful dialogue between CSOs-several CSOs worked together on a joint position paper, their interactions with the officials of the Ministry of land were unidirectional. Most of them were under the impression that their inputs will not be considered seriously. In the words of one representative of a CSO: "Consultations came as a surprise for us. We were not prepared. Moreover, the process was fake. If you look at how it is presented, it looks like they undertook the whole process seriously, but few people were consulted and are not representative of the country […] When we participated in the consultations in Morogoro, it was made very clear to us that it was the end of it. They will not be considering our suggestions". 7

I FRA

In addition, several CSOs' requests were left unanswered or ignored. For example, the Ministry of land refused to disclose key documents related to the New Land Policy, such as the strategic implementation plan of the strategy. 8

The Ministry of land also remained silent on the next steps, leaving ambiguity about what will happen next. As stated by one of my informants: "They are [Ministry of land and the government] speeding up the whole process. We have no luxury of time. We cannot stop or exert any control over the process. And we have not seen the strategic plan of implementation. In the end, we don't really know what will be in the policy." 9

MAPPING VILLAGES

In with MKUKUTA's strategy, titling and mapping land in Tanzania are depicted as crucial to ensure a productive use of land assets by increasing tenure security for citizens and investors. The Ministry of Land's Program for planning, surveying and land titling in Tanzania (PPSL) is expected to reduce unplanned settlements, decrease land conflicts, increase the use of land as a collateral for loans among smallholders, and facilitate land administration (registration and revenue collection). 10

Regarding village land, PPSL envisages a sequence that starts with providing villages with land-use maps, before issuing titles in the form of certificate of customary rights of occupancy (CCROs) to villagers. Land-use maps are conceived as tools for planning and guiding land development, and they usually have a ten-years period of validity 11 . The program explicitly states that processes of planning, surveying and titling rural land will be undertaken in partnership with local communities, thereby "increasing public awareness on land related policies, laws and guidelines", and contribute to local "capacity building and changes in the requirements for the land sector". 3 Although it mentions population growth as a factor contributing to change in land dynamics, it does not explain further any other changing dynamic.

Officially, this new Land Policy aims at "providing a framework for articulating the central role of land in socioeconomic development" by implementing an "effective land tenure system" which ensures "equitable allocation and access to land for all citizens" for an optimal use of land resources. After conducting "wide-ranging public consultations with stakeholders", the Ministry of land released an official first draft, which was adopted by stakeholders in November 2016 4 . Even though the document states that the policy was endorsed by the Cabinet, and is currently being implemented by the Ministry of land, there is still much uncertainty about the process. Because of lack of transparency, it is not clear yet if and when the policy will be effectively replacing the previous one.

The foreword of the new Land Policy gives the impression that the review process was consultative, democratic and national in the sense that views from different actors were collected and considered. Yet, the extent to which such objectives have been achieved remains unclear.

Official figures released by the Ministry state that twodays public consultations were conducted in eight different zones 5 . While the first day was spent to consult various stakeholders, the second was reserved for consulting villagers. In total, the Ministry of land is claiming having visited twenty villages by zone. This seems high, given the time spent by zone. Moreover, the process of consultation with villagers is not explained, meaning that it is difficult to evaluate the authenticity of consultative processes at village level and to which degree freedom of speech was allowed. In the Tanzanian context, where only one party has been ruling the country since Independence, one wonders to what extent people have been able to express themselves freely, especially in front of state officials. In the words of one of my informants: "if villagers were consulted with government officials and leaders in the same room, of course, they will not speak up [they will be scared]" 6 .

Regarding consultations with Civil Society Organizations (CSOs), it appears that several were not informed in advance of the consultations giving them a very short window of time to prepare their comments on the New Land Policy. Sometimes, they were notified very late, which made it practically impossible for them to assist to the meeting.

3 Ministry of Lands, Housing and Human settlements development, "The National Land Policy (Official Draft)", Government of Tanzania, 2016, pp. 6-7. 4 Ibid. 5 The zones are not further specified in the official draft of new Land Policy. 6 Interview 81: Tanzanian CSO 1, February 2017. 7 Interview 86: Tanzanian CSO 2, February 2017. training" on land management12 .

The mapping process itself consists in determining the village GPS coordinates through a cadastral process, and in identifying and delimiting village sections for current and future land usage. Generally, village sections comprise: areas reserved for residential, agricultural, and social services, and areas for pasture. In addition, in some districts, land sections have been specifically reserved for current or future land investors.

Villagers are usually involved to some extent in drafting the land-use map, as the process requires each village to name two representatives to collaborate with the district planner and surveying experts from the District and the Ministry of land. The local reception to the mapping process is generally positive as it is perceived by villagers as a way to formalize their use of village land 13 , and to help in solving conflict issues over boundaries with other villages.

However, few villagers are informed and participate in this mapping process. Moreover, their power over it and its outcomes is very minimal. The planners and surveyors have the final authority, leading to practices of consultation that are more procedural than collaborative because villagers are not allowed to significantly contribute to the process or to contest higher authorities.

For example, in a village in Pwani region, sunshine at a specific time of the day was traditionally used to identify village's demarcations. The village representatives asked the planning team to wait for the "right timing" for them to identify correctly the boundaries of their village. Their request was refused and therefore, the village's boundaries were delimited approximately. Unsurprisingly, it led to a problematic land-use map because the identified boundaries were wrong. Despite repetitive demands from the village for correctives to their landuse map and an explanation of the problem with the mapping process, they were unable to proceed any change since the Ministry of land resolved the issue by stating that changes are not allowed.

Moreover, the Ministry of land put forward a cadastral mapping of villages that does not necessarily consider historical and local understandings of demarcations. It favors an idea of what a developed village should look like, privileging straight boundaries over real demarcations, thereby leading to villages' land dispossession. For instance, in a village of Kagera region, several villages have ended up losing parts of their land. Among other examples, one of the village received its land-use plan and village land title in 2015. The leaders quickly realized that a whole section of their village was missing from the map. The land-use map draws a straight line rather than the curve which was necessary to include this part of the village. Following a village meeting, the village decided collectively to refuse the maps provided to them. When the leaders raised the issue with the District and explained why they refused to recognize the validity of their land-use map, they were threatened. The District made clear to them that by refusing to agree with this land-use plan, their village may end up losing all its land. Powerless, the leaders were afraid of the consequences of their resistance. They complied and signed the document, preferring losing a part of their village land than risking losing it all 14 .

As demonstrated in this last example, even though some villagers may contest the state's authority in mapping and planning, they generally comply without much resistance. Villagers' reaction is anchored into the historically constituted relation of governance between the state and citizens. Tanzania is a country which has undergone right after Independence a massive state-led villagisation program which has displaced forcefully about five million rural Tanzanian during the 70s. [START_REF] Hyden | Heinemann; Christy Cannon Lorgen. "Villagisation in Ethiopia, Mozambique, and Tanzania[END_REF] People perceive government's authorities as providers but also as the ultimate decision-makers. As one farmer states eloquently about the local perception of the supreme authority of the government: "Serikali ni serikali" [government is the government]. If the government tells you that you must go, you should go. […] Government is like a machete: both sides cut" (by author translation, Interview 49, Farmer, 2017).

Therefore, I argue that current mapping practices undergoing in Tanzania share striking similarities with the way villagisation was conceived and implemented. It demonstrates the lasting impact of institutional legacies. In fact, Scott's argument is still very relevant to understand the way the state is implementing its land-mapping programme. Scott argues that villagisation in Tanzania was a large-scale engineering project undertaken to render society and territory more legible and administrable, but also privileging a rationality of simplification and standardization-based on a high modernist ideology. [START_REF] Scott | Seeing like a State How Certain Schemes to Improve the Human Condition Have Failed[END_REF] Thus, land mapping practices are informed by features of both institutions. The language and the form of practices is coherent with the official neoliberal discourse, while the process itself is influenced by a different conception of the role of the state, which has been historically constituted and enforced.

Conclusion

To conclude, as demonstrated by these two examples of consultation practices, historical institutional and ideational legacies still shape significantly governance practices in Tanzania, despite the official adoption of neoliberal discourse and policies. The state is still conceptualized the way it was during the socialist era, as the only instance that could provide guidance in development. Therefore, understanding how 'things are done' in Tanzania requires acknowledging the agency of actors in navigating these ideational divergent norms.

  In Tanzania, most recent agricultural policies -Kilimo Kwanza (2009), South Agricultural Growth Corridor (SACGOT, 2010), New Alliance for Food Security and nutrition initiative (New Alliance, 2012), Big Results Now (BRN, 2012), and the National Agriculture Policy (NAP, 2013) -are rooted within a national framework: the Tanzania's National Strategy for Growth and Reduction of Poverty I and II or MKUKUTA I and II. Both MKUKUTA I and II are in fact agreements over strategic funding of policies resulting from a dialogue undertaken between major donors, international institutions such as the International Monetary Fund (IMF), the World Bank (WB) and the Tanzanian government. 1

1

  Aili Mari Tripp, "Donor Assistance and Political Reform in Tanzania, UNU-WIDER (World Institute for Development Economics Research), Working Paper no.2012/37, April 2012, pp. 16. 2 IMF, "Tanzania: Poverty Reduction Strategy Paper" (IMF Country Report No. 11/17, 2011).

8

  Interview 81: Tanzanian CSO 1, February 2017. 9 Interview 81: Tanzanian CSO 1, February 2017. 10 Ministry of Lands, Housing and Human settlements development, "Concept Note: Programme for Planning, Surveying and Land Titling in Tanzania", Government of Tanzania, November 2015, pp. 4. 11 Interview 81: Tanzanian CSO 1, February 2017.

Ministry of Lands, Housing and Human settlements development, "Concept Note: Programme for Planning,


Anna Fichtmüller

OPTATION STATE INFLUENCE IN THE MARKET AND THE INFORMAL SECTOR IN KAMPALA

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction 1

It is said that the mobilization of particular interest groups is a vital component to the consolidation of democracy (Goodfellow 2017a). Thus, discovering a vibrant associational structure in the transport and market sectors in Uganda's capital Kampala seems to give hope for the development of an active civil society. Both sectors are vital to the city's economy. In a country that lacks industrialization trading, and particularly petty trading within the informal economy, has become an important source of income for a majority of Kampala inhabitants. Conservative estimates believe that there are about 200 markets in Kampala, of which only a quarter are gazetted by the Kampala Capital City Authority -KCCA (Monteith 2016 : 155).

This does not take the numerous hawkers and street vendors into account. It is equally evoked as a place for the "common man" and thus an important target for political support. The transport sector on the other hand is estimated to employ about 100 000 people; it services many more commuters on a daily basis and has a large geographical outreachit is an important tool for mobilization (Goodfellow 2017(Goodfellow : 1569)). Since both sectors have a political clout and are well organized into associational structures, they appear to be important counterweights, representing the interests of the employees in their industries to the government. A closer look however, reveals the proximity between past and present associations with the government through a co-optation of the leadership and -more recently -the .1.

I FRA

Nairobi

.1.

I FRA

Nairobi

Volume XIV, (5) 2017

Anna FICHTMÜLLER

Aerial view of one of Kampala's taxi parks (2012) @Anna Fichtmüller

CONTROL & CO-OPTATION STATE INFLUENCE IN THE MARKET AND THE INFORMAL SECTOR IN KAMPALA

A journalist described how one can observe the sudden increase of these associations during the election period, as they are all competing for favors from the political parties. Beyond elections, there are about five groups that remain active. The biggest and most influential association is Boda Boda 2010, which is close to the current government, and according to its representative, is in control of "the industry [...] and all boda bodas in Kampala." Other groups, such as KAMBE and KAMBA or Century Boda Boda have been created in opposition to Boda Boda 2010. KAMBE, the second largest group is said to be under opposition control. There is apparently an ongoing effort to unite all competing associations of the industry under the umbrella of the government allied Boda Boda 2010. This is being done through a call for unity, leaving "politics" behind, in favor of "development". "de-democratization" of these structures by integrating their mandates into the work of the KCCA, a body controlled and appointed by the central government.

The Kampala City Council (KCC) was the body of the municipal government, approximately 80% of services were provided through the KCC. With the introduction of multipartism, Kampala became a stronghold of the opposition and the KCC has since then been governed by the opposition. In 2010 the Kampala Capital City Act paved way for the creation of the KCCA, which was widely considered as a mechanism to restrain opposition control over the city. This move dis-empowered the KCC and reduced the elected mayor to a mere ceremonial function. Instead, the management of the city became the responsibility of the executive director of KCCA, who was directly appointed by President Museveni.

This paper aims at highlighting some recurring tendencies of government control over seemingly independent bodies by combining experiences from the market and transport sectors. These strategies are basically three-fold:

to get these people when they are organized. […] According to me, they are all under political groups." Representative of Boda Boda 2010 (Kampala, 23.02.2017) "We have been having more than like 58 associations in the industry. But mainly, they have started then, because of politics, that is one, their own interests, their own interest of getting money from other people so that is where that people have started many associations. […] So, these politicians, […] they are pushing them to set an association of which it will be very easy "Fighting each other will not be the lasting solution of this industry. So that is the maturity that we are building now. That let us sit on a round table, you join that association which is in power, and then we resolve all the problems in the industry." Representative of Boda Boda 2010 (Kampala, 23.02.2017).

The proximity with NRM is equally visible in the market leadership of the non-gazetted Park Yard market, in the New Agenda Traders Development Association (NATDA). One vendor close to opposition, criticized the allegiance of their leadership to the government, to the point that they would even be hesitant to talk to people like her, knowing about her political affiliation. A representative of the association underlined the positive relationship with the government:

"The government listens to us; we created awareness through KCCA and the ministers of Kampala. […] We are not disappointed, they are promising the best. Now we have fewer evictions, we meet on a round table." NATDA Representative (Kampala, 16.01.2017) In the case of the gazetted USAFI market, no leadership elections were held but the chairman was appointed by the KCCA executive director herself, for reasons allegedly unknown to him. The good relationship proved in the end to be futile for Park Yard, as the vendors were evicted in February 2017, about one month after the interview with the NATDA chairman, in a move led by the minister for Kampala, Bety Kamya. The closing was done under

The positioning of the associational leadership in proximity with the ruling party, the National Resistance Movement (NRM). The harassment of opposition and associations not aligned with the government. The personalization of politics due to intervention of President Museveni himself. 1.

2.

3.

The paper then looks at the shift from KCC to KCCA and how it has brought in new dynamics in the associational landscape of these sectors, by appointing instead of electing leaders, suggesting a shift from dysfunctional associations to non-representative leadership.

Control and Co-optation

As mentioned above, the transport sector is considered an important arena for political mobilization. One of its increasingly growing branches is the motorcycle taxis, commonly called boda bodas. There have been several attempts by politicians of all factions to gain support from boda boda drivers and organize them into associations. There are about 60 associations today in the sector, which have, according to an associational representative, all come into being for political reasons: heavy police surveillance, but without any remarkable protests. This sparked allegations among the vendors that their leadership has been bought off, so as not to protest against the action. In the wake of the eviction Kampala's lord mayor and three members of parliament visited the market to show support to the vendors. They eventually got arrested by police, hinting again at the power struggle between opposition and government 2 . This crackdown on opposition can also be observed among the taxi 3 drivers. Since 1986 the transport sector was firmly controlled by an elite, closely linked to the government, which used its position to lever several fees, but for which only a fraction was accounted for towards KCCA (Goodfellow 2017a). In opposition to these practices the Uganda Transport Development Agency (UTRADA) was created by drivers from within the sector, but its efforts yielded no results and were met with a lot of violence. It was only in 2011 when the KCCA took over the taxi business that the fee extortion decreased. At the same time however, it drastically limited the power of any existing organization. Because in 2015 a ministerial order declared the formation of any association as illegal, UTRADA has since struggled to gain clout, as they have no legal right to assemble or mobilize. Recruitment thus becomes a challenge, and many drivers fear disadvantages if they join. According to the UTRADA representative, the government put crime preventer at the taxi stages for surveillance and intimidation, who violently oppose activists within the sector. The reign of fear and the lack of capital to pay for legal fees in case of detention have made activism in the taxi sector increasingly difficult, leaving no avenues for representation other than through KCCA bodies.

By and large, within the boda boda sector and in the case of Park Yard and USAFI market we have seen the proximity of the main associations and the NRM government. Goodfellow, Titeca and Monteith have shown similar patterns in the markets of Nakasero and Kisekka (Goodfellow & Titeca 2012; Monteith 2016). On the contrary, the case of UTRADA shows how an organization is put under pressure if it does not follow the government line.

The tendency of replacing elected leaders with appointed managers and constraining free associations in the market and transport sector hints at a general de-democratization. This shift to more authoritarian leadership may make sense, when put in relation to the coming into existence of the KCCA.

The KCCA -a new player in town

One way of gaining political support for President Museveni among large urban informal groups, has been through presidential intervention to prevent policy changes that may be unfavorable to these groups, as long as the city was governed by KCC. For example, there have been several attempts to introduce taxation and regulation in the largely deregulated sector of motorcycle taxis, commonly known as boda bodas. These attempts included regulations on: the number of passengers to carry, the obligation to wear helmets and safety jackets, or to only ride with a valid driving permit. They have been stopped through the personal interference of the president (See Goodfellow 2017b : 127). The same could be observed in the market sector (Monteith 2016). The direct appeal to the president by the various informal groups provided an opportunity for Museveni to sabotage policy implementation of the city government governed by opposition, and at the same time gain support among these groups. Goodfellow and Titeca observe:

2 "Row at Kla Park Yard: Lord Mayor Lukwago, MPs arrested". The Spear, 22.02.2017. Accessed December 14, 2017. Internet link.

3 The minibuses used for public transport are called "taxis" in Uganda, whereas regular taxis are referred to as "special hire".

"Although the city government was formally empowered by decentralization, it was at the same time disempowered by underfunding, the privatization of many of its functions andabove all -the constant intervention of central government." (Goodfellow & Titeca 2012 : 266) Being able to call on the president and have him act in their interest was thus an important political capital to have, and yet another reason to foster close links between the associational leaderships and the ruling government.

With the creation of KCCA however, the strategies needed to be revised. Contravening unpopular policy decisions is not an option anymore, if the president does not want to discredit his own executive body. Certainly, the president's influence can still be seen in the city; for example, in the run up to the 2016 election, street hawkers were allowed on the street, despite previous severe crackdowns by KCCA. This is believed to have been a campaign strategy, to gain voter support. The fact that regulations of street hawkers continued as usual after the election supports this contention. He may still receive interest groups to present their stakes, but action does not necessarily have to follow, as in the case of UTRADA. After calling for a strike in protest of "multiple taxation" through KCCA, President Museveni met with the UTRADA leadership and KCCA, ordering the authority to stop illegal fees, however the charging continued. The UTRADA representative assumes that the President is not genuine: "because, in my knowing, if the president says so, it is an order, I think he is calling them [KCCA] back after we have gone, says 'You leave it!'" However, interference has decreased, which has gained KCCA the reputation of being an efficient organization, particularly among middle and high-income earners in the city. Yet, the question remains, how is discontentment being addressed? For one, it seems to be the opposition who tries to have a political gain out of KCCA regulations, as seen in the case of the Park Yard closure.

As the eviction of the Park Yard vendors has shown, their scope of action is nevertheless much more limited. Secondly, the de-democratization as implemented by KCCA results in the closing of space for voicing discontent. Through the direct appointment of market leaders and transport committee members the KCCA is able to choose leaders in the guise of the ruling party, who are not accountable to those they are supposed to represent. Closing down associational structures additionally limits the capacities of the workers and vendors to mobilize against legislation they deem unfavorable and thus diminishes their possibilities to voice discontent. In return, an increase in small, localized riots over the past ten years can be observed; it remains to be seen whether the shrinking space fostered through KCCA hegemony will further exacerbate this tendency.

Conclusion

While organizations representing group interests are usually seen as a positive sign of democratic consolidation, the example of Uganda has shown that they can also have detrimental effects. The proximity of the leadership to government, even though vital in personalized politics to guarantee a good connection with high ranking politicians, compromises the ability of the management to act on behalf of their members. This is exacerbated if the leadership receives personal gains, such as gifts or handouts4 . Coupled with repression of opposing points of view, the politics of the "carrot and the stick" widely used by the NRM government can also be observed in these sectors. In that light, the shift from dysfunctional to no representation, as a result of KCCA hegemony, gives little hope that workers in these sectors will be sufficiently heard in the debate on the development of the city.

IFRA Laikipia Road, PO Box 58480 -00200 Nairobi, Kenya E-mail: info@ifra-nairobi.net -Web site: www.ifra-nairobi.net

This paper is a "byproduct" of a two months field research in the Ugandan capital Kampala, in

2017, supported by IFRA Nairobi. I call it a "byproduct" because the original scope of my research concentrated on the link between individual economic emergence and social mobilization. The data I refer to, has been collected through five expert interviews with two market chairmen, two representatives of associations in the transport sectors, and a KCCA official, as well as several informal discussions with workers in these sectors.

The tendency to "buy off" leaders can not only be observed in the sectors mentioned above, but also among other trade unions, as the recent case of the abrupt ending of the strike of the Makerere University Academic Staff Association (MUASA) has shown and in the wider spheres of society, for example among religious leaders (see Fichtmüller 2014).

Bibliography

Fichtmüller Anna, 'For God and My Country' -Le religieux dans la sphère politique contemporaine en Ouganda, Observatoire des enjeux politiques et sécuritaires dans la corne de l'Afrique, 2014.

Goodfellow Tom, "'Double Capture' and De-Democratisation: Interest Group Politics and Uganda's 'Transport Mafia'", The Journal of Development Studies, vol. 53, no 10, 2017a, pp. 1568-1583. Goodfellow


Michel De Lara
email: lara@cermics.enpc.fr

A Mathematical Framework for Resilience: Dynamics, Uncertainties, Strategies and Recovery Regimes

Keywords: resilience, control theory, uncertainty, risk measures, recovery regimes

Resilience is a rehashed concept in natural hazard managementresilience of cities to earthquakes, to floods, to fire, etc. In a word, a system is said to be resilient if there exists a strategy that can drive the system state back to "normal" after any perturbation. What formal flesh can we put on such a malleable notion? We propose to frame the concept of resilience in the mathematical garbs of control theory under uncertainty. Our setting covers dynamical systems both in discrete or continuous time, deterministic or subject to uncertainties. We will say that a system state is resilient if there exists an adaptive strategy such that the generated state and control paths, contingent on uncertainties, lay within an acceptable domain of random processes, called recovery regimes. We point out how such recovery regimes can be delineated thanks to so called risk measures, making the connection with resilience indicators. Our definition of resilience extends others, be they "à la Holling" or rooted in viability theory. Indeed, our definition of resilience is a form of controlability for whole random processes (regimes), whereas others require that the state values must belong to an acceptable subset of the state set.

Introduction

Consider a system whose state evolves with time, being subject to a dynamics driven both by controls and by external perturbations. The system is said to be resilient if there exists a strategy that can drive the system state towards a normal regime, whatever the perturbations. Basic references are [START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF][START_REF] Martin | Defining resilience mathematically: from attractors to viability[END_REF][START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF][START_REF] Arnoldi | Resilience, reactivity and variability: A mathematical comparison of ecological stability measures[END_REF].

In the case of fisheries, the state can be a vector of abundances at ages of one or several species; the control can be fishing efforts; the external perturbations can affect mortality rates or birth functions appearing in the dynamics (an extreme perturbation could be an El Niño event, affecting the populations renewal). In the case of a city exposed to earthquakes, floods or other climatic events, the state can be a vector of capital stocks (energy reserves, energy production units, water treatment plants, health units, etc.); the controls would be the different investments in capital as well as current operations (flows in and out capital stocks); the dynamics would express the changes in the stocks due to investment and to day to day operations; external perturbations (rain, wind, climatic events, etc.) would affect the stocks by reducing them, possibly down to zero.

In Sect. 2, we introduce basic ingredients from the mathematical framework of control theory under uncertainty. Thus equipped, we frame the concept of resilience in mathematical garbs in Sect. 3. Then, in Sect. [START_REF] Lara | Sustainable Management of Natural Resources[END_REF], we provide illustrations of the abstract general framework and compare our approach with others, "à la Holling" or the stochastic viability theory approach to resilience. In Sect. 5, we sketch how concepts from risk measures (introduced initially in mathematical finance) can be imported to tackle resilience issues. Finally, we discuss pros and cons of our approach to resilience in Sect. 6.

Ingredients for an abstract control system with uncertainties

We outline the mathematical formulations of time, controls, states, Nature (uncertainties), flow (dynamics) and strategies. As the reference [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF] is the more mathematically driven paper on resilience, we will systematically emphasize in what our approach differs from that of Rougé, Mathias and Deffuant.

Time, states, controls, Nature and flow

We lay out the basic ingredients of control theory: time, states, controls, Nature (uncertainties) and flow (dynamics).

Time

The time set T is a (nonempty) subset of the real line R. The set T holds a minimal element t 0 ∈ T and an upper bound T , which is either a maximal element when T < +∞ (that is, T ∈ T) or not when T = +∞ (that is, +∞ ∈ T). For any couple (s, t) ∈ R ∪ {+∞} 2 , we use the notation

s : t = {r ∈ T | s ≤ r ≤ t} (1)
for the segment that joins s to t (when s > t, s : t = ∅).

Special cases of discrete and continuous time. This setting includes the discrete time case when T is a discrete set, be it infinite like T = {t 0 + k∆t, k ∈ N} (with ∆t > 0), or finite like T = {t 0 + k∆t, k = 0, 1, . . . , K} (with K ∈ N). Of course, in the continuous time case, T is an interval of R, like [t 0 , T ] when T < +∞ or [t 0 , +∞[ when T = +∞. But the setting makes it possible to consider interval of continuous times separated by discrete times corresponding to jumps. For these reasons, our setting is more general than the one in [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF], which considers discrete time systems.

Environmental illustration. In fisheries, investment decisions (boats, equipment) are made at large time scale (years), regulations quotas are generally annual, boat operations are daily. By contrast, populations and external perturbations evolve in continuous time. Depending on the issues at hand, the modeler will choose the proper time scales, symbolized by the set T. In an energy system, like a micro-grid with battery and solar panels, investment decisions in equipment (buying or renewal) occur at large time scale, whereas flows inside the system have to be decided at short time scales (minutes).

States, controls, Nature

At each time t ∈ T,

• the system under consideration can be described by an element x t of the state set X t ,

• the decision-maker (DM) makes a decision u t , taken within a control set U t .

A state of Nature ω affects the system, drawn within a sample set Ω, also called Nature. No probabilistic structure is imposed on the set Ω.

Environmental illustration. In the case of dengue epidemics control at daily time steps, the state can be a vector of abundances of healthy and deseased individuals (possibly at ages), together with the same description for the mosquito vector; the control can be the daily fumigation effort, mosquito larva removal, quarantine measures, or the opening and closing of sanitary facilities; Nature represents unknown factors that affect the dengue dynamics, like rains, humidity, mosquito biting rates, individual susceptibilities, etc. Some of these factirs (like rain) can be progressively unfolded as times passes.

Special case where the sample set is a set of scenarios. In many cases, at each time t ∈ T, an uncertainty w t affects the system, drawn within an uncertainty set W t . Hence, a state of Nature has the form ω = {w t } t∈Tand is called a scenario -drawn within a product sample set Ω = t∈T W t .

Environmental illustration. The above definition of scenarios is in phase with the vocable of scenarios in climate change mitigation; it represents sequences of uncertainties that affect the climate evolution. In our framework, a scenario is not in the hands of the decision-maker; for instance, a scenario does not include investment decisions.

Relevance for resilience. In the case of scenarios, as the uncertainty sets W t depend on t, we cover the case where

• an uncertainty w t ∈ W t affects the system at each time t, possibly progressively revealed to the DM, hence available when he makes decisions;

• other uncertainties, that are present from the start (like parameters), hence are part of the set W t 0 ; such uncertainties are not necessarily revealed to the DM as times passes, and remain unknown.

Our setting is more general than the one in [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF]. First, we do not restrict the sample set to be made of scenarios as Rougé, Mathias and Deffuant do.

Second, even in the case of scenarios, no probabilistic structure is imposed on the set t∈T W t whereas Rougé, Mathias and Deffuant require that it be equipped with a probability distribution having a density (with respect to an, unspecified, measure, likely the Lebesgue measure on a Euclidian space).

State and control paths

With the basic set T and the basic families of sets {X t } t∈T and {U t } t∈T , we define Relevance for resilience. We introduce paths because, as stated in the abstract, our (forthcoming) definition of resilience requires that, after any perturbation, the system returns to an acceptable "regime", that is, that the state-control path as a whole must return to a set of acceptable paths (and not only the state values must belong to an acceptable subset of the state set). We introduce tail paths because resilience encapsulates the idea that recovery is possible after some time, and that the system remains "normal" after that time.

• the set t∈T X t of

Dynamics/flow

We now introduce a dynamics under the form of a flow {φ s:t } (s,t)∈T 2 , that is, a family of mappings

φ s:t : X s × t r=s U r × Ω → t r=s X r . (2)
When s > t, all these expressions are void because s : t = ∅. The flow φ s:t maps an initial state xs ∈ X s at time s, a tail control path {u r } r∈s:t and a state of Nature ω towards a tail state path {x r } r∈s:t = φ s:t xs , {u r } r∈s:t , ω ,

with the property that x s = xs .

Relevance for resilience. Our setting is more general than the one in [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF]: as illustrated below, we cover differential and stochastic differential systems, in addition to iterated dynamics in discrete time (which is the scope of Rougé, Mathias and Deffuant). Our approach thus allows for a general treatment of resilience.

Cemetery point to take into account either analytical properties or bounds on the controls. In general, a state path cannot be determined by ( 3) for any state of Nature or for any control path, for analytical reasons (measurability, continuity) or because of bounds on the controls. To circumvent this difficulty, one can use a mathematical trick and add to any state set X t a cemetery point ∂. Any time a state cannot be properly defined by the flow by (3), we attribute the value ∂. The vocable "cemetery" expresses the property that, once in the state ∂, the future state values, yielded by the flow, will all be ∂. Therefore, the stationary state path with value ∂ will be the image of those scenarios and control paths for which no state path can be determined by [START_REF] Carpentier | Stochastic Multi-Stage Optimization[END_REF].

Special case of an iterated dynamics in discrete time. In discrete time, when T = N, the flow is generally produced by the iterations of a dynamic

x t = x , x s+1 = F s (x s , u s , w s ) , t ≥ s . (4)
How do we include control constraints in this setting? Suppose given a family of nonempty set-valued mappings U s : X s ⇒ U s , s ∈ T. We want to express that only controls u s that belong to U s (x s ) are relevant. For this purpose, we add to all the state sets X s a cemetery point ∂. Then, when u r ∈ U r (x r ) in ( 4) for at least one r ∈ s : t, we set φ s:t xs , {u r } r∈s:t , {w r } r∈s:t , γ = {∂} s∈t:T in (3).

Environmental illustration. In natural resource management, many population models (anmal, plants) are given by discrete time abundance-at-age dynamical equations. Outside population models, many stock problems are also based upon discrete time dynamical equations. This is the case of dam management, where water stock balance equations are written at a daily scale (possibly less like every eight hours, or possibly more like months for long term planning); control constraints represent the properties that turbined water must be less than the current water stock and bounded by turbine capacity.

Special case of differential systems. In continuous time, the mapping φ s:t in (2) generally cannot be defined over the whole set t r=s U r × Ω. Tail control paths and states of Nature need to be restricted to subsets of t r=s U r and Ω, like the continuous ones for example when dealing with Euclidian spaces. For instance, when T = R + and the flow is produced by a smooth dynamical system on a Euclidian space X = R n

x t = x , ẋs = f s (x s , u s ) , s ≥ t , (5)
control paths {u s } s∈t:T are generally restricted to piecewise continuous ones for a solution to exist.

Special case of stochastic differential equations. Under certain technical assumptions, a stochastic differential equation

dX s = f s (X s , U s , W s )ds + g(X s , U s , W s )dW s , (6)
where {W s } s∈R + is a Brownian motion, gives rise to solutions in the strong sense. In that case, a flow can be defined (but not over the whole set

t r=s U r × Ω).
The case of the history flow. Any possible state derives from the socalled history

h t = {u r } r∈t 0 :t , ω . (7)
In that case, the flow (3) is trivially given by {h r } r∈s:t = h s , {u r } r∈s:t . We will use the notion of history when we compare our approach with the viability approach to resilience.

Adapted and admissible strategies

A control u t is an element of the control set U t . A policy (at time t) is a mapping

λ t : X t × Ω → U t (8)
with image in the control set U t . A strategy is a sequence {λ t } t∈T of policies.

Environmental illustration. In climate change mitigation, a strategy can be an investment policy in renewable energies as a function of the past observed temperatures. In epidemics control, a strategy can be quarantine measures or vector control as a function of observed infected individuals.

Admissible strategies

Suppose given a family of nonempty set-valued mappings U t :

X t × Ω ⇒ U t , t ∈ T.
An admissible strategy is a strategy {λ t } t∈t 0 :T such that control constraints are satisfied in the sense that, for all t ∈ T,

λ t (x t , ω) ∈ U t (x t , ω) , ∀(x t , ω) ∈ X t × Ω . (9)

Adapted strategies

Suppose that the sample set Ω is equipped with a filtration {F t } t∈T . Hence each F t is a σ-field and the sequence t → F t is increasing (for the inclusion order). Suppose that each state set X t , is equipped with a σ-field X t . An adapted policy is a mapping (8) which is measurable with respect to the product σ-field X t ⊗ F t . An adapted strategy is a family {λ t } t∈t 0 :T of adapted policies.

Special case where the sample set is a set of scenarios. Consider the case where Ω = t∈T W t and where each set W t is equipped with a σ-field W t (supposed to contain the singletons). The natural filtration {F t } t∈T is given by

F t = r≤t W r ⊗ s>t {∅, W s } . (10)
Then, in an adapted strategy {λ t } t∈t 0 :T , each policy can be identified with a mapping of the form [3]

λ t : X t × t r=t 0 W r → U t . (11)
In that case, our definition of adapted strategy means that the DM can, at time t, use no more than time t, current state value x t and past scenario {w s } s∈t 0 :t to make his decision u t = λ t (x t , {w s } s∈t 0 :t ).

Relevance for resilience. Though this is not the most general framework to handle information (see [START_REF] Carpentier | Stochastic Multi-Stage Optimization[END_REF] for a more general treatment of information), we hope it can enlighten the notion of adaptive response often found in the resilience literature. Our setting is more general than the one in [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF]: indeed, Rougé, Mathias and Deffuant only consider state feedbacks, that is, Markovian strategies as defined below. By contrast, our setting includes the case of corrupted and partially observed state feedback strategies, that is, the case where strategies have as input a partial observation of the state that is corrupted by noise.

Special case of Markovian or state feedback strategies. Markovian or state feedback policies are of the form

λ t : X t → U t . (12)
With this definition, we express that, at time t, the DM can only use time t and current state value x t -but not the state of Nature ω -to make his decision u t = λ t (x t ). In some cases (when dynamic programming applies for instance), it is enough to restrict to Markovian strategies, much more economical than general strategies.

Closed loop flow

From now on, when we say "strategy", we mean "adapted and admissible strategy".

Given an initial state and a state of Nature, a strategy will induce a state path thanks to the flow: this gives the closed loop flow as follows.

Let s ∈ T and t ∈ T, with s < t. Let {λ t } t∈T be a strategy. We suppose that, for any initial state xs ∈ X s and any state of Nature ω, the following system of (closed loop) equations 3 Resilience: a mathematical framework

{x r } r∈s:t = φ s:t xs , {u r } r∈s:t , ω (13a)
u r = λ r (x r , ω) ,
Equipped with the material in Sect. 2, we now frame the concept of resilience in mathematical garbs. For this purpose, we introduce the notion of recovery regime. Compared to other definitions of resilience [START_REF] Holling | Resilience and stability of ecological systems[END_REF][START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF][START_REF] Martin | Defining resilience mathematically: from attractors to viability[END_REF][START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF], our definition requires that, after any perturbation, the state-control path as a whole can be driven, by a proper strategy, to a set of acceptable paths (and not only the state values must belong to an acceptable subset of the state set, asymptotically or not). In addition, as state and control paths are contingent on uncertainties, we require that they lay within an acceptable domain of random processes, called recovery regimes.

Once again, as the reference [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF] is the more mathematically driven paper on resilience, we will systematically emphasize in what our approach differs from that of Rougé, Mathias and Deffuant.

Robustness, resilience and random processes

The notion of robustness captures a form of stability to perturbations; it is a static notion, as no explicit reference to time is required. By contrast, the concept of resilience makes reference to time (dynamics), strategies and perturbations. This is why, to speak of resilience -a notion that mixes time and randomness -we find it convenient to use the framework of random processes, although this does not mean that we require any probability.

From now on, we consider that the sample space Ω is a measurable set equipped with a σ-field F (but not necessarily equipped with a probability). When we consider a deterministic setting, Ω is reduced to a singleton (and ignored). The set of measurable mappings from Ω to any measurable set Y will be denoted by L 0 Ω, Y . Elements of L 0 Ω, Y are called random variables or random processes, although this does not imply the existence of an underlying probability. Random variables are designated with bold capital letters like Z. From now on, every state set X t is a measurable set equipped with a σ-field X t , every control set U t with a σ-field U t , and, when needed, every uncertainty set W t with a σ-field W t .

Fields are introduced when probabilities are needed. When they are not, as with the robust setting, it suffices to equip all sets with their complete σ-fields, made of all subsets. Then, measurable mappings L 0 Ω, Y from Ω to any set Y are all mappings.

Recovery regimes

Recovery regimes, starting from t ∈ T, are subsets of random processes of the form

A t ⊂ L 0 Ω, T s=t X s × T s=t U s . (15)
When there are no uncertainties, Ω is reduced to a singleton, so that A t ⊂ T s=t X s × T s=t U s , as in the two first following examples.

Example of recovery regimes converging to an equilibrium. Let T = R + , X t = R n and U t = R m , for all t ∈ T. Let x be an equilibrium point of the dynamical system ẋs = f (x s , ū) when the control is stationary equal to ū, that is, 0 = f (x, ū). The recovery regimes, starting from t ∈ T, converging to the equilibrium x form the set

A t = {x s } s≥t , {u s } s≥t ∈ X [t,+∞[ × U [t,+∞[ | x s → s→+∞ x . (16)
In general, the equilibrium x is supposed to be asymptotically stable, locally or globally.

A more general definition would be

A t = {x s } s≥t , {u s } s≥t lim s→+∞ x s exists , (17)
and, to account for constraints on the values taken by the controls, we can consider

A t = {x s } s≥t , {u s } s≥t lim s→+∞ x s exists and u s ∈ U s (x s ) , ∀s ≥ t , (18)
where U s : X s ⇒ U s , for all s ∈ T.

Example of bounded recovery regimes. Let

T = R + , X t = R n and U t = R m , for all t ∈ T. If B is a bounded region of X = R n , we consider A t = {x s } s≥t , {u s } s≥t | x s ∈ B , ∀s ≥ t . ( 19
)
When B is a ball of small radius ρ > 0 around the equilibrium x, we obtain state paths that remain close to x.

Example of random recovery regimes. Suppose that the measurable sample space (Ω, F) is equipped with a probability P. Let X t = R n and U t = R m , for all t ∈ T. Letting B be a bounded region of X = R n and β ∈]0, 1[, the set

A t ={ {X s } s≥t , {U s } s≥t ∈ L 0 Ω, T s=t X s × T s=t U s | P ∃s ≥ t | X s ∈ B ≤ β} (20)
represents state paths that get at least once outside the bounded region B with a probability less than β.

If T is discrete, the set

A t ={ {X s } s≥t , {U s } s≥t ∈ L 0 Ω, T s=t X s × T s=t U s | P ∃s 1 ≥ t , s 2 ≥ t , s 3 ≥ t | X s 1 ∈ B , X s 2 ∈ B , X s 3 ∈ B = 0} (21)
represents state paths that get no more than two times outside the bounded region B.

Resilient strategies and resilient states

Consider a starting time t ∈ T and an initial state xt ∈ X t . We say that the strategy λ is a resilient strategy starting from time t in state xt if the random process {X s } s∈t:T , {U s } s∈t:T given by

{X s (ω)} s∈t:T = φ λ t:T x t , ω (22a) U s (ω) = λ s (X s (ω), ω) , ∀s ∈ t : T , (22b)
where the closed loop flow φ λ t:T is given in ( 14), is such that

{X s } s∈t:T , {U s } s∈t:T ∈ A t . ( 23
)
Notice that we do not use the part {λ r } r<t of the strategy λ = {λ r } r∈t 0 :T . With this definition, a resilient strategy is able to drive the state-control random process into an acceptable regime. As a resilient strategy is adapted, it can "adapt" to the past values of the randomness but no to its future values (hence, our notion of resilience does not require clairvoyance of the DM). Our definition of resilience is a form of controlability for whole random processes (regimes): a resilient strategy has the property to shape the closed loop flow φ λ t:T so that it belongs to a given subset of random processes. We denote by Λ R t (x t ) the set of resilient strategies at time t, starting from state xt . The set of resilient states at time t is

X R t = xt ∈ X t Λ R t (x t ) = ∅ . ( 24
)
4 Illustrations

In Sect. 3, we have provided some illustrations in the course of the exposition. Now, we make the connection between the previous setting and two other settings, the resilience "à la Holling" [START_REF] Holling | Resilience and stability of ecological systems[END_REF] in §4.1 and the resilience-viability framework [START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF][START_REF] Martin | Defining resilience mathematically: from attractors to viability[END_REF][START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF] in §4.2.

Deterministic control dynamical system with attractor

As the paper [START_REF] Holling | Resilience and stability of ecological systems[END_REF] does not contain a single equation, it is bit risky to force the seminal Holling's contribution into our setting. However, it is likely that it corresponds to T = R + and to recovery regimes of the form

A t = {x s } s≥t , {u s } s≥t | x s converges towards an attractor . ( 25
)
Note that, as often in the ecological literature on resilience [START_REF] Arnoldi | Resilience, reactivity and variability: A mathematical comparison of ecological stability measures[END_REF], the underlying dynamical system is not controlled.

Resilience and viability

Some authors [START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF][START_REF] Martin | Defining resilience mathematically: from attractors to viability[END_REF][START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF] propose to frame resilience within the mathematical theory of viability [START_REF] Aubin | Viability Theory[END_REF]. Let X t = X and U t = U, for all t ∈ T. Let A ⊂ X denote a set made of "acceptable states". Let U s : X ⇒ U, s ∈ T be a family of set-valued mappings that represent control constraints.

Deterministic viability

Consider a starting time t ∈ T and the recovery regimes

A t = { {x s } s≥t , {u s } s≥t | x s ∈ A , u s ∈ U s (x s ) , ∀s ≥ t} . (26)
Then, a resilient strategy is one that is able to drive the state towards the set A of acceptable states.

Robust viability

When there are no uncertainties, we just established a connection between recovery regimes and viability. But, with uncertainties, as resilience requires a form of stability "whatever the perturbations", we are in the realm of robust viability [START_REF] Lara | Sustainable Management of Natural Resources[END_REF], as follows.

Let Ω ⊂ Ω, corresponding to the (nonempty) subset of states of Nature with respect to which the DM expects the system to be robust. Consider a starting time t ∈ T and the recovery regimes

A t = { {X s } s∈t:T {U s } s∈t:T ∈ L 0 Ω, T s=t X s × T s=t U s | ∃τ ∈ L 0 Ω, T , ∀ω ∈ Ω , τ (ω) ≥ t , X s (ω) ∈ A , U s (ω) ∈ U s (X s (ω)) , ∀s ≥ τ (ω)} . (27)
Then, a resilient strategy is one that is able to drive the state towards the set A of acceptable states, after a random time τ , whatever the perturbations in Ω.

Robust viability and recovery time attached to a resilient strategy

Let Ω ⊂ Ω, whose elements can be interpreted as shocks. Consider a starting time t ∈ T and an initial state xt ∈ X t . If λ = {λ s } s∈t:T is a resilient strategy for the recovery regimes (27), the recovery time is the random time defined by

τ (ω) = inf{r ∈ t : T | {X s (ω)} s∈t:T = φ λ t:T xt , ω U s (ω) = λ s (X s (ω), ω) , ∀s ∈ t : T X s (ω) ∈ A , U s (ω) ∈ U s (X s (ω)) , ∀s ≥ r} , (28)
for all ω ∈ Ω, with the convention that inf ∅ = +∞.

Thus, the resilient strategy drives the state towards the set A of acceptable states, after the random time τ , whatever the perturbations (shocks) in Ω. By contrast, the so-called time of crisis occurs before τ [6].

Stochastic viability

Suppose that the measurable sample space (Ω, F) is equipped with a probability P and let β ∈ [0, 1], represent a probability level. Consider a starting time t ∈ T and the recovery regimes

A t = { {X s } s∈t:T , {U s } s∈t:T ∈ L 0 Ω, T s=t X s × T s=t U s | P X s ∈ A , U s ∈ U s (X s ) , ∀s ≥ t ≥ β} . (29)
With these recovery regimes, we express that the probability to satisfy state and control constraints after time t is at least β [5].

Discussion and comparison with the viability theory approach for resilience.

Our setting is more general than the viability theory approach for resilience as introduced in [START_REF] Martin | The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication[END_REF][START_REF] Martin | Defining resilience mathematically: from attractors to viability[END_REF][START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF]. Indeed, the viability approach to resilience deals with constraints time by time; our approach does not. To illustrate our point, consider the deterministic case with discrete and finite time, and scalar controls, to make things easy. It is clear that the recovery regimes given by

A t = { {x s } s≥t , {u s } s≥t | min s≥t u s ≤ 0} (30a) = { {x s } s≥t , {u s } s≥t | ∃s ≥ t , u s ≤ 0} . (30b)
cannot be expressed as time by time constraints on the controls. Of course, the viability approach could handle such a case, but at the price of extending the state and the dynamics, to turn an intertemporal constraint into a time by time constraint. For instance, with the history state introduced at the end of §2.1.4, we can always express any recovery regimes set as viability constraints. In the example above, we do not need the whole history to turn the set A t into one described by time by time constraints: it suffices to introduce an additional component to the state like s≥t 1 {us≤0} in discrete time and impose the final constraint that this new part of an extended state be non zero.

To sum up, our approach to resilience covers more recovery regimes, described with the original states and controls, than those captured by the time by time constraints that make the specificity of the viability approach to resilience.

Resilience and risk

We now sketch how concepts from risk measures (introduced initially in mathematical finance [START_REF] Föllmer | Stochastic Finance. An Introduction in Discrete Time[END_REF]) can be imported to tackle resilience issues. This again is a novelty with respect to the stochastic viability theory approach for resilience as in [START_REF] Rougé | Extending the viability theory framework of resilience to uncertain dynamics, and application to lake eutrophication[END_REF]. Risk measures are potential candidates as indicators of resilience.

Recovery regimes given by risk measures

We start by a definition of recovery regimes given by risk measures, then we provide examples.

Definition of recovery regimes given by extended risk measures

Suppose that T ⊂ R is equipped with the trace T of the Borel field of R. Then, T × Ω is a measurable space when equipped with the product σfield T ⊗ F. Then, any random process in L 0 Ω, T s=t X s × T s=t U s can be identified with a random variable in L 0 t : T × Ω, T s=t X s T s=t U s . We call extended risk measure any G t that maps random variables in L 0 t : T × Ω, T s=t X s T s=t U s towards the real numbers [START_REF] Föllmer | Stochastic Finance. An Introduction in Discrete Time[END_REF]. The lower the risk measure G t , the better.

The basic example of a risk measure is the mathematical expectation under a given probability distribution. A celebrated risk measure in mathematical finance is the tail/average/conditional value-at-risk.

With G t an extended risk measure and α ∈ R a given risk level, we define recovery regimes by

A t = { {X s } s∈t:T , {U s } s∈t:T ∈ L 0 Ω, T s=t X s × T s=t U s | G t {X s } s∈t:T , {U s } s∈t:T ≤ α} . (31)
The quantity G t {X s } s∈t:T , {U s } s∈t:T measures the "risk" borne by the random process {X s } s∈t:T , {U s } s∈t:T . Therfore, recovery regimes like in (31) represent a form of "risk containment" under the level α.

Robust viability and the worst case risk measure

The robust viability inspired definition of resilience in §4.2.2 corresponds to (31) with α < 1 and the worst case risk measure

G s {X s } s∈t:T , {U s } s∈t:T = sup s∈t:T sup ω∈Ω 1 A c X s (ω) , (32)
where Ω ⊂ Ω. Indeed, G t {X s } s∈t:T , {U s } s∈t:T ≤ α < 1 means that 1 A c X s (ω) ≡ 0, that is, the state X s (ω) always belongs to A (as A c is the complementary set of A in X) for all ω ∈ Ω.

Here, the worst case risk measure captures that the state X s (ω) belongs to A both for all times -the core of viability, here handled by the term sup s≥t -and for all states of Nature in Ω -the core of robustness, here handled by the term sup ω∈Ω .

Stochastic viability and beyond: ambiguity

The stochastic viability inspired definition of resilience in §4.2.4 corresponds to (31) with α = 1β and the risk measure

G s {X s } s∈t:T , {U s } s∈t:T = P ∃s ≥ t | X s ∈ A or U s ∈ U s (X s ) . (33)
Now, suppose that different risk-holders do not share the same beliefs and let P denote a set of probabilities on (Ω, F). We can arrive at an ambiguity viability inspired definition of resilience using the risk measure

G s {X s } s∈t:T , {U s } s∈t:T = sup P∈P P ∃s ≥ t | X s ∈ A or U s ∈ U s (X s ) . (34) Here, G t {X s } s∈t:T , {U s } s∈t:T ≤ α = 1 -β means that P X s ∈ A , U s ∈ U s (X s ) , ∀s ≥ t ≥ β,
for all P ∈ P.

Random exit time and viability

Let µ be a measure on the time set T like, for instance, the counting measure when T = N or the Lebesgue measure when T = R + . Then, the random quantity

µ{s ≥ t , X s ∈ A or U s ∈ U s (X s )} (35)
measures the number of times that the state-control path {X s } s∈t:T , {U s } s∈t:T exits from the viability constraints.

Using risk measures -like the tail/average/conditional value-at-risk [7] -or stochastic orders [START_REF] Muller | Comparison Methods for Stochastic Models and Risk[END_REF][START_REF] Shaked | Stochastic Orders[END_REF], we have differents ways to express that this random quantity remains "small".

The general umbrella of cost functions

All the examples above, and many more [START_REF] Rougé | Vulnerability: From the conceptual to the operational using a dynamical system perspective[END_REF], fall under the general umbrella of cost functions as follows.

Consider a starting time t ∈ T and a measurable function

Ψ t : T s=t X s × T s=t U s × Ω → R , (36)
that attachs a disutility or cost -the opposite of value, utility, payoffto any tail state and control path, starting from time t, and to any state of Nature.

Let F be a risk measure that maps random variables on Ω towards the real numbers. Then, an extended risk measure is given by

G t {X s } s∈t:T , {U s } s∈t:T = F Ψ t {X s (•)} s∈t:T , {U s (•)} s∈t:T , • .
(37)

Resilience and risk minimization

When the set Λ R t (x t ) of resilient strategies at time t in §3.3 is not empty, how can we select one among the many? Here is a possible way that makes use of risk measures for risk minimization purposes.

Indicators of resilience

Let G t be an extended risk measure. We can look for resilient strategies that minimize risk, solution of min λ∈Λ R t (xt)

G t φ λ t:T (x t , •), • .

The minimum of the risk measure is a potential candidate as an indicator of resilience. Indeed, it is the best achievable measure of residual risk under a resilient strategy.

Examples

Using cost functions as in §5.1.

where F is a risk measure that maps random variables on Ω towards the real numbers [START_REF] Föllmer | Stochastic Finance. An Introduction in Discrete Time[END_REF]. For instance, in the robust viability setting of §4.2.3, an indicator of resilience could be the minimum (over all resilient strategies) of the maximal (over all states of Nature in Ω) recovery time.

Conclusion

Resilience is a rehashed concept in natural hazard management. Most of the formalizations of the concept require that, after any perturbation, the state of a system returns to an acceptable subset of the state set. Equipped with tools from control theory under uncertainty, we have proposed that resilience is the ability for the state-control random process as a whole to be driven to an acceptable "recovery regime" by a proper resilient strategy (adaptive). Our definition of resilience is a form of controlability: a resilient strategy has the property to shape the closed loop flow so that the resulting state and control random process belongs to a given subset of random processes, the acceptable recovery regimes.

We have proposed to handle risk thanks to risk measures 1 , by defining recovery regimes that represent a form of "risk containment". In addition, risk measures are potential candidates as indicators of resilience as they measure the residual risk under a resilient strategy.

Our contribution is formal, with its pros and cons: by its generality, our approach covers a large scope of notions of resilience; however, such generality makes it difficult to propose resolution methods. For instance, the possibility to use dynamic programing in stochastic viability relies upon a white noise assumption that we have not supposed. Much would remain to be done regarding applications and numerical implementation.

  state paths, made of sequences {x t } t∈T where x t ∈ X t for all t ∈ T; tail state paths {x r } r∈s:t (starting at time s < t) are elements of t r=s X r ; • the set t∈T U t of control paths, made of sequences {u t } t∈T where u t ∈ U t for all t ∈ T; tail control paths {u r } r∈s:t (starting at time s < t) are elements of t r=s U r .

  ∀r ∈ s : t (13b) has a unique solution {x r } r∈s:t , {u r } r∈s:t . Quite naturally, we define the closed loop flow φ λ s:t by φ λ s:t xs , ω = {x s } s∈s:t , {u s } s∈s:t .

	(14)

  5, we can look for resilient strategies that minimize expected costs

	min λ∈Λ R t (xt)
	min λ∈Λ R t (xt)	ω∈Ω sup	Ψ t φ λ t:T (x t , ω), ω ,	(40)
	or, more generally, that minimize
	min λ∈Λ R t (xt)

E Ψ t φ λ t:T (x t , •), • ,

(39)

or that minimize worst case costs, where Ω ⊂ Ω,

F Ψ t φ λ t:T (x t , •), • ,

Acknowledgement. The author is indebted to the editor-in-chief, the advisory editor and two reviewers. They supplied detailed critique, comments and inputs which, ultimately, contributed to an improved version of the manuscript.