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Abstract: Uncertainty occupies a major place in the literature, both when it is being defined and when it is being measured. The mathematical viability theory framework provides useful concepts to go beyond the probabilistic views of uncertainty that are not able to capture all forms of uncertainty, particularly in the absence of statistical regularities. Bounded‐set uncertainty can help to analyze changes in the evolutionary conditions that affect ecological and social systems with probabilistic or nonprobabilistic distributions. This paper introduces uncertainty into a viability model by associating stochastic and bounded‐set uncertainty related to sales price as a driver of economic uncertainty. Based on an illustration of agroecosystems, the model identifies decision rules that guarantee viability. The larger the initial crop portfolio for sequential diversification is, the lower the costs to belong to the viability kernels will be, regardless of economic and environmental constraints. The diversification of farming systems transforms their complexity into an advantage against uncertainty, which opens up numerous viability trajectories. Recommedations for Resource Managers: Considering the relevance of a bounded set of economic uncertainty, restoring soil quality and choosing diversification guarantee viability and allow a single generation of farmers to relax both their environmental and economic constraints. When the time available to conform to both their environmental and economic constraints decreases, sequential diversification can be used as a tool to relax the economic constraint of self‐financing soil restoration. Sequential diversification is an asset, especially for low‐quality soil. Even under uncertainty and very low soil quality, diversification drastically reduces the value‐at‐risk for viable farming investment. Eligibility for agricultural subsidies should be conditional on soil quality to motivate practices of soil restoration that provide a sort of insurance against economic uncertainty. [ABSTRACT FROM AUTHOR]

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Our contribution aims to revisit the well known Goodwin's model in macroeconomics by the light of set-valued analysis taking into account state and regulation constraints in a viability program. The model of Goodwin (1967) deals with dynamic interactions between employment and salary levels. It provides endogenous explanations of cyclical trends in dynamical economy. Viability methods enable investigating model properties and revealing appropriate regulation allowing the evolution to fulfill some prescribed qualitative objective. Then, applying computational methods derived from the Viability Kernel Algorithm, one can stretch the traditional Goodwin model analysis up to the institutional framework of the economy including monetary and budgetary aspects of the regulation policy from the public authorities, namely the state government, the central bank and eventually the rivalry between the two boards thanks to dynamical games. [ABSTRACT FROM AUTHOR]

Copyright of Natures Sciences Sociétés is the property of EDP Sciences and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article focuses on the approaches of viability theory and tychastic in problems in asset and liability management (ALM). It states that viability theory is use for ALM mechanism that maximize utility function such as Constant Proportional Portfolio Insurance (CPPI). It mentions the use of three uncertainties in management rules for portfolio that require scarcity constraint such as tychastic, contingent, and impulse. It argues on the justification of using tychastic system because of its characteristic of stochastic viability.

We adopt viability theory to assess the sustainability of the world's forests while taking into account some of the competing economic, social, and environmental uses of these forests, namely, timber production, poverty alleviation through agriculture, and air quality as well as the negative externalities that these uses create. We provide insights on the different trade-offs faced to achieve sustainability and draw some policy implications as to what is the path leading to sustainability in the long run. [ABSTRACT FROM AUTHOR]

We use viability techniques for solving Dirichlet problems with inequality constraints (obstacles) for a class of Hamilton-Jacobi equations. The hypograph of the "solution" is defined as the "capture basin" under an auxiliary control system of a target associated with the initial and boundary conditions, viable in an environment associated with the inequality constraint. From the tangential condition characterizing capture basins, we prove that this solution is the unique "upper semicontinuous" solution to the Hamilton-Jacobi-Bellman partial differential equation in the Barron-Jensen/Frankowska sense. We show how this framework allows us to translate properties of capture basins into corresponding properties of the solutions to this problem. For instance, this approach provides a representation formula of the solution which boils down to the Lax-Hopf formula in the absence of constraints. [ABSTRACT FROM AUTHOR]

This paper deals with inertia functions in control theory introduced in Aubin, Bernardo and Saint-Pierre (2004, 2005) and their adaptation to dynamical games. The inertia function associates with any initial state-control pair the smallest of the worst norms over time of the velocities of the controls regulating viable evolutions. For tychastic systems (parameterized systems where the parameters are tyches, disturbances, perturbations, etc.), the palicinesia of a tyche measure the worst norm over time of the velocities of the tyches. The palicinesia function is the largest palicinesia threshold c such that all evolutions with palicinesia smaller than or equal to c are viable. For dynamical games where one parameter is the control and the other one is a tyche (games against nature or robust control), we define the guaranteed inertia function associated with any initial state-control-tyche triple the best of the worst of the norms of the velocities of the controls and of the tyches and study their properties. Viability Characterizations and Hamilton-Jacobi equations of which these inertia and palicinesia functions are solutions are provided. [ABSTRACT FROM AUTHOR]

Some issues on fluctuation of solutions to Lorenz and Rossler systems, for instance, are related to viability kernels of subsets under continuous time systems, or in the case of Julia sets, for instance, under discrete time systems. It happens that viability kernels of subsets, capture basins of targets and the combination of the twos provide tools for the analysis of the local behavior around equilibria (local stable and unstable manifolds), the asymptotic behavior and the functuation of evolutions between two areas of a domain, etc. Since algorithms and softwares do exist for computing the viability kernels and the capture basins, as well as evolutions viable in the viability kernel until they converge to a target infinite time, we are able to localize fluctuation basins. [ABSTRACT FROM AUTHOR]

Viability theory can be applied for determining viable capture basin for control problem in presence of uncertainty. We first recall the concepts of viability theory which allow to develop numerical methods for computing viable capture basin for control problems and guaranteed control problems. Recent developments of option pricing in the framework of dynamical games with constraints lead to the formulation of guaranteed valuation in terms of guaranteed viable-capture basin of a dynamical game. As an application we show how the viability/capturability algorithm evaluates and manages portfolios. Regarding viability/capturability issues, stochastic control is a particular use of tychastic control. We replace the standard translation of uncertainty by stochastic control problem by tychastic ones and the concept of stochastic viability by the one of guaranteed viability kernel. Considering the Cox—Rubinstein model, we extend algorithms for hedging portfolios in the presence of transaction costs and dividends using recent developments on hybrid calculus. [ABSTRACT FROM AUTHOR]

Fertilities varying in time in an unpredictable manner raise the question of the maintenance of polymorphism, and the subsequent question of the minimal total number of generations spent by the system in impoverished polymorphism. In the one-locus two-allele model, level surfaces of this minimal time out of a given set K of constraints defining polymorphism are delineated. The surface of null minimal time is also the largest set of genotype frequencies from which there exists at least one route remaining in K forever. The influence of the range of admissible fertilities clarifies the trade-off between homozygotes and heterozygotes. Notably, ranges of fertility forbidding a system leaving K to ever return to it are determined. Generally, keeping a chance to regain polymorphism demands a sufficiently relatively high fertility for matings involving heterozygotes. [ABSTRACT FROM AUTHOR]

Examines the viability of pay-as-you-go social security systems. Technique in controlling pay-as-you-go social security systems; Operations used for the control of trajectories; Inference of flexibility from the timing and magnitude of the operation.

We prove the existence of a value for pursuit games with state constraints. We also prove that this value is lower semicontinuous. [ABSTRACT FROM AUTHOR]

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem. [ABSTRACT FROM AUTHOR]

In an unpredictable environment, the distributions of alleles from which polymorphism can be maintained forever belong to a certain set, the C-viability kernel. Such a set is calculated in the two-locus haploid model, as well as the corresponding fitnesses at any time which make this maintenance possible. The dependence of the C-viability kernel on the set U of admissible fitnesses and on the recombination rate r is studied. Notably, the C-viability kernel varies rapidly in the neighborhood of equal fitness of AB and ab; it becomes empty when ab has a fitness below a certain function, which is delineated, of the recombination rate. The properties of the two-locus model under constraints, out of equilibrium and with unpredictable selection are thus presented. [ABSTRACT FROM AUTHOR]

Focuses on the development of a planning procedures in economies with externalities. Advantages of the planning procedures with prices; Properties of monotonicity and feasibility in planning procedures; Usefulness of the Monte Carlo experiment on the speeds of convergence for the different planning procedures.

We study recursive inclusions x ε G(x). For instance, such systems appear for discrete finite-difference inclusions x ε G( x) where G:=1+ ρF. The discrete viability kernel of G, i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel of K under F. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel of K under F can be approached by a sequence of discrete viability kernels associated with x ε Г(x) where Г(x) = x + ρF(x) + (ML/2) ρℬ. Secondly, we show that it can be approached by finite viability kernels associated with x ε ( Г(x) + α(hℬ) ∩ X. [ABSTRACT FROM AUTHOR]

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convex K, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. When K is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation. [ABSTRACT FROM AUTHOR]

Viability theory provides an efficient framework for looking for zeros of multivalued equations: 0 ∈ F(x). The main idea is to consider solutions of a suitable differential inclusion, viable in graph of F. The choice of the differential inclusion is guided necessarily by the fact that any solution should converge or go through a zero of the multivalued equation. We investigate here a new understanding of the well-known Newton's method, generalizing it to set-valued equations and set up a class of algorithms which involve generalization of some homotopic path algorithms. [ABSTRACT FROM AUTHOR]

The present paper deals with some methods of decomposition where the numerical convergence for non linear differential equations is studied in the case of adapted decompositions of the right hand side. [ABSTRACT FROM AUTHOR]

To cope with ever-increasing demand and ensure food security, agronomic systems have shifted over time from traditional agriculture, based on the organic fertilization of soils, to intensive and specialized farming that use chemical fertilization. This resulted in increased soil productivity in the short term, but caused serious ecological drawbacks over time (degradation of soil quality, pollution of water and air, loss of biodiversity, erosion, etc.), and even reversed the trend of agricultural productivity. In this paper, we propose a viability theory–based model to study the sustainability of an agricultural system subject to climate uncertainty. Our objective is to determine what farming practices and activity sequences restore soil quality to a desired level while ensuring an acceptable level of productivity in the presence of the risk of major climatic disasters. The model is applied to Guadeloupe, an island in the West French Indies. We found that the results are highly sensitive to the direct effect of hurricanes on the soil's quality, which, in turn, strongly affects the impact of the other parameters and that the export oriented sector is more vulnerable and less resilient to climatic uncertainties than the sector aimed at the local market. [ABSTRACT FROM AUTHOR]