T -viable states in a closed set K under a certain set-valued dynamic are states from which there exists at least one solution remaining in K until a given time horizon T . Minimizing the cost to constraints lets us determine whether a given state is T -viable or not, and this is implementable in large dimension for the statespace. Minimizing on the initial condition itself lets find viable states. Quincampoix's semi-permeability property helps find other states located close to the viability boundary, which is then gradually delineated. The algorithm is particularly suited to the identification of specific trajectories, such as the heavy viable solution, or to the computation of viability kernels associated with delayed dynamics. The volume of the viability kernel and its confidence interval can be estimated by randomly drawing states and checking their viability status. Examples are given.
When cheap fossil energy is polluting and pollutant no longer absorbed beyond a certain concentration, there is a moment when the * bonneuil@ined.fr † Raouf.Boucekkine@uclouvain.be 1 introduction of a cleaner renewable energy, although onerous, is optimal with respect to inter-temporal utility. The cleaner technology is adopted either instantaneously or gradually at a controlled rate. The problem of optimum under viability constraints is 6-dimensional under a continuous-discrete dynamic controlled by energy consumption and investment into production of renewable energy. Viable optima are obtained either with gradual or with instantaneous adoption. A longer time horizon increases the probability of adoption of renewable energy and the time for starting this adoption. It also increases maximal utility and the probability to cross the threshold of irreversible pollution. Exploiting a renewable energy starts sooner when adoption is gradual rather than instantaneous. The shorter the period remaining after adoption until the time horizon, the higher the investment into renewable energy.
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