Viability theory is the study of dynamical systems that asks what set of initial conditions will generate evolutions that obey the laws of motion of a system and some state constraints, for the length of the evolution. We apply viability theory to Judd's dynamic tax model [The welfare cost of factor taxation in a perfect-foresight model, Journal of Political Economy 95(4), 675–709 (1987)] to identify which economic states today are sustainable under only slightly constrained tax-rate adjustments in the future, when the dynamic budget constraint and the consumers' transversality condition at infinity are satisfied. We call the set of such states the economic viability kernel. In broad terms, knowledge of the viability kernel can tell the planner what economic objectives are achievable and assist in the choice of suitable controls to realize them. We observe that high consumption levels can only be sustained when capital is abundant and, unsurprisingly, that a very high consumption economy lies outside such kernels, at least for annual tax-adjustment levels limited by 20 percentage points. Furthermore, we notice that by and large the sizes of the kernel slices do not diminish as the tax rate rises; hence high-taxation economies are not necessarily more prone to explode, or implode, than their low-taxation counterparts. In fact, higher tax rates are necessary to keep many consumption choices viable, especially when capital approaches the constraint-set boundaries.

In European grassland landscapes, grazing and mowing play a key role for the maintenance of high-quality habitats that host important bird populations. As grasslands are also key resources for cattle feeding, there is a need to develop management strategies that achieve the double objective of production and biodiversity conservation. The objective of this study was to use a modelling approach to generate recognisable patterns of bird dynamics in farms composed of different land use proportions, and to compare their production and ecological dimensions. We developed a dynamic model, which linked grassland management to bird population dynamics at the field and farm levels. The model was parameterised for two types of suckling farms corresponding to contrasting levels of grassland intensification and for two bird species of high conservation value. A viability algorithm was used to define and assess viable management strategies for production and ecological performance so as to draw the shape of the relationship between both types of performances for the two types of farms. Our results indicated that, at the farm level, there was a farming system effect with a negative and non-linear relationship linking performance. Improving bird population maintenance was less costly in extensive farms compared with intensive farms. At the field level, the model predicted the timing and intensity of land use, maximising either production or ecological performance. The results suggested that multi-objective grassland management would benefit from public policies that consider levels of organisation higher than the field level, such as the farm or the landscape.

In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].

This paper provides a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems in which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce a set-valued return function V and show that the epigraph of V equals the viability kernel of a certain related augmented dynamical system. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function equals to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre. Birkhauser, Boston (1999) 177–247. P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Set-Valued Analysis 8 (2000) 111–126]. As a result, the epigraph of the approximate set-valued return function converges to the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples illustrate our approach.

Herbert A. Simon, 1978 Economics Nobel Prize laureate, talked about satisficing (his neologism) rather than optimizing as being what economists really need. Indeed, optimization might be an unsuitable solution procedure (in that it suggests a unique “optimal” solution) for problems where many solutions could be satisfactory. We think that looking for an applicable monetary policy is a problem of this kind because there is no unique way in which a central bank can achieve a desired inflation (unemployment, etc.) path. We think that it is viability theory, which is a relatively young area of mathematics, that rigorously captures the essence of satisficing. We aim to use viability analysis to analyze a simple macro policy model and show how some robust adjustment rules can be endogenously obtained.

We investigate the value function of the Bolza problem of the Calculus of Variations
$$ V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\},$$ with a lower semicontinuous Lagrangian L and a final cost $ \varphi $, and show that it is locally Lipschitz for t>0 whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities.