Narrow Results

We give complete proofs to the verification theorems announced recently by the author for the "pairs of relatively optimal feedback strategies" of an autonomous differential game. These concepts are considered to describe the possibly optimal solutions of a differential game while the corresponding value functions are used as "instruments" for proving the relative optimality and also as "auxiliary characteristics" of the differential game. The 6 verification theorems in the paper are proved under different regularity assumptions accompanied by suitable differential inequalities verified by the generalized derivatives, mainly of contingent type, of the value function.

Examples with zero-sum linear differential games of fixed terminal time and a convex terminal payoff function depending on two components of the phase vector are considered. Such games can have an indifferent zone with constant value function. The level set of the value function associated with the indifferent zone is called the "critical" tube. In the selected examples, the critical tube and the neighboring level sets exhibit "narrow throats". Presence of such throats requires extremely precise computations for constructing the level sets. The paper presents different forms of critical tubes with narrow throats and indicates the combinations of problem parameters that can produce them.

The paper discusses the existence of the value function of a time-optimal differential game with lifeline in a quite general formulation. In such a game, the first player tries to lead the trajectory of the system to a terminal set as soon as possible avoiding the lifeline. The aim of the second player is to lead the trajectory to the lifeline or, if it is not possible, to keep it outside the terminal set, or, if the latter is impossible too, to postpone maximally reaching the terminal set.

This paper considers the evaluation step in a decision-making process that follows decision-making goals setting, feasible alternatives and attributes or criteria that characterize them determination steps. Evaluation step must establish a model or algorithm to evaluate alternatives taking into account their performances with regard to criteria as well as decision makers or stakeholders preferences. Though this problem is rather a classic one, researches related to evaluation model construction continue to be active to find models that cope with more realities or that fit well how human beings behave in group and proceed when facing the problem of choosing, ranking or sorting alternatives or options. The purpose of this paper is to construct an evaluation model that integrate the performances of alternatives with regard to attributes or criteria and decision makers or agents opinions with regard to the importance to assign to each criterion in order to obtain a value function. As any decision problem is almost always a matter of tradeoff, among attributes characterizing alternatives there will be those acting toward the achievement of decision makers goal (benefit) and those that decision makers would like to reduce as much as possible (cost); we will designate the first ones as positive attributes and the later ones as negative attributes. The process of dividing attributes into positive attributes and negative attributes is beyond the scope of this paper and this partition will be considered as a part of the problem specification. The model is constructed in two steps: firstly, satisfiability (selectability and rejectability) measures or functions are obtained for each alternative using attributes values (positive attributes will contribute to selectability measure whereas negative ones are used in the derivation of rejectability measure) and agents opinions in the framework of satisficing game theory and secondly a value function is built on that measures. Agents opinions with regard to attributes will be expressed locally by weighting them by category (positive/negative).

We consider the initial-boundary value problem for scalar conservation laws on the strip (0, ∞) × [0, 1] with strictly convex smooth flux of a superlinear growth. We show that an associated Hamilton–Jacobi equation with initial and (appropriately defined) boundary conditions has a unique generalized solution V that can be obtained as minimum of three value functions of the calculus of variation. Each of these functions, in turn, can be expressed using Lax's formula. The traces of the gradients V_x satisfy generalized boundary conditions (as in LeFloch (1988)) in a pointwise manner when the initial and boundary data are continuous and in a weak sense when they are discontinuous. It is also shown that V_x is continuous almost everywhere, and a result concerning the traces of the sign of f′(V_x(t, ⋅)) is proven.

We observe that the standard variant of Prospect Theory cannot describe very risk-averse choices in simple lotteries. This makes it difficult to accommodate it with experimental data. Using an exponential value function can solve this problem and allows to cover the whole spectrum of risk-averse behavior. Further evidence in favor of the exponential value function comes from the evaluation of data from a large scale survey on preferences over lotteries where the exponential value function produces the best fits. The results enhance the understanding on what types of lotteries pose potential problems for the classical value function.

We give some conditions satisfied by the solution of utility maximization problems under several constraints, with incomes normalized to one. We generalize the one constraint Slutsky conditions in consumer theory. We note that the necessary conditions are not sufficient in contrary to the one constraint case.

Optimal control under state constraints has brought new mathematical challenges that have led to new techniques and new theories. We survey some recent results related to issues of regularity of optimal trajectories, optimal controls and the value function, and discuss optimal synthesis and necessary optimality conditions. We also show how abstract inverse mapping theorems of set-valued analysis can be applied to study state constrained control systems.