Catálogo de publicaciones - libros

Among optimal control problems, singular arcs problems are interesting and difficult to solve with indirect methods, as they involve a multi-valued control and differential inclusions. Multiple shooting is an efficient way to solve this kind of problems, but typically requires some a priori knowledge of the control structure. We limit here ourselves to the case where the Hamiltonian is linear with respect to the control , and primarily use a quadratic () perturbation of the criterion. The aim of this continuation approach is to obtain an approximate solution that can provide reliable information concerning the singular structure. We choose to use a PL (simplicial) continuation method, which can be more easily adapted to the multi-valued case. We will first present some convergence results regarding the continuation, and then study the numerical resolution of two example problems. All numerical experiments were conducted with the Simplicial package we developed.

A nonlinear elliptic control problem with pointwise control-state constraints is considered. Existence of regular Lagrange multipliers, first-order necessary and and second-order sufficient optimality conditions are derived. The theory is verified by numerical examples.

The aim of this paper is to extend and generalize the known necessary optimality conditions to non-smooth as well as higher-order setting concerning the optimization problem (which is called an abstract control problem) where is an open set of a Banach space is a nonempty set, and the data fulfill a certain convexity condition which can often be verified in the context of ordinary optimal control problems.

In this paper we formulate and use the duality concept of Klötzler (1977) for infinite horizon optimal control problems. The main idea is choosing weighted Sobolev and weighted spaces as the state and control spaces, respectively. Different criteria of optimality are known for specific problems, e.g. the overtaking criterion of von Weizsäcker (1965), the catching up criterion of Gale (1967) and the sporadically catching up criterion of Halkin (1974). Corresponding to these criteria we develop the duality theory and prove sufficient conditions for local optimality. Here we use some remarkable properties of weighted spaces. An example is presented where the solution is obtained in the framework of these weighted spaces, but which does not belong to standard Sobolev spaces.

We provide two examples concerning the relaxation properties of a model problem in multidimensional control: , ⊂ ℝ, ∈ {sk0/1,∞} (, ℝ), () ∈ ⊂ ℝ a. e. where ≤ 2, ≤ 2, () is the Jacobian of , and is a convex body. The first example justifies the use of quasiconvex functions with infinite values in the relaxation process. In the second one, we examinate the relaxation properties of a restricted quasiconvex envelope function ƒ* introduced by Dacorogna/Marcellini.

In this paper we consider autonomous discrete time optimal control problems. We discuss the reduction to finite cost and the representation formula, the existence of optimal solutions on infinite horizon and their structure, and the structure of optimal solutions on finite intervals.

We investigate possibilities to deal with optimal control problems that have special integer restrictions on the time dependent control functions, namely to take only the values of 0 or 1 on given time intervals. A heuristic penalty term homotopy and a Branch and Bound approach are presented, both in the context of the direct multiple shooting method for optimal control. A tutorial example from population dynamics is introduced as a benchmark problem for optimal control with 0 –1 controls and used to compare the numerical results of the different approaches.

Several characterizations of convexity for totally balanced games are presented. As a preliminary result, it is first shown that the core of any subgame of a nonnegative totally balanced game can be easily obtained from the maximum average value (MAV) function of the game. This result is then used to get a characterization of convex games in terms of MAV functions. It is also proved that a game is convex if and only if all of its marginal games are totally balanced.

A new way is presented to define for minimum cost spanning tree (mcst) games the irreducible core, which is introduced by Bird in 1976. The Bird core correspondence turns out to have interesting monotonicity and additivity properties and each stable cost monotonic allocation rule for mcst-problems is a selection of the Bird core correspondence. Using the additivity property an axiomatic characterization of the Bird core correspondence is obtained.

We introduce here a family of mixed coalitional values. They extend the binomial semivalues to games endowed with a coalition structure, satisfy the property of symmetry in the quotient game and the quotient game property, generalize the symmetric coalitional Banzhaf value introduced by Alonso and Fiestras and link and merge the Shapley value and the binomial semivalues. A computational procedure in terms of the multilinear extension of the original game is also provided and an application to political science is sketched.