Young-measure solutions for multidimensional systems of conservation laws

We explore Young measure solutions of systems of conservation laws through an alternative variational method that introduces a suitable, non-negative error functional to measure departure of feasible fields from being a weak solution. Young measure solutions are then understood as being generated by minimizing sequences for such functional much in the same way as in non-convex, vector variational problems. We establish an existence result for such generalized solutions based on an appropriate structural condition on the system. We finally discuss how the classic concept of a Young measure solution can be improved, and support our arguments by considering a scalar, single equation in dimension one

Optimal feedback control, linear first-order PDE systems, and obstacle problems

We introduce an alternative approach for the analysis and numerical approximation of the optimal feedback control mapping. It consists in looking at a typical optimal control problem in such a way that feasible controls are mappings depending both in time and space. In this way, the feedback form of the problem is built-in from the very beginning. Optimality conditions are derived for one such optimal mapping, which by construction is the optimal feedback mapping of the problem. In formulating optimality conditions, costates in feedback form are solutions of linear, first-order transport systems, while optimal descent directions are solutions of appropriate obstacle problems. We treat situations with no constraint-sets for control and state, as well as the more general case where a constraint-set is considered for the control variable. Constraints for the state variable are deferred to a coming contribution

Some evidence in favor of Morrey's conjecture

We provide further evidence to favor the fact that rank-one convexity does not imply quasiconvexity for two-component maps in dimension two. We provide an explicit family of maps parametrized by $\tau$, and argue that, for small $\tau$, they cannot be achievable by lamination. In this way, Morrey's conjecture might turn out to be correct in all cases.Comment: If has been shown (Sebesty\'en, G., Sz\'ekelyhidi, L., Jr., Laminates supported on cubes, J. Convex Anal. 24 (2017), no. 4, 1217--1237) that the example in this preprint cannot give rise to a true counterexample to the fact that rank-one convexity does not imply quasiconvexity in the 2x2 cas

Constrained optimization through fixed point techniques

We introduce an alternative approach for constrained mathematical programming problems. It rests on two main aspects: an efficient way to compute optimal solutions for unconstrained problems, and multipliers regarded as variables for a certain map. Contrary to typical dual strategies, optimal vectors of multipliers are sought as fixed points for that map. Two distinctive features are worth highlighting: its simplicity and flexibility for the implementation, and its convergence properties.Comment: 14 pages, 2 figure

A dynamical-system approach to mathematical programming

We explore how to build a vector field from the various functions involved in a given mathematical program, and show that locally-stable equilibria of the underlying dynamical system are precisely the local solutions of the optimization problem. The general situation in which explicit inequality constraints are present is especially interesting as the vector field has to be discontinuous, and so one is led to consider discontinuous dynamical systems and their equilibria.Comment: 13 page

Rank-one convexity implies quasiconvexity for two-component maps

We prove that, for two-component maps, rank-one convexity is equivalent to quasiconvexity. The essential tool for the proof is a fixed-point argument for a suitable set-valued map going from one component to the other and preserving decomposition directions in the $(H_n)$-condition formalism; the existence of a fixed point ensures that, in addition to keeping decomposition directions, joint volume fractions are preserved as well. When maps have more than two components, then fixed points exist for every combination of two components, but they do not match in general

Hilbert's 16th problem. II. Pfaffian equations and variational methods

Starting from a Pfaffian equation in dimension $N$ and focusing on compact solutions for it, we place in perspective the variational method used in [29] to solve Hilbert's 16th problem. In addition to exploring how this viewpoint can help in detecting and finding approximations for limit cycles of planar systems, we recall some of the initial important facts of the full program developed in [29] to motivate that the same proposal could eventually be used in other situations. In particular, we make some initial interesting calculations in dimension $N=3$ that lead to some similar initial conclusions as with the case $N=2$

A different look at controllability

We explore further controllability problems through a standard least square approach. By setting up a suitable error functional $E$, and putting $m(\ge0)$ for the infimum, we interpret approximate controllability by asking $m=0$, while exact controllability corresponds, in addition, to demanding that $m$ is attained. We also provide a condition, formulated entirely in terms of the error $E$, which turns out to be equivalent to the unique continuation property, and to approximate controllability. Though we restrict attention here to the 1D, homogeneous heat equation to explain the main ideas, they can be extended in a similar way to many other scenarios some of which have already been explored numerically, due to the flexibility of the procedure for the numerical approximation

Hilbert's 16th problem. I. When differential systems meet variational methods

We provide an upper bound for the number of limit cycles that planar polynomial differential systems of a given degree may have. The bound turns out to be a polynomial of degree four in the degree of the system. The strategy brings together variational and dynamical system techniques by transforming the task of counting limit cycles into counting critical points for a certain smooth, non-negative functional, through Morse inequalities, for which limit cycles are global minimizers. We thus solve the second part of Hilbert's 16th problem providing a uniform upper bound for the number of limit cycles which only depends on the degree of the polynomial differential system.Comment: An improvement on the upper bound of limit cycles of planar polynomial differential systems has been foun

Existence results for optimal control problems with some special non-linear dependence on state and control

We present a general approach to prove existence of solutions for optimal control problems not based on typical convexity conditions which quite often are very hard, if not impossible, to check. By taking advantage of several relaxations of the problem, we isolate an assumption which guarantees the existence of solutions of the original optimal control problem. To show the validity of this crucial hypothesis through various means and in various contexts is the main body of this contribution. In each such situation, we end up with some existence result. In particular, we would like to stress a general result that takes advantage of the particular structure of both the cost functional and the state equation. One main motivation for our work here comes from a model for guidance and control of ocean vehicles. Some explicit existence results and comparison examples are given.Comment: The revised version was published in SIAM J. Control Opti