arXiv subjects

Pablo Pedregal

Publications and source records attributed to Pablo Pedregal.

23 records · Page 2Linked to original sources

arXiv2015

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.

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arXiv2015

About least-squares type approach to address direct and controllability problems

- We discuss the approximation of distributed null controls for partial differential equations. The main purpose is to determine an approximation of controls that drives the solution from a prescribed initial state at the initial time to the zero target at a prescribed final time. As a non trivial example, we mainly focus on the Stokes system for which the existence of square-integrable controls have been obtained in [Fursikov \& Imanuvilov, Controllability of Evolution Equations, 1996]) via Carleman type estimates. We introduce a least-squares formulation of the controllability problem, and we show that it allows the construction of strong convergent sequences of functions toward null controls for the Stokes system. The approach consists first in introducing a class of functions satisfying a priori the boundary conditions in space and time-in particular the null controllability condition at time T-, and then finding among this class one element satisfying the system. This second step is done by minimizing a quadratic functional, among the admissible corrector functions of the Stokes system. We also discuss briefly the direct problem for the steady Navier-Stokes system. The method does not make use of any duality arguments and therefore avoid the ill-posedness of dual methods, when parabolic type equation are considered.

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arXiv2014

Weak limits in non-linear conductivity

The problem of characterizing weak limits of sequences of solutions for a non-linear diffusion equation of $p$-laplacian type is addressed. It is formulated in terms of certain moments of underlying Young measures associated with main fields corresponding to the non-linear equation. We show two main results. One is a necessary condition; the other one a sufficient counterpart. Unlike the linear case, those two conditions do not match.

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arXiv2014

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.

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arXiv2014

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.

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