Luis A. Fernández

Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state

In this paper we are concerned with optimal control problems governed by an elliptic semilinear equation, the control being distributed in Ω. The existence of constraints on the control as well as pointwise constraints on the gradient of the state is assumed. A convenient choice of the control space permits us to derive the optimality conditions and study the adjoint state equation, which has derivatives of measures as data. In order to carry out this study, we prove a trace theorem and state Greens formula by using the transposition method.

Optimal control of quasilinear parabolic equations

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekelands variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

Dealing with Integral State Constraints in BoundaryControl Problems of Quasilinear Elliptic Equations

This paper deals with state-constrained optimal control problems governed by a quasilinear elliptic equation. These constraints are given in an integral form and depend on the state and its gradient. Equality and inequality constraints are simultaneously considered. Existence of a solution is investigated and some optimality conditions are obtained with the aid of Ekelands variational principle.

Relativistic mechanical–thermodynamical formalism—description of inelastic collisions

We present a relativistic formalism inspired on the Minkowski four-vectors that also includes conservation laws such as the first law of thermodynamics. It remains close to the relativistic four-vector formalism developed for a single particle, but it is also related to the classical treatment of problems that imperatively require both the Newtons second law and the energy conservation law. We apply the developed formalism to inelastic collisions to better show how it works.

The principle of relativity and the de Broglie relation

The de Broglie relation is revisited in connection with an ab initio relativistic description of particles and waves, which is the same treatment that historically led to this famous relation. In the same context of the Minkowski four-vector formalism, we also discuss the phase and the group velocity of a matter wave, explicitly showing that both transform as ordinary velocities under a Lorentz transformation. We show that such a transformation rule is a necessary condition for the covariance of the de Broglie relation, and stress the pedagogical value of the Einstein-Minkowski-Lorentz relativistic context in the presentation of the de Broglie relation.

Optimal cooling strategies in polymer crystallization

An optimal control problem for cooling strategies in polymer crystallization processes described by a deterministic model is solved in the framework of a free boundary problem. The strategy of cooling both sides of a one dimensional sample is introduced for the first time in this model, and is shown to be well approximated by the sum of the solutions of two one-phase Stefan problems, even for arbitrary applied temperature profiles. This result is then used to show that cooling both sides is always more effective in polymer production than injecting the same amount of cold through only one side. The optimal cooling strategy, focused in avoiding low temperatures and in shortening cooling times, is derived, and consists in applying the same constant temperature at both sides. Explicit expressions of the optimal controls in terms of the parameters of the material are also obtained.

State Constrained Optimal Control for some Quasilinear Parabolic Equations

We study optimal control problems where the system is governed by some quasilinear parabolic equations and there are restrictions on the control as well as on the state. The distributed control can appear in all the coefficients of the operator. State constraints of integral type and also pointwise in time are considered. Our main interest is the derivation of the first order optimality conditions. Finally, an application to exact controllability in finite dimensional subspaces is given.

Classical One-Phase Stefan Problems for Describing Polymer Crystallization Processes

A free boundary problem framework is proposed to approximate the solution of a deterministic nonisothermal polymer crystallization model in which crystallization fronts appear as the result of the combination of two heat transfer processes: the heat conduction due to the application of a cooling temperature below the polymer melting temperature threshold, and the latent heat production due to the phase change. When the latent heat is larger than the sensible heat of the crystallization process, a classical one-phase Stefan problem can be formulated which allows one to derive analytical approximations describing, for arbitrary applied cooling temperature profiles, the main features of the crystallization process: the relation between the latent heat and the specific heat capacity, the evolution of the temperature distribution, and the advance of the crystallization front. Analytical expressions of magnitudes of industrial interest such as the crystallization time are also derived, allowing the design of op...

Free boundary problems and optimal control of axisymmetric polymer crystallization processes

Abstract The non-isothermal crystallization of a hollow cylindrical polymer sample with radial symmetry is studied. Three radial cooling strategies are considered: cooling from inside (outward cooling), cooling from outside (inward cooling), and cooling from both sides (double cooling). When the initial and boundary conditions are axisymmetric, the crystallization problem can be reduced to a one-dimensional formulation where a free boundary problem framework can be used. The solution is approximated by appropriate one-phase Stefan problems for which the analytical solution is provided. These results are compared to direct numerical simulations of the crystallization process, finding an excellent agreement in the approximation of the time-evolution of the crystallization front, the temperature distribution and the crystallization time. In a second part, the corresponding optimal control problems are formulated for a cost functional assessing the use of low temperatures and the duration of the crystallization process. Analytical expressions of the approximated optimal controls are derived for each cooling strategy. In particular, the double cooling case presents special difficulties that we are able to overcome by extending the technique that we previously developed for the case of homogeneous rectangular samples.

Controllability properties for some semilinear parabolic PDE with a quadratic gradient term

Abstract We study several controllability properties for some semilinear parabolic PDE with a quadratic gradient term. For internal distributed controls, it is shown that the system is approximately and null controllable. The proof relies on the Cole–Hopf transformation. The same approach is used to deal with initial controls.