Error estimates for the numerical approximation of a distributed optimal control problem governed by the von K\'arm\'an equations

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Karman equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Numerical results that justify the theoretical results are presented

Feedback stabilization of a boundary layer equation

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence results in the literature of solutions to algebraic Riccati equations do not apply to this class of problems. Here taking advantage of the fact that the semigroup of the state equation is exponentially stable and that the observation operator is a Hilbert-Schmidt operator, we are able to prove the existence and uniqueness of solution to the A.R.E. satisfied by the kernel of the operator which associates the 'optimal adjoint state' with the 'optimal state'. In part 2 [Buchot and Raymond, Appl. Math. Res. eXpress (2010) doi:10.1093/amrx/abp00

Exact controllability in fluid-solid structure: the Helmholtz model

A model representing the vibrations of a fluid-solid coupled structure is considered. Following Hilbert Uniqueness Method (HUM) introduced by Lions, we establish exact controllability results for this model with an internal control in the fluid part and there is no control in the solid part. Novel features which arise because of the coupling are pointed out. It is a source of difficulty in the proof of observability inequalities, definition of weak solutions and the proof of controllability results

Neumann Boundary Control of Hyperbolic Equations with Pointwise State Constraints

We consider optimal control problems for hyperbolic systems with controls in Neumann boundary conditions with pointwise (hard) constraints on control and state functions. Focusing on hyperbolic dynamics governed by the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the pointwise form of the Pontryagin Maximum Principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allows us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems

A tight Erd\H{o}s-P\'osa function for wheel minors

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.Comment: 15 pages, 1 figur

Optimal Boundary Control of Hyperbolic Equations with Pointwise State Constraints

In this paper we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them

Feedback stabilization of a 3D fluid-structure model with a boundary control

We study a system coupling the incompressible Navier-Stokes equations in a 3D parallelepiped type domain with a damped plate equation. The plate is located in a part of the upper boundary of the fluid domain. The fluid domain depends on the deformation of the plate, and therefore it depends on time. We are interested in the stabilization, with a prescribed decay rate, of such a system in a neighborhood of a stationary solution, by a Dirichlet control acting at the boundary of the fluid domain. For that, we first study the stabilizability of the corresponding linearized system and we determine a finite-dimensional feedback control able to stabilize the linearized model. A crucial step in the analysis consists in showing that this linearized system can be rewritten thanks to an analytic semigroup, the infinitesimal generator of which has a compact resolvent. A fixed-point argument is used to prove the local stabilization of the original nonlinear system. The main difficulties come from the coupling between the fluid and plate equations, and the fact that the fluid domain varies with time, giving rise to geometric nonlinearities. The results of the paper may be adapted to other more complex geometrical configurations for the same type of system. Ongoing research concerns the numerics of the control problem

Évaluation de la validité prédictive de neuf instruments chez les agresseurs sexuels adultes

Un large éventail d’instruments s’offre aux cliniciens pour évaluer le risque que posent les délinquants sexuels. La présente étude vise à comparer la validité prédictive de huit instruments (VRAG, SORAG, RRASOR, Statique-99, Statique-2002, RM2000, MnSOST–R et SVR-20) et un outil de référence (PCL-R) selon quatre types de récidive (nuisance sexuelle, récidive sexuelle, récidive violente non sexuelle et récidive non violente et non sexuelle) et pour trois groupes d’agresseurs sexuels. Les résultats indiquent que ces outils ont une validité prédictive marginale à modeste en ce qui concerne la récidive sexuelle. Cependant, une étude plus détaillée en fonction du type d’agresseurs indique une plus grande efficacité de ces instruments pour la récidive sexuelle chez les agresseurs d’enfants et pour la récidive non sexuelle chez les agresseurs de femmes. Les outils actuariels ne permettent de prédire aucune forme de récidive chez les agresseurs mixtes.This study compares the predictive accuracy of eight commonly used risk assessment instruments (the VRAG, the SORAG, the RRASOR, the Static-99, the Static-2002, the Risk Matrix 2000, the MnSOST–R and the SVR-20) and the PCL–R. Four types of recidivism were used (nuisance, sexual, nonsexual victim-involved and nonsexual victimless) and offenders were divided into three subgroups (rapists, child molesters and mixed offenders). Results indicate that instruments showed marginal to modest predictive accuracy for sexual recidivism. However, when we take account of the subgroups, the predictive accuracy for sexual recidivism was higher with child molesters and for non-sexual recidivism with rapists. Most instruments showed predictions no better than chance for mixed offenders and for predicting non-contact sexual reoffending.En la actualidad, se dispone de una amplia gama de herramientas para evaluar el riesgo de reincidencia de los delincuentes sexuales. El presente estudio compara la validez predictiva de ocho métodos (VRAG, SORAG, RRASOR, Static-99, Static-2002, RM2000, MnSOST-R y SVR-20) y un método de referencia (PCL-R) que evalúa cuatro tipos de reincidencia (reincidencia sexual violenta, reincidencia sexual, reincidencia violenta no sexual y reincidencia no violenta y no sexual) para tres grupos de agresores sexuales. Los resultados indican que estos métodos tienen una validez predictiva que va de marginal a modesta en lo que respecta a la reincidencia sexual. Sin embargo, un estudio más detallado en función del tipo de agresores indica una mayor eficacia de estos métodos para evaluar la probabilidad de reincidencia sexual en el caso de agresores de menores y la reincidencia no sexual de los agresores de mujeres. Los instrumentos estadísticos no permiten predecir ninguna forma de reincidencia en el grupo de los agresores mixtos