scholarly journals Global exact controllability of ideal incompressible magnetohydrodynamic flows through a planar duct

2021 ◽  
Author(s):  
Manuel Rissel ◽  
Ya-Guang Wang
Keyword(s):  
Ad Hoc ◽  
Fluid Velocity ◽  
Type System ◽  
Exact Controllability ◽  
Rectangular Domain ◽  
Magnetohydrodynamic Flows ◽  
Induction Equation ◽  
Vertical Walls ◽  
Boundary Controls ◽  
Return Method

This article is concerned with the global exact controllability for ideal incompressible magnetohydrodynamics in a rectangular domain where the controls are situated in both vertical walls. First, global exact controllability via boundary controls is established for a related Elsässer type system by applying the return method, introduced in [Coron J.M., Math. Control Signals Systems, 5(3) (1992) 295--312]. Similar results are then inferred for the original magnetohydrodynamics system with the help of a special pressure-like corrector in the induction equation. Overall, the main difficulties stem from the nonlinear coupling between the fluid velocity and the magnetic field in combination with the aim of exactly controlling the system. In order to overcome some of the obstacles, we introduce ad-hoc constructions, such as suitable initial data extensions outside of the physical part of the domain and a certain weighted space.

scholarly journals On the uniform controllability for a family of non-viscous and viscous Burgers-α systems

2021 ◽  
Author(s):  
Diego Araujo de Souza ◽  
Raul K. C. Araujo ◽  
Enrique Fernández-Cara
Keyword(s):  
Burgers Equation ◽  
Wave Length ◽  
Exact Controllability ◽  
Convective Velocity ◽  
Distributed Controls ◽  
Target States ◽  
Controllability Property ◽  
Return Method ◽  
Uniform Controllability ◽  
Controllability Result

In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers-α systems by using boundary and space independent distributed controls. In these equations, the usual convective velocity of the Burgers equation is replaced by a regularized velocity, induced by a Helmholtz filtered of characteristic wave-length α. First, we prove a global exact controllability result (uniform with respect to α) for the non-viscous Burgers-α system, using the return method and a fixed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we first prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states.

scholarly journals How to make ad hoc proof automation less ad hoc

2013 ◽  
Vol 23 (4) ◽  
pp. 357-401 ◽  
Author(s):  
GEORGES GONTHIER ◽  
BETA ZILIANI ◽  
ALEKSANDAR NANEVSKI ◽  
DEREK DREYER
Keyword(s):  
Design Patterns ◽  
Ad Hoc ◽  
Type System ◽  
Type Inference ◽  
Interactive Theorem Provers ◽  
Proof Automation ◽  
Novel Approach ◽  
Type Classes ◽  
Structure Programming ◽  
Base Logic

AbstractMost interactive theorem provers provide support for some form of user-customizable proof automation. In a number of popular systems, such as Coq and Isabelle, this automation is achieved primarily through tactics, which are programmed in a separate language from that of the prover's base logic. While tactics are clearly useful in practice, they can be difficult to maintain and compose because, unlike lemmas, their behavior cannot be specified within the expressive type system of the prover itself.We propose a novel approach to proof automation in Coq that allows the user to specify the behavior of custom automated routines in terms of Coq's own type system. Our approach involves a sophisticated application of Coq's canonical structures, which generalize Haskell type classes and facilitate a flexible style of dependently-typed logic programming. Specifically, just as Haskell type classes are used to infer the canonical implementation of an overloaded term at a given type, canonical structures can be used to infer the canonical proof of an overloaded lemma for a given instantiation of its parameters. We present a series of design patterns for canonical structure programming that enable one to carefully and predictably coax Coq's type inference engine into triggering the execution of user-supplied algorithms during unification, and we illustrate these patterns through several realistic examples drawn from Hoare Type Theory. We assume no prior knowledge of Coq and describe the relevant aspects of Coq type inference from first principles.

scholarly journals Exact controllability for a semilinear wave equation with both interior and boundary controls

2005 ◽  
Vol 2005 (6) ◽  
pp. 619-637 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Wave Equation ◽  
Exact Controllability ◽  
Open Domain ◽  
Semilinear Wave Equation ◽  
Boundary Controls ◽  
Feedback Laws

The exact controllability of a semilinear wave equation in a bounded open domain ofRn, with controls on a part of the boundary and in the interior, is shown. Feedback laws are established.

scholarly journals Featherweight generic confinement

2006 ◽  
Vol 16 (6) ◽  
pp. 793-811 ◽  
Author(s):  
ALEX POTANIN ◽  
JAMES NOBLE ◽  
DAVE CLARKE ◽  
ROBERT BIDDLE
Keyword(s):  
Programming Languages ◽  
Ad Hoc ◽  
Type System ◽  
Type Systems ◽  
General Purpose ◽  
Ownership Type ◽  
Essential Change ◽  
Polymorphic Type ◽  
Syntactic Restrictions

Existing approaches to object encapsulation either rely on ad hoc syntactic restrictions or require the use of specialised type systems. Syntactic restrictions are difficult to scale and to prove correct, while specialised type systems require extensive changes to programming languages. We demonstrate that confinement can be enforced cheaply in Featherweight Generic Java, with no essential change to the underlying language or type system. This result demonstrates that polymorphic type parameters can simultaneously act as ownership parameters and should facilitate the adoption of confinement and ownership type systems in general-purpose programming languages.

scholarly journals Generalized control with compact support for systems with distributed parameters

2015 ◽  
Vol 25 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Asatur ZH. Khurshudyan
Keyword(s):  
Compact Support ◽  
Exact Controllability ◽  
Time Interval ◽  
State Function ◽  
Infinite Dimensional ◽  
Distributed Parameters ◽  
Boundary Controls ◽  
Systems With Distributed Parameters ◽  
Distributed And Boundary Controls ◽  
Controllability Conditions

Abstract We propose a generalization of the Butkovskiy's method of control with compact support [1] allowing to derive exact controllability conditions and construct explicit solutions in control problems for systems with distributed parameters. The idea is the introduction of a new state function which is supported in considered bounded time interval and coincides with the original one therein. By means of techniques of the distributions theory the problem is reduced to an interpolation problem for Fourier image of unknown function or to corresponding system of integral equalities. Treating it as infinite dimensional problem of moments, its L1, L2 and L∞-optimal solutions are constructed explicitly. The technique is explained for semilinear wave equation with distributed and boundary controls. Particular cases are discussed.

scholarly journals CONTROL AND STABILIZATION OF THE NONLINEAR SCHRÖDINGER EQUATION ON RECTANGLES

2010 ◽  
Vol 20 (12) ◽  
pp. 2293-2347 ◽  
Author(s):  
LIONEL ROSIER ◽  
BING-YU ZHANG
Keyword(s):  
Schrödinger Equation ◽  
Boundary Conditions ◽  
Control Region ◽  
Internal Control ◽  
Schrodinger Equation ◽  
Exact Controllability ◽  
Neumann Boundary ◽  
Local Controllability ◽  
Boundary Controls ◽  
Semilinear Schrödinger Equation

This paper studies the local exact controllability and the local stabilization of the semilinear Schrödinger equation posed on a product of n intervals (n ≥ 1). Both internal and boundary controls are considered, and the results are given with periodic (resp. Dirichlet or Neumann) boundary conditions. In the case of internal control, we obtain local controllability results which are sharp as far as the localization of the control region and the smoothness of the state space are concerned. It is also proved that for the linear Schrödinger equation with Dirichlet control, the exact controllability holds in H-1(Ω) whenever the control region contains a neighborhood of a vertex.

scholarly journals Control of a Boussinesq system of KdV–KdV type on a bounded interval

2019 ◽  
Vol 25 ◽  
pp. 58 ◽  
Author(s):  
Roberto A. Capistrano–Filho ◽  
Ademir F. Pazoto ◽  
Lionel Rosier
Keyword(s):  
Kdv Equation ◽  
Exact Controllability ◽  
Boussinesq System ◽  
Smoothing Effect ◽  
Controllability Problem ◽  
Bounded Interval ◽  
Neumann Type ◽  
Linearized System ◽  
Boundary Controls ◽  
Propagation Of Waves

We consider a Boussinesq system of KdV–KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley–Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.

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