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1. We describe, at first in a very formaI manner, our essential
aim. n Let m be an op en subset of R , with boundary am. In m and
on am we introduce, respectively, linear differential operators P
and Qj' 0 ~ i ~ 'V. By "non-homogeneous boundary value problem" we
mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be
given in function space s F and G , F being a space" on m" and the
G/ s spaces" on am" ; j we seek u in a function space u/t "on m"
satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v"])). Qj
may be identically zero on part of am, so that the number of
boundary conditions may depend on the part of am considered 2. We
take as "working hypothesis" that, for fEF and gjEG , j the problem
(1), (2) admits a unique solution u E U/t, which depends 3
continuously on the data . But for alllinear probIems, there is a
large number of choiees for the space s u/t and {F; G} (naturally
linke d together). j Generally speaking, our aim is to determine
families of spaces 'ft and {F; G}, associated in a "natural" way
with problem (1), (2) and con j venient for applications, and also
all possible choiees for u/t and {F; G} j in these families.
I. In this second volume, we continue at first the study of non
homogeneous boundary value problems for particular classes of evolu
tion equations. 1 In Chapter 4 , we study parabolic operators by
the method of Agranovitch-Vishik [lJ; this is step (i)
(Introduction to Volume I, Section 4), i.e. the study of
regularity. The next steps: (ii) transposition, (iii)
interpolation, are similar in principle to those of Chapter 2, but
involve rather considerable additional technical difficulties. In
Chapter 5, we study hyperbolic operators or operators well defined
in thesense of Petrowski or Schroedinger. Our regularity results
(step (i)) seem to be new. Steps (ii) and (iii) are all3.logous to
those of the parabolic case, except for certain technical
differences. In Chapter 6, the results of Chapter'> 4 and 5 are
applied to the study of optimal control problems for systems
governed by evolution equations, when the control appears in the
boundary conditions (so that non-homogeneous boundary value
problems are the basic tool of this theory). Another type of
application, to the characterization of "all" well-posed problems
for the operators in question, is given in the Ap pendix. Still
other applications, for example to numerical analysis, will be
given in Volume 3.
1. The development of a theory of optimal control (deterministic)
requires the following initial data: (i) a control u belonging to
some set ilIi ad (the set of 'admissible controls') which is at our
disposition, (ii) for a given control u, the state y(u) of the
system which is to be controlled is given by the solution of an
equation (*) Ay(u)=given function ofu where A is an operator
(assumed known) which specifies the system to be controlled (A is
the 'model' of the system), (iii) the observation z(u) which is a
function of y(u) (assumed to be known exactly; we consider only
deterministic problems in this book), (iv) the "cost function" J(u)
("economic function") which is defined in terms of a numerical
function z-+
1. Our essential objective is the study of the linear,
non-homogeneous problems: (1) Pu = I in CD, an open set in RN, (2)
fQjtl = gj on am (boundary of m), lor on a subset of the boundm"J
am 1 v, where Pis a linear differential operator in m and where the
Q/s are linear differential operators on am. In Volumes 1 and 2, we
studied, for particular c1asses of systems {P, Qj}, problem (1),
(2) in c1asses of Sobolev spaces (in general constructed starting
from P) of positive integer or (by interpolation) non-integer
order; then, by transposition, in c1asses of Sobolev spaces of
negative order, until, by passage to the limit on the order, we
reached the spaces of distributions of finite order. In this
volume, we study the analogous problems in spaces of inlinitely
dilferentiable or analytic Itlnctions or of Gevrey-type I~mctions
and by duality, in spaces 01 distribtltions, of analytic
Itlnctionals or of Gevrey- type ultra-distributions. In this
manner, we obtain a c1ear vision (at least we hope so) of the
various possible formulations of the boundary value problems (1),
(2) for the systems {P, Qj} considered here.
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Numerical Analysis of Partial Differential Equations - Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Ispra (Varese), Italy, July 3-11, 1967 (English, French, Italian, Paperback, Reprint of the 1st. Ed. C.I.M.E., Ed. Cremonese, Roma, 1968.)
Jacques-Louis Lions
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R2,181
Discovery Miles 21 810
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Ships in 10 - 15 working days
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S. Albertoni: Alcuni metodi di calcolo nella teoria della
diffusione dei neutroni.- I. Babuska: Optimization and numerical
stability in computations.- J.H. Bramble: Error estimates in
elliptic boundary value problems.- G. Capriz: The numerical
approach to hydrodynamic problems.- A. Dou: Energy inequalities in
an elastic cylinder.- T. Doupont: On the existence of an iterative
method for the solution of elliptic difference equation with an
improved work estimate.- J. Douglas, J.R. Cannon: The approximation
of harmonic and parabolic functions of half-spaces from interior
data.- B.E. Hubbard: Error estimates in the fixed Membrane
problem.- K. Jorgens: Calculation of the spectrum of a Schrodinger
operator.- A. Lasota: Contingent equations and boundary value
problems.- J.L. Lions: Reduction a des problemes du type
Cauchy-Kowalewska.- J.L. Lions: Problemes aux limites non homogenes
a donnees irregulieres; une methode d'approximation.- J.L. Lions:
Remarques sur l'approximation regularisee de problemes aux
limites.- W.V. Petryshyn: On the approximation-solvability of
nonlinear functional equations in normed linear spaces.- P.A.
Raviart: Approximation des equations d'evolution par des methodes
variationnelles.- M. Sibony, H. Brezis: Methodes d'approximation et
d'iteration pour les operateurs monotones.- V. Thomee: Some topics
in stability theory for partial difference operators."
Dans un espace de BANACH H soit A(t) une famille d'operateurs non
bornes, tE [0, TJ pour fixer les idees. On appelle equation
difterentielle operationneUe (lineaire) une equation de la forme
A(t)u(t)]u'(t) =f(t), la fonction f Hant donnee continue de [0, TJ
dans H, la fonction u Hant une fois continument differentiable dans
[0, TJ a valeurs dans H, u(t) appartenant a D(A(t)) (domaine de
A(t)) pour chaque tE[O, T]. Les exemples les plus importants sont
ceux Oll A(t) est un systeme differentiel, le domaine de A(t) Hant
alors fixe par des conditions aux limites. Le probleme de CAUCHY
consiste a trouver une solution de (*), verifiant la condition
initiale u(O) = u, U donne (dans D(A(O))). o o Mais il est
classique que, pour bien des applications, le probleme pose sous la
forme precedente impose des conditions trop restrictives a u. Il
faut introduire alors la notion de solution faible de ce probleme;
il y a un tres grand nombre de telles notions; une classification
en est donnee au Chap. 1. Les Chap. IV, V, VII, IX, X donnent
diverses con- ditions suffisantes portant sur les A (t) pour que
tel ou tel probleme faible admette une solution et une seule; on y
Hudie la regularite de ces solutions, et les meilleurs domaines Oll
l' on doit prendre les donnees initiales.
The behaviour of systems occurring in real life is often modelled
by partial differential equations. This book investigates how a
user or observer can influence the behaviour of such systems
mathematically and computationally. A thorough mathematical
analysis of controllability problems is combined with a detailed
investigation of methods used to solve them numerically, these
methods being validated by the results of numerical experiments. In
Part I of the book the authors discuss the mathematics and numerics
relating to the controllability of systems modelled by linear and
non-linear diffusion equations; Part II is dedicated to the
controllability of vibrating systems, typical ones being those
modelled by linear wave equations; finally, Part III covers flow
control for systems governed by the Navier-Stokes equations
modelling incompressible viscous flow. The book is accessible to
graduate students in applied and computational mathematics,
engineering and physics; it will also be of use to more advanced
practitioners.
The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. The objective of the present work is to compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form.
The advent of high-speed computers has made it possible for the
first time to calculate values from models accurately and rapidly.
Researchers and engineers thus have a crucial means of using
numerical results to modify and adapt arguments and experiments
along the way. Every facet of technical and industrial activity has
been affected by these developments. The objective of the present
work is to compile the mathematical knowledge required by
researchers in mechanics, physics, engineering, chemistry and other
branches of application of mathematics for the theoretical and
numerical resolution of physical models on computers. Since the
publication in 1924 of the "Methoden der mathematischen Physik" by
Courant and Hilbert, there has been no other comprehensive and
up-to-date publication presenting the mathematical tools needed in
applications of mathematics in directly implementable form.
These 6 volumes - the result of a 10 year collaboration between the
authors, two of France's leading scientists and both distinguished
international figures - compile the mathematical knowledge required
by researchers in mechanics, physics, engineering, chemistry and
other branches of application of mathematics for the theoretical
and numerical resolution of physical models on computers. Since the
publication in 1924 of the "Methoden der mathematischen Physik" by
Courant and Hilbert, there has been no other comprehensive and
up-to-date publication presenting the mathematical tools needed in
applications of mathematics in directly implementable form. The
advent of large computers has in the meantime revolutionised
methods of computation and made this gap in the literature
intolerable: the objective of the present work is to fill just this
gap. Many phenomena in physical mathematics may be modeled by a
system of partial differential equations in distributed systems: a
model here means a set of equations, which together with given
boundary data and, if the phenomenon is evolving in time, initial
data, defines the system. The advent of high-speed computers has
made it possible for the first time to calculate values from models
accurately and rapidly. Researchers and engineers thus have a
crucial means of using numerical results to modify and adapt
arguments and experiments along the way. Every facet of technical
and industrial activity has been affected by these developments.
Modeling by distributed systems now also supports work in many
areas of physics (plasmas, new materials, astrophysics,
geophysics), chemistry and mechanics and is finding increasing use
in the life sciences.
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