Krastanov, M. I., N. K. Ribarska.
Viability and an Olech type result
(pp. 423–446)
A B S T R A C T S
ASYMPTOTIC BEHAVIOR IN SLIDING MODE CONTROL SYSTEMS
Tullio Zolezzi
zolezzi@dima.unige.it
2010 Mathematics Subject Classification: 93B12, 93D15.
Key words:
sliding mode control, practical stability, asymptotic convergence.
Practical stability of real states of nonlinear sliding mode control systems is related to asymptotic vanishing of the corresponding sliding errors. Conditions are found such that, if the equivalent control achieves exponential stability, then real states are practically stable. In special cases, their exponential stability is obtained. A link between convergence of regularization procedures and metric regularity is pointed out.
A SECOND-ORDER MAXIMUM PRINCIPLE IN OPTIMAL CONTROL UNDER STATE CONSTRAINTS
Hélène Frankowska
frankowska@math.jussieu.fr
Daniel Hoehener
daniel-hoehener@bluewin.ch
Daniela Tonon
tonon@ceremade.dauphine.fr
2010 Mathematics Subject Classification: 49K15, 49K21, 34A60, 34K35.
Key words:
optimal control, second-order necessary
optimality conditions, second-order tangents.
A second-order variational inclusion for control systems under
state constraints is derived and applied to investigate necessary
optimality conditions for the Mayer optimal control problem. A new
pointwise condition verified by the adjoint state of the maximum
principle is obtained as well as a second-order necessary
optimality condition in the integral form. Finally, a new
sufficient condition for normality of the maximum principle is
proposed. Some extensions to the Mayer optimization problem
involving a differential inclusion under state constraints are
also provided.
NEWTON-SECANT METHOD FOR FUNCTIONS WITH VALUES IN A CONE
Alain Pietrus
apietrus@univ--ag.fr
Célia Jean-Alexis
cjeanale@univ-ag.fr
2010 Mathematics Subject Classification: 49J53, 47H04, 65K10, 14P15.
Key words:
Variational inclusion, set-valued map, pseudo-Lipschitz map, divided differences, closed convex cone, majorized sequences, normed convex process.
This paper deals with variational inclusions of the form 0 ∈ K − f(x) − g(x)
where f is a smooth function from a reflexive Banach space X into a Banach space Y, g is a function from X into Y admitting divided differences and K is a nonempty closed convex cone in the space Y. We show that the previous problem can be solved by a combination of two methods: the Newton and the Secant methods. We show that the order of the semilocal method obtained is equal to (1 + √5)/2. Numerical results are also given to illustrate the convergence at the end of the paper.
ABOUT UNIFORM REGULARITY OF COLLECTIONS OF SETS
Alexander Y. Kruger
a.kruger@ballarat.edu.au
Nguyen H. Thao
hieuthaonguyen@students.ballarat.edu.au,
nhthao@ctu.edu.vn
2010 Mathematics Subject Classification:
49J53, 41A25, 74S30.
Key words:
uniform regularity, projection method.
We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems.
RECENT RESULTS ON DOUGLAS--RACHFORD METHODS
Francisco J. Aragón Artacho
francisco.aragon@ua.es
Jonathan M. Borwein
jon.borwein@gmail.com
Matthew K. Tam
matthew.k.tam@gmail.com
2010 Mathematics Subject Classification:
90C27, 90C59, 47N10.
Key words:
Douglas-Rachford, projections, reflections, combinatorial optimization, modelling, feasibility, satisfiability, Sudoku, Nonograms.
Recent positive experiences applying convex feasibility algorithms of Douglas-Rachford type to highly combinatorial and far from convex problems are described.
ON THE OPTIMAL CONTROL OF SOME PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS ARISING IN ECONOMICS
R. Boucekkine
raouf.boucekkine@univ-amu.fr
C. Camacho
maria.camacho-perez@univ-paris1.fr
G. Fabbri
giorgio.fabbri@univ-evry.fr
2010 Mathematics Subject Classification:
91B62, 91B72, 49K20, 49L20.
Key words:
Parabolic partial differential equations, optimal control, infinite dimensional problems, infinite time horizons, ill-posedness, dynamic programming.
We review an emerging application field to parabolic partial differential equations (PDEs), that's economic growth theory. After a short presentation of concrete applications, we highlight the peculiarities of optimal control problems of parabolic PDEs with infinite time horizons. In particular, the heuristic application of the maximum principle to the latter leads to single out a serious ill-posedness problem, which is, in our view, a barrier to the use of parabolic PDEs in economic growth studies as the latter are interested in long-run asymptotic solutions, thus requiring the solution to infinite time horizon optimal control problems. Adapted dynamic programming methods are used to dig deeper into the identified ill-posedness issue.
ON CLUSTER POINTS OF ALTERNATING PROJECTIONS
Heinz H. Bauschke
heinz.bauschke@ubc.ca
Dominikus Noll
noll@mip.ups-tlse.fr
2010 Mathematics Subject Classification:
Primary 65K10; Secondary 47H04, 49M20, 49M37, 65K05,
90C26, 90C30.
Key words:
Cluster point, convex set, continuum, method of alternating projections,
nonconvex set, projection.
Suppose that A and B are closed subsets of a
Euclidean space such that A ∩ B ≠ ∅,
and we aim to find a point in this intersection
with the help of the sequences (a_{n})_{ n∈N} and (b_{n})_{n∈N} generated by the
method of alternating projections.
It is well known that if A and B are convex, then
(a_{n})_{ n∈N} and (b_{n})_{n∈N} converge to some point in A ∩ B. The situation in the nonconvex case is much more delicate. In 1990, Combettes and Trussell presented a dichotomy result
that guarantees either convergence to a point in the intersection
or a nondegenerate compact continuum as the set of cluster points.
In this note, we construct two sets in the Euclidean plane
illustrating the continuum case. The sets A and B can be
chosen as countably infinite unions of closed convex sets. In contrast,
we also show that such behaviour is impossible for finite unions.
REGULARITY OF SET-VALUED MAPS AND THEIR SELECTIONS THROUGH SET DIFFERENCES.
PART 1: LIPSCHITZ CONTINUITY
Robert Baier
robert.baier@uni-bayreuth.de
Elza Farkhi
elza@post.tau.ac.il
2010 Mathematics Subject Classification:
54C65, 54C60, 26E25.
Key words:
Lipschitz continuous set-valued maps, selections, generalized Steiner selection, metric selection, set differences, Demyanov metric, Demyanov difference, metric difference.
We introduce Lipschitz continuity of set-valued maps with respect to a given set difference. The existence of Lipschitz selections that pass through any point of the graph of the map and inherit its Lipschitz constant is studied. We show that the Lipschitz property of the set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of its generalized Steiner selections. For a univariate multifunction with only compact values in R^{n}, we characterize its Lipschitz continuity in the Hausdorff metric (with respect to the metric difference) by the same property of its metric selections with the same constant.
REGULARITY OF SET-VALUED MAPS AND THEIR SELECTIONS THROUGH SET DIFFERENCES.
PART 2: ONE-SIDED LIPSCHITZ PROPERTIES
Robert Baier
robert.baier@uni-bayreuth.de
Elza Farkhi
elza@post.tau.ac.il
2010 Mathematics Subject Classification:
47H06, 54C65, 47H04, 54C60, 26E25.
Key words:
one-sided Lipschitzian set-valued maps, selections, generalized Steiner selection, metric selection, set differences, Demyanov difference, metric difference.
We introduce one-sided Lipschitz (OSL) conditions of set-valued maps with respect to given set differences. The existence of selections of such maps that pass through any point of their graphs and inherit uniformly their OSL constants is studied. We show that the OSL property of a convex-valued set-valued map with respect to the Demyanov difference with a given constant is characterized by the same property of the generalized Steiner selections. We prove that an univariate OSL map with compact images in R^{1} has OSL selections with the same OSL constant. For such a multifunction which is OSL with respect to the metric difference, one-sided Lipschitz metric selections exist through every point of its graph with the same OSL constant.
VIABILITY AND AN OLECH TYPE RESULT
M. I. Krastanov
krastanov@fmi.uni-sofia.bg
N. K. Ribarska
ribarska@fmi.uni-sofia.bg
2010 Mathematics Subject Classification:
34A36, 34A60.
Key words:
differential inclusions with nonconvex right-hand side, existence of solutions, colliding on a set.
We study the existence of solutions of differential inclusions
with upper semicontinuous right-hand side. The presented approach
is based on the directionally continuous selections techniques
developed by Bressan and on Srivatsa's Baire class 1 selectors
for upper semicontinuous set-valued maps. We propose a concept for
invariant ε-approximation of an upper semicontinuous
set-valued map on the elements of a relatively open partitioning.
We prove a result of Olech type where the assumptions on the set
of lower semicontinuity is G_{δ} in contrast to the usual
openness assumption. The proof is based on a generalization of
invariant ε-approximations.
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