154 research outputs found
Robust Control of Constrained Parabolic Systems with Neumann Boundary Conditions
This paper presents recent results by the authors on minimax robust control design of parabolic systems with uncertain perturbations under pointwise state and control constraints. The design procedure involves multi-step approximations and essentially employs monotonicity properties of the parabolic dynamics as well as its asymptotics on the infinite horizon. The results obtained justify a suboptimal three-positional structure of feedback controllers in the Neumann boundary conditions and provide calculations of their optimal parameters to ensure the required state performance and stability under any admissible perturbations. The problem under consideration was originally motivated by control design in water resources but certainly admits a much broader spectrum of applications
Minimax Control of Constrained Parabolic Systems
In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions
Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions
In this paper we intend to give some calculus rules for tangent sets in the
sense of Bouligand and Ursescu, as well as for corresponding derivatives of
set-valued maps. Both first and second order objects are envisaged and the
assumptions we impose in order to get the calculus are in terms of metric
subregularity of the assembly of the initial data. This approach is different
from those used in alternative recent papers in literature and allows us to
avoid compactness conditions. A special attention is paid for the case of
perturbation set-valued maps which appear naturally in optimization problems.Comment: 17 page
Prox-regularity of rank constraint sets and implications for algorithms
We present an analysis of sets of matrices with rank less than or equal to a
specified number . We provide a simple formula for the normal cone to such
sets, and use this to show that these sets are prox-regular at all points with
rank exactly equal to . The normal cone formula appears to be new. This
allows for easy application of prior results guaranteeing local linear
convergence of the fundamental alternating projection algorithm between sets,
one of which is a rank constraint set. We apply this to show local linear
convergence of another fundamental algorithm, approximate steepest descent. Our
results apply not only to linear systems with rank constraints, as has been
treated extensively in the literature, but also nonconvex systems with rank
constraints.Comment: 12 pages, 24 references. Revised manuscript to appear in the Journal
of Mathematical Imaging and Visio
Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem
We study a system of nonlinear partial differential equations resulting from
the traditional modelling of oil engineering within the framework of the
mechanics of a continuous medium. Recent results on the problem provide
existence, uniqueness and regularity of the optimal solution. Here we obtain
the first necessary optimality conditions.Comment: 9 page
Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales
We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201
Regularity of a kind of marginal functions in Hilbert spaces
We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient
Nonsmooth analysis of doubly nonlinear evolution equations
In this paper we analyze a broad class of abstract doubly nonlinear evolution
equations in Banach spaces, driven by nonsmooth and nonconvex energies. We
provide some general sufficient conditions, on the dissipation potential and
the energy functional,for existence of solutions to the related Cauchy problem.
We prove our main existence result by passing to the limit in a
time-discretization scheme with variational techniques. Finally, we discuss an
application to a material model in finite-strain elasticity.Comment: 45 page
- …