Catálogo de publicaciones - libros

The application of state-discontinuous feedback laws to infinite-dimensional control systems, with particular reference to sliding motions, is discussed for linear systems with distributed control. Using differential inclusions a definition of generalized solutions for the discontinuous closed loop system is introduced. Sliding modes can both be defined as viable generalized solutions or by extending the equivalent control method to infinite dimensional systems. Regularity properties of the sliding manifold under which the two methods are equivalent are investigated. Then, a comparison between classical results obtained for finite dimensional spaces and properties of infinite dimensional sliding modes is made.

The argument of Razumikhin-type has been well developed and showed significant advantage for the stability of stochastic functional differential equations in finite dimensions. However, so far there have been almost no results of Razumikhin-type on the stability of mild solutions of stochastic functional differential equations in infinite dimensions. The main aim of this paper is to establish Razumikhin-type stability theorems for stochastic functional differential equations in infinite dimensions. By virtue of these new criteria, we can establish the exponential stability of stochastic delay differential equations and stochastic delay partial differential equations.

As a starting point of this paper we present a problem from mammographic image processing. We show how it can be formulated as an optimal control problem for PDEs and illustrate that it leads to penalty terms which are non-standard in the theory of optimal control of PDEs. To solve this control problem we use a generalization of the conditional gradient method which is especially suitable for non-convex problems. We apply this method to our control problem and illustrate that this method also covers the recently proposed method of surrogate functionals from the theory of inverse problems.

Two types of problems for parabolic inclusions, namely, problems of robust control under the action of uncontrolled disturbances and problems of dynamical identification of inputs, are discussed. Algorithms for solving such problems stable with respect to informational noises and computational errors are presented. The algorithms oriented to computer realization allow one to simulate a solving process in the “real time” mode. They adaptively take into account inaccurate measurements of phase trajectories and are regularizing in the following sense, the more precise is incoming information, the better is algorithm’s output. The algorithms are based on the method of auxiliary positionally-controlled models [ 1 ], [ 2 ], [ 4 ], [ 5 ], [ 6 ], [ 7 ]. The basic elements of the algorithms are represented by stabilization procedures (functioning by the feedback principle) for appropriate Lyapunov functionals.

In many engineering and biomedical problems there is a need to identify parameters of the systems from experimental data. A typical example is the biochemical-kinetics systems describing oncogenic hyperplasia where the dynamical model is nonlinear and the number of the parameters to be identified can reach a few hundreds. Solving these large-scale identification problems by the local- or global-search methods can not be practical because of the complexity and prohibitive computing time. These difficulties can be overcome by application of the non-search techniques which are much less computation- demanding. The present work proposes key components of the corresponding mathematical formulation of the nonsearch paradigm. This new framework for the nonlinear large-scale parameter identification specifies and further develops the ideas of the well-known approach of A. Krasovskii. The issues are illustrated with a concise analytical example. The new results and a few directions for future research are summarized in a dedicated section.

We state uniqueness and stability results for the inverse problem of determining a rigid inclusion inside an isotropic elastic body Ω, from a single measurement of traction and displacement taken on the boundary of Ω.

This paper deals with the formulation of a necessary optimality condition for a shape optimization problem of a viscoelastic body in unilateral dynamic contact with a rigid foundation. The contact with Coulomb friction is assumed to occur at a portion of the boundary of the body. The contact condition is described in velocities. The friction coefficient is assumed to be bounded and Lipschitz continuous with respect to a slip velocity. The evolution of the displacement of the viscoelastic body in unilateral contact is governed by a hemivariational inequality of the second order. The shape optimization problem for a viscoelastic body in contact consists in finding, in a contact region, such shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. It is assumed, that the volume of the body is constant. Using material derivative method, we calculate the directional derivative of the cost functional and we formulate a necessary optimality condition for this problem.

Second order necessary and sufficient optimality conditions for bang-bang control problems in a very general form have been obtained by the first author. These conditions require the positive (semi)-definiteness of a certain quadratic form on the finite-dimensional critical cone. In the present paper we formulate a generalization of these results to optimal control problems where the control variable has two components: a continuous unconstrained control appearing nonlinearly and a bang-bang control appearing linearly and belonging to a convex polyhedron. Many examples of control of this kind may be found in the literature.

Using the concept of well-posedness under perturbations we give an answer to the following question posed by Stanislaw Ulam : “when it is true that solutions of two problems in the calculus of variations which corresponds to “close” physical data must be close to each other?”(“A collection of mathematical problems”, 1960).

We study optimal control problems with vector-valued controls. In the article, we propose a solution strategy to solve optimal control problems with pointwise convex control constraints. It involves a SQP-like step with an imbedded active-set algorithm.