Controllability of Sobolev-type semilinear integrodifferential systems in Banach spaces

AbstractSufficient conditions for controllability of Sobolev-type semilinear integrodifferential systems in a Banach space are established. The results are obtained by using the Schaefer fixed-point theorem

NULL CONTROLLABILITY FOR THE SEMILINEAR HEAT EQUATION IN UNBOUNDED DOMAINS

In this paper, we consider the null controllability problem for the semilinear heat equation in an unbounded domain Ω of RN with Dirichlet boundary conditions. The control ís assumed to be distributed along a subdomain w such that the uncontrolled regian Ω\w ís bounded. Using Carleman inequalities we first prove the null controllabitity of the linearized equation. Then, by a fixed point method, we obtain the main result for the semilinear case. This result asserts that, when the nonlinearity is globally Lipschitz, the system is null controllable

Some applications of projections in nonlinear control and estimation

The present thesis uses an approach which regards nonlinear systems as a pair (z(·),zf) trajectory-final state, for the case of controlability or (zO,z(·)) initial state-trajectory, for the case of state estimation, in a space M which is the cross product 1x Z or Z x F between the space of trajectories F and the state space Z. In this setting some mappings F:M + M are constructed using projections P onto specific subsets S of M (i.e. p2 = P and Rep} = S). The solution of the problems of nonlinear controllability, state estimation and state and parameter estimation are obtained via the fixed points of such F's. Fixed points theorems have been used in [8, 9, 25, 35 and 15] to provide global controllability, state estimation ,and joint state and parameter estimation. Some theoretical results are presented here, which show that it is possible to eliminate some of the assumptions which restricted the systems treated in these papers and at the same time to , obtain mappings with fixed points which contain all the possible solutions for the problem of nonlinear controllability, state estimation and the joint state and parameter estimation. Among the relaxations allowed now are, for example, in the problem of control, the possibility of a set of admissible controls Uad different from the set U of all input controls of the system. In order to obtain continuous projections P, S must be closed in M. This will occur naturally in the case of state estimation however, for the control case, in general, it will be necessary to adjust the topologies of the spaces U and/or M in order to achieve this. A comprehensive theory which shows that this adjustment is always possible as well as a complete procedure for obtaining the adjusted spaces, U and M are presented here

Control and Coordination in Hierarchical Systems

This book presents the applied theory of control and cooordination in hierarchical systems which are those where decision making has been divided in a certain way. It concentrates on various aspects of optimal control in large scale systems and covers a range of topics from multilevel methods for optimizing by interactive feedback procedures to methods for sequential, hierarchical control in large dynamic systems

Functional analysis and aspects of non-linear control theory

This thesis studies the state reconstruction problem for a class of non-linear systems. This class is that of perturbed linear systems. The properties of the linear part are used to arrive at results for the complete system. Whilst this is a common technique in mathematics and physics its use in non-linear infinite dimensional systems theory has not been extensively investigated. The present work makes such an investigation with a view to indicating the successes, and limitations, of such a treatment. As to contribution, as far as the author is aware, many of the results are new both in precise statement and general approach. Chapter 1 introduces, and motivates, the formulation adopted. Chapter 2 provides some useful information on linear infinite dimensional control theory. Chapter 3 gives, subject to certain, perhaps restrictive, conditions, a rigorous statement, and proof, of the basic theorems. Here, as elsewhere, the standard fixed point results are used. Parts of this chapter are extracts from, as yet unpublished, joint work with A.J. Pritchard and M.D. Quinn. Chapter 4 relaxes some of the conditions in 3 and applies the same techniques to other areas. Chapter 5 surveys, in a formal fashion the more constructive, numerical aspects of the preceding results with a view to indicating directions for this important area of further research. It is concluded that the "perturbed linear" approach used here can give results that are both theoretically and computationally useful. The strength of the requirements placed on the linear part, however, indicates a challenging area for future investigations: a constructive approach to intrins1cally non-linear problems

Equations of structured population dynamics.

Guo Bao Zhu.Thesis (Ph.D.)--Chinese University of Hong Kong.Includes bibliographical references.Abstract --- p.1Introduction --- p.3Chapter Chapter 1. --- Semigroup for Age-Dependent Population Equations with Time DelayChapter 1.1 --- Introduction --- p.13Chapter 1.2 --- Problem Statement and Linear Theory --- p.14Chapter 1.3 --- Spectral Properties of the Infinitesimal Generator --- p.20Chapter 1.4 --- A Nonlinear Semigroup of the Logistic Age-Dependent Model with Delay --- p.26References --- p.34Chapter Chapter 2. --- Global Behaviour of Logistic Model of Age-Dependent Population GrowthChapter 2.1 --- Introduction --- p.35Chapter 2.2 --- Global Behaviour of the Solutions --- p.37Chapter 2.3 --- Oscillatory Properties --- p.47References --- p.51Chapter Chapter 3. --- Semigroups for Age-Size Dependent Population Equations with Spatial DiffusionChapter 3. 1 --- Introduction --- p.52Chapter 3.2 --- Properties of the Infinitesimal Generator --- p.54Chapter 3.3 --- Properties of the Semigroup --- p.59Chapter 3.4 --- Dynamics with Age-Size Structures --- p.61Chapter 3.5 --- Logistic Model with Diffusion --- p.66References --- p.70Chapter Chapter 4. --- Semi-Discrete Population Equations with Time DelayChapter 4. 1 --- Introduction --- p.72Chapter 4.2 --- Linear Semi-Discrete Model with Time Delay --- p.74Chapter 4.3 --- Nonlinear Semi-Discrete Model with Time Delay --- p.88References --- p.98Chapter Chapter 5. --- A Finite Difference Scheme for the Equations of Population DynamicsChapter 5.1 --- Introduction --- p.99Chapter 5.2 --- The Discrete System --- p.102Chapter 5.3 --- The Main Results --- p.107Chapter 5.4 --- A Finite Difference Scheme for Logistic Population Model --- p.113Chapter 5.5 --- Numerical Simulation --- p.116References --- p.119Chapter Chapter 6. --- Optimal Birth Control Policies IChapter 6. 1 --- Introduction --- p.120Chapter 6.2 --- Fixed Horizon and Free Point Problem --- p.120Chapter 6.3 --- Time Optimal Control Problem --- p.129Chapter 6.4 --- Infinite Horizon Problem --- p.130Chapter 6.5 --- Results of the Nonlinear System with Logistic Term --- p.143Reference --- p.148Chapter Chapter 7. --- Optimal Birth Control Policies IIChapter 7. 1 --- Free Final Time Problems --- p.149Chapter 7.2 --- Systems with Phase Constraints --- p.160Chapter 7.3 --- Mini-Max Problems --- p.166References --- p.168Chapter Chapter 8. --- Perato Optimal Birth Control PoliciesChapter 8.1 --- Introduction --- p.169Chapter 8.2 --- The Duboviskii-Mi1yutin Theorem --- p.171Chapter 8.3 --- Week Pareto Minimum Principle --- p.172Chapter 8.4 --- Problem with Nonsmooth Criteria --- p.175References --- p.181Chapter Chapter 9. --- Overtaking Optimal Control Problems with Infinite HorizonChapter 9. 1 --- Introduction --- p.182Chapter 9.2 --- Problem Statement --- p.183Chapter 9.3 --- The Turnpike Property --- p.190Chapter 9.4 --- Existence of Overtaking Optimal Solutions --- p.196References --- p.198Chapter Chapter 10. --- Viable Control in Logistic Populatiuon ModelChapter 10. 1 --- Introduction --- p.199Chapter 10. 2 --- Viable Control --- p.200Chapter 10.3 --- Minimum Time Problem --- p.205References --- p.208Author's Publications During the Candidature --- p.20

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute