Strict Dissipativity for LQ Problem with Left-invertible Underlying Semigroups

The aim of this paper is to find a feedback law which steers the initial state to the origin in uniform finite time for the infinite dimensional singular discrete control systems where the state and control spaces are Hilbert spaces.

LetA, ℬ be evolution operators (possibly nonlinear) which act within a Banach spaceB andu(·) a measurable, real valued, control function. We study control systems of the form ∂υt/∂t=Aυt +u(t)ℬυt, υ0=ϕ ∈B. An observation of this system is defined to be a continuous linear mapg:B→ℝk. Our main result gives a computable sufficient condition to assure that fort > 0 and sufficiently small, the observation of the reference solution (which corresponds tou(t)≡0) at timet is interior to the set of observations of all solutions at timet. An example to illustrate the theory is the local controllability, via tension, of various observations of a vibrating string.

We consider the infinite dimensional linear control system described by the population dynamics model of Lotka-McKendrick with spatial diffusion. Considering control functions localized with respect to the spatial variable but active for all ages, we prove that the whole population can be steered to zero in any positive time. The main novelty we bring is that, unlike the existing results in the literature, we can also control the population of ages very close to 0. Another novelty brought in is the employed methodology: as far as we know, the present work is the first one remarking that the null controllability of the considered system can be obtained by using the Lebeau-Robbiano strategy, originally developed for the null-controllability of the heat equation.

In this work we study the structure of “approximate” solutions for a nonautonomous infinite dimensional discrete-time control system determined by a sequence of continuous functions \(v_i: X \times X \to R^1\), \(i=0,\pm 1,\pm 2,\dots \) where X is a metric space.

Optimal control of stochastic functional neutral differential equations with time lag in control

The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in [1] for finite dimensional systems. We show that this result extends to infinite dimensional systems, i.e. control of linear partial differential equations. The quadratic objective paired with the linear dynamics encode the relevant kernel, defining a Hilbert space of controlled trajectories, for which we obtain a concise formula based on the solution of the differential Riccati equation. This paves the way to applying representer theorems from kernel methods to solve infinite dimensional optimal control problems.

Weak convergence of random sets in Banach spaces

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

This paper deals with the reduced order controller design for infinite dimensional port Hamiltonian systems (IDPHS). Firstly, a structure preserving and passive LQG control design equivalent to Control by Interconnection is proposed. Based on this LQG controller, a structure preserving reduction method is used to approximate both the closed loop IDPHS and the LQG controller. This closed loop reduction guarantees that the reduced order controller will ensure acceptable closed loop performances on the infinite dimensional system. The proposed methods is applied to the control of a vibro-acoustic system.

The concept of zeros at infinity is generalized to some particular cases of infinite dimensional control systems: those described with bounded operators and having a finite number of inputs and outputs. These zeros are characterized with the help of four equivalent descriptions. Two geometric characterizations are provided as well as a matricial one using some particular Toeplitz matrices. The last one is directly deduced from the Structure Algorithm. Finally, for systems having a finite number of disturbance inputs, we show that the Disturbance Decoupling Problem with Measurement of the Disturbance is solvable if and only if the orders of the zeros at infinity are the same for the system with and without the disturbance. Applications to invertibility tests or to ideal observability are also given.

On Lipschitz conditions of infinite dimensional systems

A general perturbation theory of the Kolmogorov-Arnold-Moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems. The key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures. This approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems.

In this paper first we establish the existence of extremal solutions for evolution inclusions driven by time dependent convex subdifferentials, and then show that they are dense for the topology of uniform convergence on T=[0, b] to the solutions of the original system. Then these results are applied to establish “bang‐bang” type results for some classes of nonlinear infinite dimensional control systems. A few examples are also presented.

Approximate Controllability of Strongly Degenerate Fractional Order System of Distributed Control

Atomic force microscopy (AFM) has been extensively used and AFM based nanomanipulation has also been studied for many years. But efficiency and accuracy of AFM based nanomanipulation are still major issues. In order to accurately control the tip position, which is important to prevent the tip from missing the objects, the deformation of cantilever has to be compensated or eliminated during manipulation. An active AFM probe is used to eliminate the deformation of cantilever by directly controlling the cantilever's flexibility or rigidity during manipulation. In this paper, the active probe is modeled as an infinite dimensional system based on Euler-Bernoulli theory. A periodic-output-feedback (POF) controller is designed and simulations are carried out to optimize the parameters of the controller. The POF controller is implemented in a real time Linux system. Experimental results demonstrated the validity of infinite dimensional model and POF controller. Experimental results on nanomanipulation have also shown that the control accuracy of tip position is improved by using this controller of active probe.