Null controllability of the Lotka-McKendrick system with spatial diffusion

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

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.

Strict Dissipativity for LQ Problem with Left-invertible Underlying Semigroups

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.

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

Full probabilistic solution of a finite dimensional linear control system with random initial and final conditions

The question of power and asymptotic stability of infinite dimensional discrete-time state space systems is investigated. It is shown that every balanced realization is asymptotically stable. Conditions are given for balanced, input normal, or output normal realization to be asymptotically and/or power stable. >

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.

In this paper we attempt to describe a circle of ideas which makes possible a unified exposition of a large part of linear algezbra, operator theory and both finite and infinite dimensional multivariable linear systems. As the title suggests the unifying conzcepts will be those of functional models, module theory and various factorizations of polynomial and analytic matrix valued functions. The full exposition of the ideas presented here will be the theme of a forthcoming monograph [20].

This paper provides a general theory for characterizing and constructing a decoherence-free (DF) subsystem for an infinite dimensional linear open quantum system. The main idea is that, based on the Heisenberg picture of the dynamics rather than the commonly-taken Schrodinger picture, the notions of controllability and observability in control theory are employed to characterize a DF subsystem. A particularly useful result is a general if and only if condition for a linear system to have a DF component; this condition is used to demonstrate how to actually construct a DF dynamics in some specific examples. It is also shown that, as in the finite dimensional case, we are able to do coherent manipulation and preservation of a state of a DF subsystem.

The minimal time function of a class of semilinear control systems is considered in Banach spaces, with the target set being a closed ball. It is shown that the minimal time functions of the Yosida approximation equations converge to the minimal time function of the semilinear control system. Complete characterization is established for the subdifferential of the minimal time function satisfying the Hamilton---Jacobi---Bellman equation. These results extend the theory of finite dimensional linear control systems to infinite dimensional semilinear control systems.

Weak convergence of random sets in Banach spaces

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.

This paper describes the classification problem of continuous families of infinite dimensional linear systems. We are concerned with a family of controllable and observable linear systems $(F(s),G(s),H(s))$ in Banach spaces, which depend continuously on a parameter s in a Hausdorff topological space S. Our problem is to classify such families of systems by isomorphism relation and parametrize them by “moduli.” It is known that there exist “moduli” for finite dimensional controllable and observable systems. We show the result fails in the infinite dimensional case, and derive some conditions under which there exist “moduli,” i.e., a fine moduli theorem for infinite dimensional systems. The theory of Banach bundles plays an important role in our study.