IMA Journal of Mathematical Control and Information - current issue

Properties of a subalgebra of H{infty}(D) and stabilization

Sasane, A. — 2008-03-06

Let D denote the open unit disc in C. Let T denote the unit circle and let S T. We denote by A_S(D) the set of all functions f : D S -> C that are holomorphic in D and are bounded and continuous in D S. Equipped with the supremum norm, A_S(D) is a Banach algebra, and it lies between the extreme cases of the disc algebra A(D) and the Hardy space H(D). We show that A_S(D) has the following properties:

P1. The corona theorem holds for A_S(D).

P2. The integral domain A_S(D) is not a Bézout domain, but it is a Hermite ring.

P3. The stable rank of A_S(D) is 1.

P4. The Banach algebra A_S(D) has topological stable rank 2.

The classes A_S(D) serve as appropriate transfer function classes for infinite-dimensional systems that are not exponentially stable, but stable only in some weaker sense. Consequences of the above properties to stabilizing controller synthesis using a coprime factorization approach are discussed.

Gain reduction in switched sliding-mode control

Ferrara, A., Scattolini, R. — 2008-03-06

A switched sliding-mode control strategy for a class of nonlinear uncertain systems is presented in this paper. It is characterized by an event-driven gain reduction mechanism which relies on a decomposition of the system state into regions. By enforcing sliding-mode behaviours on a suitable set of sliding manifolds, while avoiding the generation of limit cycles, the proposed strategy proves to globally asymptotically stabilize the origin of the system state space.

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Bahaa, G. M. — 2008-03-06

Optimal control problems of systems governed by parabolic equations with an infinite number of variables and with additional equality constraints are considered. The extremum principle, as well as sufficient condition of optimality, is formulated for the Neumann problem by using certain extensions of Dubovitskii–Milyutin method.

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

Bahaa, G. M., Kotarski, W. — 2008-03-06

A distributed control problem for n x n parabolic coupled systems involving operators with infinite order is considered. The performance index is more general than the quadratic one and has an integral form. Constraints on controls are imposed. Making use of the Dubovitskii–Milyutin theorem, the necessary and sufficient conditions of optimality are derived for the Dirichlet problem. Yet, the problem considered here is more general than the problems in El-Saify & Bahaa (2002, Optimal control for n x n hyperbolic systems involving operators of infinite order. Math. Slovaca, 52, 409–424), El-Zahaby (2002, Optimal control of systems governed by infinite order operators. Proceeding (Abstracts) of the International Conference of Mathematics (Trends and Developments) of the Egyptian Mathematical Society, Cairo, Egypt, 28–31 December 2002. J. Egypt. Math. Soc. (submitted)), Gali & El-Saify (1983, Control of system governed by infinite order equation of hyperbolic type. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributed control of a system governed by Dirichlet and Neumann problems for elliptic equations of infinite order. Proceeding of the International Conference on Functional-Differential Systems and Related Topics, vol. III. Poland, pp. 83–87) and Kotarski et al. (200b, Optimal control problem for a hyperbolic system with mixed control-state constraints involving operator of infinite order. Int. J. Pure Appl. Math., 1, 241–254).

Weakening the strengthened condition of Weierstrass for the isoperimetric problem in the calculus of variations

Licea, G. S. — 2008-03-06

An alternate sufficiency proof for the fixed end-point isoperimetric problem in the calculus of variations is presented. This technique not only shows how the problem need not be transformed into a problem of Lagrange but also shows how we can weaken the classical strengthened condition of Weierstrass. The usefulness of this sufficiency result is illustrated with an example which cannot be transformed into a problem of Lagrange and for which it is possible to apply the alternate sufficiency theorem in order to conclude that a given extremal affords a strict strong minimum. On the other hand, we show that the classical sufficiency theorem does not respond for this case.

Properties of the weighted logarithmic matrix norms

Hu, G.-D., Liu, M. — 2008-03-06

In this paper, we are concerned with the properties of the weighted logarithmic matrix norms. A relation between the elliptic logarithmic matrix norm and the weighted logarithmic matrix norm is given. Based on Lyapunov equations, two weighted logarithmic matrix norms are constructed which are less than 1-logarithmic matrix norm and -logarithmic matrix norm, respectively. Then, an iterative scheme is presented to obtain the logarithmically -efficient matrix norm. Numerical examples are given to illustrate the results.

Feedback theory for time-varying regular linear systems with input and state delays

Hadd, S., Rhandi, A., Schnaubelt, R. — 2008-03-06

We show that the class of regular time-varying systems is invariant under perturbations by time-varying state and input delays. In particular, we give explicit formulas of the resulting input, output and input–output maps. This result is used to solve the feedback problem for the delayed system. The relationship between the open- and the closed-loop system is investigated. Our results are applied to a parabolic boundary control problem with input and state delays.

Optimal boundary feedback stabilization of a string with moving boundary

Gugat, M. — 2008-03-06

We consider a finite string that is fixed at one end and subject to a feedback control at the other end which is allowed to move. We show that the behaviour is similar to the situation where both ends are fixed: As long as the movement is not too fast, the energy decays exponentially and for a certain parameter in the feedback law it vanishes in finite time. We consider movements of the boundary that are continuously differentiable with a derivative whose absolute value is smaller than the wave speed. We solve a problem of worst-case optimal feedback control, where the parameter in the feedback law is chosen such that the worst-case L^p-norm of the space derivative at the fixed end of the string is minimized (p [1, )). We consider the worst case both with respect to the initial conditions and with respect to the boundary movement. It turns out that the parameter for which the energy vanishes in finite time is optimal in this sense for all p.

Open-loop linearization of non-linear discrete input-output systems through simplification algorithms

Kotsios, S. — 2008-03-06

The problem of linear equivalence for a general class of non-linear systems is examined throughout this paper. A relevant algorithm is developed based on a factorization procedure. This factorization is based on the star product, an operation corresponding to the cascade connection of systems.