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1

The motion planning problem and exponential stabilization of a heavy chain. Part II

Grabowski P.

Opuscula Mathematica

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2008

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Vol. 28, no. 4

481-505

EN

This is the second part of paper [8], where a model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations was interpreted as an abstract semigroup system and then analysed on a Hilbert state space. In particular, in [8] we have formulated the problem of exponential stabilizability of a heavy chain in a given position. It was also shown that the exponential stability can be achieved by applying a stabilizer of the colocated-type. The proof used the method of Lyapunov functionals. In the present paper, we give other two proofs of the exponential stability, which provides an additional intrinsic insight into the exponential stabilizability mechanism. The first proof makes use of some spectral properties of the system. In the second proof, we employ some relationships between exponential stability and exact observability.

2

Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory

Grabowski P.

Opuscula Mathematica

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2006

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Vol. 26, no. 1

45-97

EN

The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [12, 13] two groups of similarity criteria for a given hybrid closed-loop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions.

3

On the circle criterion for boundary control systems in factor form

Grabowski P.

Opuscula Mathematica

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2003

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Vol. 23

25-47

EN

In this paper we return to the origins of the circle criterion initiated by Irwin Sandberg nearly forthy years ago. A version of the Leray-Schauder alternative is applied to get an existence of an abstract Hammerstein output equation for the closed-loop system. This existence result completes Sandberg's method based on using the Banach fixed-point theorem. It is shown that the assertion of the circle criterion can be strengthened by adding a characterization of an asymptotic behaviour of the state trajectories. Results are being compared with a recent version of the circle criterion for boundary control systems in factor form. Some prospects for further studies are also suggested.

4

Constrained Controllability of Semilinear Delayed Systems

Klamka J.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2001

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vol. 49, nr 3

505-515

EN

In the paper infinite-dimensional dynamical control systems described by semilinear differential equations with delays in the state variables are considered. Using a general sufficient conditions for constrained exact controllability for infinite-dimentional dynamical systems sufficient conditions for constrained exact absolute local controllability are formulated and proved. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero. As an illustrative example, constrained exact absolute local controllability problem for semilinear dynamical system with one constant delay in the state variable is solved in details. Some remarks and comments on the existing results for controllability of nonlinear dynamical systems are also presented.