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Mathematical Control & Related Fields

2013 , Volume 3 , Issue 1

Special Issue dedicated to Professor George Leitmann on the occasion of his 88th birthday (Part I)

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Compositions of passive boundary control systems
Atte Aalto and  Jarmo Malinen
2013, 3(1): 1-19 doi: 10.3934/mcrf.2013.3.1 +[Abstract](68) +[PDF](500.3KB)
Abstract:
We show under mild assumptions that a composition of internally well-posed, impedance passive (or conservative) boundary control systems through Kirchhoff type connections is also an internally well-posed, impedance passive (resp., conservative) boundary control system. The proof is based on results of Malinen and Staffans [21]. We also present an example of such composition involving Webster's equation on a Y-shaped graph.
Stability estimates for a Robin coefficient in the two-dimensional Stokes system
Muriel Boulakia , Anne-Claire Egloffe and  Céline Grandmont
2013, 3(1): 21-49 doi: 10.3934/mcrf.2013.3.21 +[Abstract](60) +[PDF](695.4KB)
Abstract:
In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [11] and under the assumption that the velocity of a given reference solution stays far from $0$ on a part of the boundary where Robin conditions are prescribed.
On the minimum time function around the origin
Giovanni Colombo and  Khai T. Nguyen
2013, 3(1): 51-82 doi: 10.3934/mcrf.2013.3.51 +[Abstract](37) +[PDF](521.9KB)
Abstract:
We deal with finite dimensional linear and nonlinear control systems. If the system is linear and autonomous and satisfies the classical normality assumption, we improve the well known result on the strict convexity of the reachable set from the origin by giving a polynomial estimate. The result is based on a careful analysis of the switching function. We extend this result to nonautonomous linear systems, provided the time dependent system is not too far from the autonomous system obtained by taking the time to be $0$ in the dynamics.
    Using a linearization approach, we prove a bang-bang principle, valid in dimensions $2$ and $3$ for a class of nonlinear systems, affine and symmetric with respect to the control. Moreover we show that, for two dimensional systems, the reachable set from the origin satisfies the same polynomial strict convexity property as for the linearized dynamics, provided the nonlinearity is small enough. Finally, under the same assumptions we show that the epigraph of the minimum time function has positive reach, hence proving the first result of this type in a nonlinear setting. In all the above results, we require that the linearization at the origin be normal. We provide examples showing the sharpness of our assumptions.
Bounded real and positive real balanced truncation for infinite-dimensional systems
Chris Guiver and  Mark R. Opmeer
2013, 3(1): 83-119 doi: 10.3934/mcrf.2013.3.83 +[Abstract](37) +[PDF](658.6KB)
Abstract:
Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations.

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