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Equivalent one-dimensional first-order linear hyperbolic systems and range of the minimal null control time with respect to the internal coupling matrix
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Abstract
In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical LU-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.
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This paper is devoted to study exact controllability of two one-dimensional coupled wave equations with first-order coupling terms with coefficients depending on space and time. We give a necessary and sufficient condition for both exact controllability in high frequency in the general case and the unique continuation in the cascade case.
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.
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In this article we study the controllability properties of general compactly perturbed exactly controlled linear systems with admissible control operators. Firstly, we show that approximate and exact controllability are equivalent properties for such systems. Then, and more importantly, we provide for the perturbed system a complete characterization of the set of reachable states in terms of the Fattorini-Hautus test. The results rely on the Peetre lemma.
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In this work, we consider the problem of boundary stabilization for a
quasilinear 2X2 system of first-order hyperbolic PDEs. We design a new
full-state feedback control law, with actuation on only one end of the domain,
which achieves H^2 exponential stability of the closedloop system. Our proof
uses a backstepping transformation to find new variables for which a strict
Lyapunov function can be constructed. The kernels of the transformation are
found to verify a Goursat-type 4X4 system of first-order hyperbolic PDEs, whose
well-posedness is shown using the method of characteristics and successive
approximations. Once the kernels are computed, the stabilizing feedback law can
be explicitly constructed from them.
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Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.
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n\times n inhomogeneous quasilinear hyperbolic systems. A backstepping method
is developed. The main result supplements the previous works on how to design
multi-boundary feedback controllers to realize exponential stability of the
original nonlinear system in the spatial H^2 sense.
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Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems.
In a recent survey paper D. L. Russell reports (among other things) on controllability results for symmetric hyperbolic systems is one space dimension. Russell explicitly gives two numbers T//0 and T//1 such that the system in question is exactly controllable in times T greater than equivalent to T//1 but not even approximately controllable in times T less than T//0. It is remarked that there must exist some T//c such that the same results hold if both T//0 and T//1 are replaced by T//c. In the present paper it is shown what this ″critical time″ T//c is.
In this paper, the authors define the strong (weak) exact boundary controllability and the strong (weak) exact boundary observability
for first order quasilinear hyperbolic systems, and study their properties and the relationship between them.
KeywordsStrong (weak) exact boundary controllability-Strong (weak) exact boundary observability-First order quasilinear hyperbolic system
2000 MR Subject Classification35B37-93B05-93B07
Null-controllability of linear hyperbolic systems in one dimensional space
, Null-controllability of linear hyperbolic systems in one dimensional space,
Systems Control Lett. 148 (2021), 104851.
- Jean-Michel Coron
Jean-Michel Coron, Control and nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007.
Minimal time for the exact controllability of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls
, Minimal time for the exact controllability of one-dimensional first-order
linear hyperbolic systems by one-sided boundary controls, J. Math. Pures Appl. (9)
148 (2021), 24-74.
- Harry Hochstadt
Harry Hochstadt, Integral equations, John Wiley & Sons, New York-London-Sydney, 1973, Pure and Applied Mathematics.
Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions
, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), no. 4, 639-739.