Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A hierarchic multi-level energy method for the control of bi-diagonal
  and mixed n-coupled cascade systems of PDE's by a reduced number of controls   This work is concerned with the exact controllability/observability of
abstract cascade hyperbolic systems by a reduced number of
controls/observations. We prove that the observation of the last component of
the vector state allows to recover the initial energies of all of its
components in suitable functional spaces under a necessary and sufficient
condition on the coupling operators for cascade bi-diagonal systems. The
approach is based on a multi-level energy method which involves $n$-levels of
weakened energies.
  We establish this result for the case of bounded as well as unbounded dual
control operators and under the hypotheses of partial coercivity of the $n-1$
coupling operators on the sub-diagonal of the system.
  We further extend our observability result to mixed bi-diagonal and non
bi-diagonal $n+p$-coupled cascade systems by $p+1$ observations. Applying the
HUM method, we derive the corresponding exact controllability results for
$n$-coupled bi-diagonal cascade and $n+p$-coupled mixed cascade systems. Using
the transmutation method for the wave operator, we prove that the corresponding
heat (resp. Schr\"odinger) multi-dimensional cascade systems are
null-controllable for control regions and coupling regions which are disjoint
from each other and for any positive time for $n \le 5$ for dimensions larger
than $2$, and for any $n \ge 2$ in the one-dimensional case. The controls can
be localized on a subdomain or on the boundary and in the one-dimensional case
the coupling coefficients can be supported in any non-empty subset of the
domain.

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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat