Controllability cost of conservative systems: resolvent condition and transmutation

By Luc Miller. 2004

142 citations recorded by the source corpus.

Abstract This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second-order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like exp( αL 2 / T ) as T tends to 0. In the application to the cost of fast controls for the Schrodinger equation, L is the length of the longest ray of geometric optics which does not intersect the control region. This article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost, and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/422c9631f1c0d895765653b5fea127fda1a7ba4e

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Citations in the corpus

Resolvent conditions for the control of unitary groups and their approximations (2012, methodology): “We refer to [22] for the history of this latter issue since [40] and the extension of the results on unitary groups in [30] to more general groups.”
On the observability of time-discrete conservative linear systems (2008, background): “It was proved in [2] and [17] that system (1.”
The Wave Equation: Control and Numerics (2012, methodology): “To overcome this drawback, more recently in [36], we have dev eloped a discrete transmutation technique, inspired on previous works , in particular by Miller [80, 79], which establishes a connection between solutions of the time continuous systems and the time-discrete ones.”
Sharp polynomial decay rates for the damped wave equation on the torus (2012, methodology): “In particular [Miller 2005, Theorem 5.1] (or [Tucsnak and Weiss 2009, Theorem 6.6.1]) yields that system (2-1) is observable in some time T > 0 if and only if there exists a constant C > 0 such that we have ‖u‖2H ≤ C ( ‖(A− λ Id)u‖2H +‖B ∗u‖2Y ) for all λ ∈ R and u ∈ D(A).”
New blow-up rates for fast controls of Schrödinger and heat equations (2007, methodology): “This result has been extended and made more precise in [15] and [ 17 ], where it is shown that, for the constant coefficients Schrodinger and heat equations on the interval [0 ,a ] ,w e have”
Uniformly exponentially stable approximations for a class of damped systems (2009, background): “4) is the so-called resolvent estimate, which has been proved in [4,22] to be equivalent to the continuous observability inequality (1.”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, methodology): “This approach, the so-called transmutation method according to the terminology in [149], plays the role in the physical space of that used by Russell [173] which consists in performing a change of variable in the frequency domain.”
Non-uniform stability for bounded semi-groups on Banach spaces (2008, background): “Such spectral criteria have also proved efficient in control theory [29, 7, 21, 27].”
Indirect controllability of locally coupled wave-type systems and applications (2013, methodology): “Ervedoza for having pointed out the papers [23,24], and L.”