Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $ y_0 \in H$ we associate the minimal “energy” needed to transfer $ y_0 $ to $ 0 $ in a time $ T \ge T_0$ (``energy” of a control being the square of its $ L^2 $ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $ 0 $ for $ T\to+\infty $. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk [SIAM J. Control Optim., 42 (2003), pp. 1013--1032] for distributed systems. The proofs in the Priola--Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability a...

[1]  Jim Hefferon,et al.  Linear Algebra , 2012 .

[2]  Marius Tucsnak,et al.  On the null-controllability of diffusion equations , 2011 .

[3]  Enrique Fernández-Cara,et al.  Boundary controllability of parabolic coupled equations , 2010 .

[4]  Luc Miller.  A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups , 2010 .

[5]  Xu Zhang,et al.  A unified controllability/observability theory for some stochastic and deterministic partial differential equations , 2010, 1003.5819.

[6]  G. Weiss,et al.  Observation and Control for Operator Semigroups , 2009 .

[7]  Kim Dang Phung,et al.  On the existence of time optimal controls for linear evolution equations , 2007 .

[8]  A. Ichikawa.  Null controllability with vanishing energy for discrete-time systems in Hilbert space , 2007, 2008 47th IEEE Conference on Decision and Control.

[9]  Luciano Pandolfi,et al.  A quadratic regulator problem related to identification problems and singular systems , 2006, 2006 14th Mediterranean Conference on Control and Automation.

[10]  W. Li,et al.  Controllability of Parabolic and Hyperbolic Equations: Toward a Unified Theory , 2005 .

[11]  Vilmos Komornik,et al.  Fourier Series in Control Theory , 2005 .

[12]  R. Nagel,et al.  Functional Analytic Methods for Evolution Equations , 2004 .

[13]  J. Zabczyk,et al.  Liouville theorems for non-local operators , 2004 .

[14]  J. M. A. M. van Neerven,et al.  Null Controllability and the Algebraic Riccati Equation in Banach Spaces , 2004, SIAM J. Control. Optim..

[15]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[16]  Luciano Pandolfi,et al.  The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems , 1999 .

[17]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[18]  Luciano Pandolfi,et al.  Dissipativity and the Lur'e Problem for Parabolic Boundary Control Systems , 1998 .

[19]  P. Halmos.  Introduction to Hilbert Space: And the Theory of Spectral Multiplicity , 1998 .

[20]  L. Pandolfi.  The Kalman-Popov-Yakubovich Theorem: an overview and new results for hyperbolic control systems , 1997 .

[21]  Sergei Avdonin,et al.  Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems , 1995 .

[22]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .

[23]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[24]  Jerzy Zabczyk,et al.  Mathematical control theory - an introduction , 1992, Systems & Control: Foundations & Applications.

[25]  Roberto Triggiani,et al.  Exact boundary controllability onL2(Ω) ×H−1(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundary∂Ω, and related problems , 1988 .

[26]  R. Triggiani,et al.  Regularity of hyperbolic equations underL2(0,T; L2(Γ))-Dirichlet boundary terms , 1983 .

[27]  P. Fuhrmann.  Exact controllability and observability and realization theory in Hilbert space , 1976 .

[28]  Paul A. Fuhrmann,et al.  On weak and strong reachability and controllability of infinite-dimensional linear systems , 1972 .

[29]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[30]  Jianhong Wu,et al.  Introduction to Functional Differential Equations , 2013 .

[31]  Enrique Zuazua,et al.  Controllability and Observability of Partial Differential Equations: Some Results and Open Problems , 2007 .

[32]  Irena Lasiecka,et al.  Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems , 2004 .

[33]  J. Zabczyk,et al.  Null Controllability with Vanishing Energy , 2003, SIAM J. Control. Optim..

[34]  I. Lasiecka,et al.  Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation , 2003 .

[35]  S. Ivanov.  CONTROL NORMS FOR LARGE CONTROL TIMES , 1999 .

[36]  L. Pandolfi.  A frequency domain approach to the boundary control problem for parabolic equations , 1996 .

[37]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[38]  Luciano Pandolfi,et al.  Null controllability of a class of functional differential systems , 1988 .

[39]  J. Lions.  Controlabilite exacte, perturbations et stabilisation de systemes distribues , 1988 .

[40]  Roberto Triggiani,et al.  Exact boundary controllability on L2(Ω)×H−1(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary ∫Ω, and related problems , 1987 .

[41]  S. Rolewicz.  On universal time for the controllability of time-depending linear control systems , 1976 .

[42]  D. L. Russell,et al.  Exact controllability theorems for linear parabolic equations in one space dimension , 1971 .

[43]  M. Hasse Dresden,et al.  W. H. Greub, Linear Algebra. (Die Grundlehren der mathematischen Wissenschaften, Band 97). XVI + 434 S. Berlin/Heidelberg/New York 1967. Springer‐Verlag. Preis geb. DM 39,20 , 1969 .

[44]  S. Goldberg.  Unbounded linear operators : theory and applications , 1966 .

[45]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[46]  Laurent Schwartz,et al.  Étude des sommes d'exponentielles , 1959 .