Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability

Johnson, Russell; Novo, Sylvia; Núñez, Carmen; Obaya, Rafael

doi:10.1007/s10884-015-9495-1

Cited by 11 publications

(11 citation statements)

References 28 publications

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“…The setting that we consider throughout the paper, that is, the existence of systems of the family (1.2) which are not null controllable, is closely related to the existence of abnormal systems of the family. It is proved in Johnson et al [8] that there are minimal subsets Ω * ⊆ Ω for which all the systems (1.6) are abnormal (i.e., they have solutions of the form 0 z2(t) for t ∈ R), and such that at least one of the associated Lagrange planes l ± (ω) lies on the vertical Maslov cycle C, defined in Subsection 2.2.1, for all ω ∈ Ω * . A more precise description of this connection will also be included at the beginning of Section 4.…”

mentioning

confidence: 78%

“…This concept will be used in Section 4, in which will just work with families of Hamiltonian systems z = H1 H3 H2 −H T 1 z for which H 3 takes positive semidefinite values (H 3 ≥ 0). This fact allows us to choose, among the many equivalent definition of the rotation number (see Chapter 2 of [10], which contains and extend the previous results of [16], [3] and [8]), that based on the characteristics of the so-called proper focal points, which is valid just for those Hamiltonian systems with H 3 ≥ 0. This definition involves several concepts and properties which will be useful later.…”

Section: 22mentioning

confidence: 95%

“…for t ∈ R , and define d(ω) = dim Λ(ω) and d ± (ω) = dim Λ ± (ω). A complete analysis of the functions d, d + and d − is carried out in [8]. Proof.…”

Section: Null Controllability and Exponential Dichotomymentioning

confidence: 99%

“…The interested reader can find in Chapter 7 of Johnson et al [10] an exhaustive description of these results. Similar conditions are used in the description of the dissipativity of the LQ problems in the papers [26,5,9,8] and in Chapter 8 of [10].…”

mentioning

confidence: 99%

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Null controllable sets and reachable sets for nonautonomous linear control systems

Fabbri¹,

Novo²,

Núñez³

et al. 2016

Discrete &Amp; Continuous Dynamical Systems - S

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mentioning

confidence: 78%

Section: 22mentioning

confidence: 95%

“…for t ∈ R , and define d(ω) = dim Λ(ω) and d ± (ω) = dim Λ ± (ω). A complete analysis of the functions d, d + and d − is carried out in [8]. Proof.…”

Section: Null Controllability and Exponential Dichotomymentioning

confidence: 99%

mentioning

confidence: 99%

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Null controllable sets and reachable sets for nonautonomous linear control systems

Fabbri¹,

Novo²,

Núñez³

et al. 2016

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

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“…Note that, in situation O2 * , each one of the families (3.8) λ has an abnormal system both at +∞ and at −∞. The reader is referred to [36,37,46,47,48,10,24] and references therein for an analysis of abnormal linear Hamiltonian systems.…”

Section: Global Existence Of Weyl Functionsmentioning

confidence: 99%

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

Núñez

Obaya

2018

J Dyn Diff Equat

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We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions M ± for one-parametric families of finitedimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of M + , then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of M + are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.Partly supported by MINECO/FEDER (Spain) under project MTM2015-66330-P and by European Commission under project H2020-MSCA- ITN-2014. where x ∈ R n and u ∈ R m , together with the quadratic form (supply rate)The functions A 0 , B 0 , G 0 , g 0 , and R 0 are assumed to be bounded and uniformly continuous functions on R, with values in the sets of real matrices of the appropriate dimensions. In addition, G and R are symmetric, and R(t) ≥ ρI m for a common ρ > 0 and all t ∈ R. We also fix x 0 ∈ R n and introduce the quadratic functionalthose for which u belongs to L 2 ((0, ∞), R m ) and the solution x(t) of (1.1) for this control with x(0) = x 0 belongs to L 2 ((0, ∞), R n ). The problem to consider is that of minimizing J x0 relative to the set of admissible pairs. By means of a standard construction (the so called hull or Bebutov construction, which we will summarize in Section 2), this problem can be included in a family, given by the control problemsand by the functionalsfor ω ∈ Ω and x 0 ∈ R n . Here, Ω is a compact metric space admitting a continuous flow, ω·t is the orbit of a point ω ∈ Ω, A, B, G, g, and R are bounded and uniformly continuous matrix-valued functions on Ω, G and R are symmetric, and R > 0. It is important to point out that Ω is minimal in the case of recurrence of the initial coefficients, which includes the autonomous, periodic, quasi-periodic, almost-periodic and almost-automorphic cases. The Pontryagin Maximum Principle relates the problem of minimizing J x0,ω to the properties of the family of linear Hamiltonian systemswhere z = [ x y ] for x, y ∈

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Motivation and Preliminaries

Došlý

Elyseeva

Hilscher

2019

Pathways in Mathematics

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Nonautonomous Linear-Quadratic Dissipative Control Processes Without Uniform Null Controllability

Cited by 11 publications

References 28 publications

Null controllable sets and reachable sets for nonautonomous linear control systems

Null controllable sets and reachable sets for nonautonomous linear control systems

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

Motivation and Preliminaries

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