On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem
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On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem Roberta Fabbri1 · Carmen Núñez2 Received: 28 June 2019 / Accepted: 6 December 2019 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract The Yakubovich Frequency Theorem, in its periodic version and in its general nonautonomous extension, establishes conditions which are equivalent to the global solvability of a minimization problem of infinite horizon type, given by the integral in the positive half-line of a quadratic functional subject to a control system. It also provides the unique minimizing pair “solution, control” and the value of the minimum. In this paper, we establish less restrictive conditions under which the problem is partially solvable, characterize the set of initial data for which the minimum exists, and obtain its value as well a minimizing pair. The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis. Keywords Infinite-horizon control problem · Frequency Theorem · Nonautonomous dynamical systems · Exponential dichotomy · Rotation number Mathematics Subject Classification 37B55 · 49N10 · 34F05
Partly supported by Ministerio de Economía y Competitividad/FEDER under project MTM2015-66330-P, by Ministerio de Ciencia, Innovación y Universidades under project RTI2018-096523-B-I00, by European Commission under project H2020-MSCA-ITN-2014, and by INDAM—GNAMPA Project 2018.
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Carmen Núñez [email protected] Roberta Fabbri [email protected]
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Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Università degli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy
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Departamento de Matemática Aplicada, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain
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R. Fabbri, C. Núñez
1 Introduction and main result Let us consider the control problem x = A0 (t) x + B0 (t) u
(1.1)
for x ∈ Rn and u ∈ Rm , the quadratic form (or supply rate) Q(t, x, u) :=
1 ( x, G 0 (t) x + 2 x, g0 (t) u + u, R0 (t) u ) , 2
(1.2)
and a point x0 ∈ Rn . We represent by Px0 the set of pairs (x, u) : [0, ∞) → Rn × Rm of measurable functions satisfying (1.1) with x(0) = x0 , and consider the problem of minimizing the quadratic functional ∞ Ix0 : Px0 → R ∪ {±∞}, (x, u) → Q(t, x(t), u(t)) dt. (1.3) 0
The functions A0 , B0 , G 0 , g0 , and R0 are assumed to be bounded and uniformly continuous on R, with values in the sets of real matrices of the appropriate dimensions; G 0 and R0 are symmetric, with R0 (t) ≥ ρ Im for a common ρ > 0 and all t ∈ R; and ·, · represents the Euclidean inner product in Rn or Rm . A pair (x, u) ∈ Px0 is admissible for Ix0 if (x, u) ∈ L 2 ([0, ∞), Rn ) × L 2 ([0, ∞, Rm ). That is, u : [0, ∞) → Rm belongs to L 2 ([0, ∞), Rm ), x : [0, ∞) → Rn solves (1.1) for this control with x(0) = x0 , and x belongs to L 2 ([0, ∞), Rn ). In particular, Ix0
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