Periodic solutions of linear systems coupled with relay

Bliman, Pierre‐Alexandre; Krasnoselʹskiǐ, A. M.

doi:10.1016/s0362-546x(96)00372-0

Cited by 15 publications

(22 citation statements)

References 4 publications

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“…Finally, in section 5, we consider an example in which the boundary condition is of the Neumann type and the "uniform" distribution of thermal sensors inside the domain is assumed. We show that the "mean" temperature satisfies an ordinary differential equation, which simplifies the situation (there is vast literature devoted to ordinary differential equations with hysteresis operators; see, e.g., [2,5,15,18] and others). In this case, we prove the existence of a mean-periodic solution (hence, a strong periodic solution) for the problem in question.…”

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confidence: 85%

On Periodicity of Solutions For Thermocontrol Problems with Hysteresis-Type Switches

Gurevich¹,

Jäger²,

Скубачевский³

2009

SIAM J. Math. Anal.

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Mathematical models of thermocontrol processes occurring in chemical reactors and climate control systems are considered. In the models under consideration, the temperature inside a domain is controlled by a thermostat acting on the boundary. The feedback is based on temperature measurements performed by thermal sensors inside the domain. The solvability of the corresponding nonlinear nonlocal problems and the periodicity of solutions are studied. Introduction.We consider mathematical models of thermocontrol processes in which the temperature inside a domain is controlled by a thermostat acting on the boundary. The feedback is based on temperature measurements performed by thermal sensors inside the domain. The processes under consideration occur in chemical reactors and climate control systems.The temperature distribution in the domain obeys the heat equation, while the boundary condition involves a control function (the Dirichlet, the Neumann, and the Robin boundary conditions are considered). The control function satisfies an ordinary differential equation whose right-hand side is a nonlinear operator H depending on the "mean" temperature over the domain and taking the values 0 or 1. There are two temperature thresholds w 1 and w 2 (w 1 < w 2 ). If the "mean" temperature is less than or equal to w 1 , then H = 1 (the heating is switched on); if the "mean" temperature is greater than or equal to w 2 , then H = 0 (the heating is switched off); if the "mean" temperature is greater than w 1 and less than w 2 , then H takes the same value as "just before." Thus, the presence of the operator H provides the so-called hysteresis phenomenon, while the thermal sensors inside the domain cause nonlocal effects.Thermocontrol models similar to ours were originally proposed in [8,9]. By transforming the problem into an equivalent set-valued integro-differential equation, the existence of a solution was proved. The existence and uniqueness of solutions for two-phase Stefan problems with the Robin boundary condition involving a hysteresis control were studied in [4,6,12]. Some questions related to optimal control for heat conduction problems with hysteresis were considered, e.g., in [3].The question whether periodic solutions exist turns out to be much more difficult. In [7], a one-dimensional thermocontrol problem is considered under the assumption that the temperature of the thermostat changes by jumping. Thus, there is no coupling with an ordinary differential equation in this case. The existence of a periodic solution is proved. Its uniqueness in a class of the so-called two-phase periodic solu-

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confidence: 85%

On Periodicity of Solutions For Thermocontrol Problems with Hysteresis-Type Switches

Gurevich¹,

Jäger²,

Скубачевский³

2009

SIAM J. Math. Anal.

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“…It adds a non-invasive control containing a delay term to an equation. For example, if an equation [4,17,19]), then an equation of the form 2T )), t > 0, (1.3) possesses the same periodic solution u per (t), however its stability properties can change. Note that equation (1.3) is of the same form as (1.1).…”

Section: Introductionmentioning

confidence: 99%

“…Periodic solutions naturally arise for ordinary differential equations with a hysteresis of non-ideal relay type, and were studied e.g. in [4,34]. Periodic solutions to parabolic differential equations with discontinuous hysteresis were studied mostly in the case of the thermal control problem, which was suggested in [14,15].…”

Section: Introductionmentioning

confidence: 99%

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

Gurevich

Ron²

2018

J Dyn Diff Equat

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We study an interplay between delay and discontinuous hysteresis in dynamical systems. After having established existence and uniqueness of solutions, we focus on the analysis of stability of periodic solutions. The main object we study is a Poincaré map that is infinite-dimensional due to delay and non-differentiable due to hysteresis. We propose a general functional framework based on the fractional order Sobolev-Slobodeckij spaces and explicitly obtain a formal linearization of the Poincaré map in these spaces. Furthermore, we prove that the spectrum of this formal linearization determines the stability of the periodic solution and then reduce the spectral analysis to an equivalent finite-dimensional problem.

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“…Switching systems described by ordinary differential equations with hysteresis were considered by many authors (see, e.g., [1,3,14,17,20]). …”

Section: Introductionmentioning

confidence: 99%

“…Then, due to (4.20) and (6.15), we have w(·, t) W 1 [1,2]. (8.6) Let t ∈ [2,3]. Then (w(·, t), u(t)) coincides with the solution of the thermocontrol problem, at the moment t − 1 ∈ [1,2], with the initial data…”

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confidence: 99%

Parabolic problems with the Preisach hysteresis operator in boundary conditions

Gurevich

Jäger

2009

Journal of Differential Equations

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MSC: 35K15 47J40 45M15 35B41 Keywords: Thermocontrol problem Hysteresis Preisach operator Periodicity Global attractorParabolic initial boundary-value problems coupled (via the boundary condition) with ordinary differential equations whose righthand side contains the Preisach hysteresis operator are considered. In particular, these problems model thermocontrol processes in chemical reactors, climate-control systems, biological cells, etc. For the Preisach operator with and without time delay, solvability, periodicity of solutions, and global B-attractors are studied.

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Periodic solutions of linear systems coupled with relay

Cited by 15 publications

References 4 publications

On Periodicity of Solutions For Thermocontrol Problems with Hysteresis-Type Switches

On Periodicity of Solutions For Thermocontrol Problems with Hysteresis-Type Switches

Stability of Periodic Solutions for Hysteresis-Delay Differential Equations

Parabolic problems with the Preisach hysteresis operator in boundary conditions

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