Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One-Space Dimension

Küçük, İsmail; Yildirim, Kenan

doi:10.1155/2014/493130

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…Let .x 1 ; y 1 ; t 1 /; :::; .x p ; y p ; t p / be P arbitrary points in the open region Q and " j are the coefficients such that the regions, R j D OEx j ; x j C p " j OEy j ; y j C p " j OEt j ; t j C p " j do not have any intersection for 1 Ä j Ä p. Let us define the following energy integral like in [7,25]:…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

A nonlinear plate control without linearization

Yildirim

Küçük

2017

Open Mathematics

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Abstract:In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as a penalty term. By using a maximum principle, the nonlinear control problem is transformed to solving a system of partial differential equations including state and adjoint variables linked by initial-boundary-terminal conditions. Hence, it is shown that optimal control of the nonlinear systems can be obtained without linearization of the nonlinear term and optimal control function can be obtained analytically for nonlinear systems without linearization.

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Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

A nonlinear plate control without linearization

Yildirim

Küçük

2017

Open Mathematics

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“…Since time delay leads to a decrement in the performance of the actuator, the state variable can-not reflect the changes in the system [18]. Active vibration control of mechanical systems, which are modeled as DPS without time delays, has been excessively studied in the literature by several authors, such as, but not limited to [3,7,[12][13][14][15][16]; however, the optimal control of DPS with time delay has not received considerable attention yet. Some studies of the control of DPS with time delay can be summarized by [4-6, 11, 18].…”

Section: Introductionmentioning

confidence: 99%

A time-delay equation: well-posedness to optimal control

Yildirim

Alkan

2016

Open Physics

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Abstract:In this paper, well-posedness, controllability and optimal control for a time-delay beam equation are studied. The equation of motion is modeled as a timedelayed distributed parameter system(DPS) and includes Heaviside functions and their spatial derivatives due to the finite size of piezoelectric patch actuators used to suppress the excessive vibrations based on displacement and moment conditions. The optimal control problem is defined with the performance index including a weighted quadratic functional of the displacement and velocity which is to be minimized at a given terminal time and a penalty term defined as the control voltage used in the control duration. Optimal control law is obtained by using Maximum principle and hence, the optimal control problem is transformed the into a boundary-, initial and terminal value problem.The explicit solution of the control problem is obtained by eigenfunction expansions of the state and adjoint variables. Numerical results are presented to show the effectiveness and applicability of the piezoelectric control.

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“…Later, the controllability of the system is discussed. Also, optimal control function for the beam equation system in one space dimension is obtained by employing Maximum principle(for more details about maximum principle [5,6,7,8,9]). Performance index functional of the control problem consists of a weighted quadratic functional of the dynamic responses of the system to be minimized and a penalty term defined as the control spent in the control process.…”

Section: Introductionmentioning

confidence: 99%

Control of an equation by maximum principle

Yildirim¹,

Kutlu²

2016

ntmsci

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Abstract:In this paper, some results, which are related to well posedness, controllability and optimal control of a beam equation, are presented. In order to obtain the optimal control function, maximum principle is employed. Performance index function is defined as quadratic functional of displacement and velocity and also includes a penalty in terms of control function. The solution of the control problem is formulated by using Galerkin expansion. Obtained results are given in the table and graphical forms.

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Necessary and Sufficient Conditions of Optimality for a Damped Hyperbolic Equation in One-Space Dimension

Cited by 5 publications

References 29 publications

A nonlinear plate control without linearization

A nonlinear plate control without linearization

A time-delay equation: well-posedness to optimal control

Control of an equation by maximum principle

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