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

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 212-220
Author(s):  
Kenan Yildirim ◽  
Sertan Alkan
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Finite Size ◽  
Beam Equation ◽  
Quadratic Functional ◽  
Control Voltage ◽  
Well Posedness ◽  
Penalty Term

AbstractIn this paper, well-posedness, controllability and optimal control for a time-delay beam equation are studied. The equation of motion is modeled as a time-delayed 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.

scholarly journals A nonlinear plate control without linearization

2017 ◽  
Vol 15 (1) ◽  
pp. 179-186
Author(s):  
Kenan Yildirim ◽  
Ismail Kucuk
Keyword(s):  
Optimal Control ◽  
Nonlinear Systems ◽  
Control Problem ◽  
Performance Index ◽  
Control Function ◽  
Initial Boundary ◽  
Quadratic Functional ◽  
Penalty Term ◽  
Nonlinear Plate ◽  
Optimal Control Function

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.

scholarly journals Stochastic recursive optimal control problem with time delay and applications

2015 ◽  
Vol 5 (4) ◽  
pp. 859-888 ◽  
Author(s):  
Jingtao Shi ◽  
◽  
Juanjuan Xu ◽  
Huanshui Zhang ◽  
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problem ◽  
Control Problem

scholarly journals OPTIMAL CONTROL OF SWITCHED IMPULSIVE SYSTEMS WITH TIME DELAY

2012 ◽  
Vol 53 (4) ◽  
pp. 292-307 ◽  
Author(s):  
K. H. WONG ◽  
W. M. TANG
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Time Instant ◽  
Computational Method ◽  
Impulsive Systems ◽  
Delay Differential ◽  
The Cost ◽  
System With Time Delay

AbstractWe develop a computational method for solving an optimal control problem governed by a switched impulsive dynamical system with time delay. At each time instant, only one subsystem is active. We propose a computational method for solving this optimal control problem where the time spent by the state in each subsystem is treated as a new parameter. These parameters and the jump strengths of the impulses are decision parameters to be optimized. The gradient formula of the cost function is derived in terms of solving a number of delay differential equations forward in time. Based on this, the optimal control problem can be solved as an optimization problem.

scholarly journals Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis

2021 ◽  
Author(s):  
Pierluigi Colli ◽  
Andrea Signori ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Tumor Growth ◽  
Control Problem ◽  
Growth Model ◽  
Second Order ◽  
Well Posedness ◽  
Singular Potentials ◽  
Order Analysis ◽  
Tumor Growth Model

This paper concerns a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type including chemotaxis with possibly singular potentials, where the control and state variables are nonlinearly coupled. First, we discuss the weak well-posedness of the system under very general assumptions for the potentials, which may be singular and nonsmooth. Then, we establish the strong well-posedness of the system in a reduced setting, which however admits the logarithmic potential: this analysis will lay the foundation for the study of the corresponding optimal control problem. Concerning the optimization problem, we address the existence of minimizers and establish both first-order necessary and second-order sufficient conditions for optimality. The mathematically challenging second-order analysis is completely performed here, after showing that the solution mapping is twice continuously differentiable between suitable Banach spaces via the implicit function theorem. Then, we completely identify the second-order Fr ́echet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

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