scholarly journals Distributed optimal control problems for phase field systems with singular potential

2018 ◽  
Vol 26 (2) ◽  
pp. 71-85
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Gabriela Marinoschi ◽  
Elisabetta Rocca
Keyword(s):  
Control Problem ◽  
Phase Field ◽  
Singular Potential ◽  
Necessary Optimality Conditions ◽  
Logarithmic Potential ◽  
Diffuse Interface ◽  
Distributed Optimal Control ◽  
Distributed Optimal Control Problems ◽  
Field Systems ◽  
The Cost

Abstract In this paper we review some results obtained for a distributed con- trol problem regarding a class of phase field systems of Caginalp type with logarithmic potential. The aim of the control problem is forcing the location of the diffuse interface to be as close as possible to a pre- scribed set. However, due to some discontinuity in the cost functional, we have to regularize it and solve the related control problem for the approximation. We discuss the necessary optimality conditions.

scholarly journals Optimal control of time-discrete two-phase flow driven by a diffuse-interface model

2019 ◽  
Vol 25 ◽  
pp. 13 ◽  
Author(s):  
Harald Garcke ◽  
Michael Hinze ◽  
Christian Kahle
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Time Discretization ◽  
Diffuse Interface ◽  
Interface Model ◽  
Diffuse Interface Model ◽  
Two Phase ◽  
Fully Discrete

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in Abelset al.[M3AS22(2012) 1150013]. On the basis of the stable time discretization proposed in Garckeet al.[Appl. Numer. Math.99(2016) 151] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

scholarly journals Strong well-posedness and inverse identification problem of a non-local phase field tumour model with degenerate mobilities

2021 ◽  
pp. 1-42
Author(s):  
SERGIO FRIGERI ◽  
KEI FONG LAM ◽  
ANDREA SIGNORI
Keyword(s):  
Inverse Problem ◽  
Phase Field ◽  
Necessary Optimality Conditions ◽  
Strong Solutions ◽  
Reaction Diffusion Equation ◽  
Diffuse Interface ◽  
Well Posedness ◽  
Tumour Model ◽  
Degenerate Mobility ◽  
Non Local

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

scholarly journals Solution stability to parametric distributed optimal control problems with finite unilateral constraints

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nguyen Hai Son
Keyword(s):  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Solution Stability ◽  
Unilateral Constraints ◽  
Solution Map ◽  
Distributed Optimal Control ◽  
First Order ◽  
Stability Of Solution ◽  
Distributed Optimal Control Problems ◽  
Objective Functional

<p style='text-indent:20px;'>This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations with finite unilateral constraints, where the objective functional is not convex. By using the first-order necessary optimality conditions, we derive some sufficient conditions under which the solution map is upper semicontinuous with respect to parameters.</p>

scholarly journals Optimal control for the coupled chemotaxis-fluid models in two space dimensions

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yunfei Yuan ◽  
Changchun Liu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Lagrange Multipliers ◽  
Existence And Uniqueness ◽  
Fluid Models ◽  
Distributed Optimal Control ◽  
Two Space Dimensions ◽  
The Cost ◽  
Time Existence

<p style='text-indent:20px;'>This paper deals with a distributed optimal control problem to the coupled chemotaxis-fluid models. We first explore the global-in-time existence and uniqueness of a strong solution. Then, we define the cost functional and establish the existence of Lagrange multipliers. Finally, we derive some extra regularity for the Lagrange multiplier.</p>

scholarly journals An Optimal Control Problem by a Hybrid System of Hyperbolic and Ordinary Differential Equations

Games ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 23
Author(s):  
Alexander Arguchintsev ◽  
Vasilisa Poplevko
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Ordinary Differential Equations ◽  
Hyperbolic Equations ◽  
Control Methods ◽  
The Right ◽  
The Cost ◽  
Optimal Control Methods

This paper deals with an optimal control problem for a linear system of first-order hyperbolic equations with a function on the right-hand side determined from controlled bilinear ordinary differential equations. These ordinary differential equations are linear with respect to state functions with controlled coefficients. Such problems arise in the simulation of some processes of chemical technology and population dynamics. Normally, general optimal control methods are used for these problems because of bilinear ordinary differential equations. In this paper, the problem is reduced to an optimal control problem for a system of ordinary differential equations. The reduction is based on non-classic exact increment formulas for the cost-functional. This treatment allows to use a number of efficient optimal control methods for the problem. An example illustrates the approach.

Pontryagin maximum principle and second order optimality conditions for optimal control problems governed by 2D nonlocal Cahn–Hilliard–Navier–Stokes equations

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 127-150
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stokes Equations ◽  
Minimum Principle ◽  
Immiscible Fluids ◽  
Second Order ◽  
Navier Stokes ◽  
Distributed Optimal Control ◽  
Navier Stokes Equations

AbstractIn this paper, we formulate a distributed optimal control problem related to the evolution of two isothermal, incompressible, immiscible fluids in a two-dimensional bounded domain. The distributed optimal control problem is framed as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn–Hilliard–Navier–Stokes equations. We describe the first order necessary conditions of optimality via the Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.

scholarly journals Distributed optimal control of a nonstandard nonlocal phase field system

2016 ◽  
Vol 1 (3) ◽  
pp. 225-260 ◽  
Author(s):  
Pierluigi Colli ◽  
◽  
Gianni Gilardi ◽  
Jürgen Sprekels
Keyword(s):  
Optimal Control ◽  
Phase Field ◽  
Distributed Optimal Control ◽  
Field System ◽  
Phase Field System
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