Search Results
A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions
2015, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
A boundary control problem for the pure Cahn–Hilliard equations with possibly singular potentialsand dynamic boundary conditions is studied and rst-order necessary conditions for optimality are proved.
Distributed optimal control of a nonstandard system of phase field equations : dedicated to Prof. Dr. Ingo Müller on the occasion of his 75th birthday
2011, Colli, Pierluigi, Gilardi, Gianni, Podio-Guidugli, Paolo, Sprekels, Jürgen, Müller, Ingo
We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been introduced recently in [4], on the basis of the theory developed in [15], and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth
2015, Colli, Pierluigi, Gilardi, Gianni, Rocca, Elisabetta, Sprekels, Jürgen
In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.
Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of CahnHilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called deep quench approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.
Distributed optimal control of a nonstandard nonlocal phase field system
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
Optimal distributed control of a diffuse interface model of tumor growth
2016, Colli, Pierluigi, Gilardi, Gianni, Rocca, Elisabetta, Sprekels, Jürgen
In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by HawkinsDaruud et al. in [25]. The model consists of a CahnHilliard equation for the tumor cell fraction 'coupled to a reaction-diffusion equation for a function phi representing the nutrientrich extracellular water volume fraction. The distributed control u monitors as a right-hand side the equation for sigma and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.
Regularity of the solution to a nonstandard system of phase field equations
2013, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
A nonstandard systems of differential equations describing two-species phase segregation is considered. This system naturally arises in the asymptotic analysis recently done by Colli, Gilardi, Krej¡cí, and Sprekels as the diffusion coefficient in the equation governing the evolution of the order parameter tends to zero. In particular, a well-posedness result is proved for the limit system. This paper deals with the above limit problem in a less general but still very significant framework and provides a very simple proof of further regularity for the solution. As a byproduct, a simple uniqueness proof is given as well.
Constrained evolution for a quasilinear parabolic equation
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
In the present contribution, a feedback control law is studied for a quasilinear parabolic equation. First, we prove the well-posedness and some regularity results for the CauchyNeumann problem for this equation, modified by adding an extra term which is a multiple of the subdifferential of the distance function from a closed convex set K of L2 (Omega). Then, we consider convex sets of obstacle or double-obstacle type, and we can act on the factor of the feedback control in order to be able to reach the convex set within a finite time, by proving rigorously this property.
Distributed optimal control of a nonstandard nonlocal phase field system
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
Optimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition
2016, Colli, Pierluigi, Gilardi, Gianni, Sprekels, Jürgen
In this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fréchet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.