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.
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.
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.
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 nonlocal Cahn-Hilliard/Navier-Stokes system in 2D
2014, Frigeri, Sergio Pietro, Rocca, Elisabetta, Sprekels, Jürgen
We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier-Stokes system with a convective nonlocal Cahn-Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.
Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth
2015, Colli, Pierluigi, Gilardi, Gianni, Rocca, Elisabetta, Sprekels, Jürgen
This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in gianni terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers citeCGH and citeCGRS gianni from the viewpoint of well-posedness, long time bhv and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in citeCGRS by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates
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.
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.
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.