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- ItemAnalysis and optimal boundary control of a nonstandard system of phase field equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gillardi, Gianni; Sprekels, JürgenWe investigate a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced in [16], describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been studied recently in [5], [6] for the case of homogeneous Neumann boundary conditions. In this paper, we investigate the case that the boundary condition for one of the unknowns of the system is of third kind and nonhomogeneous. For the resulting system, we show well-posedness, and we study optimal boundary control problems. Existence of optimal controls is shown, and the first-order necessary optimality conditions are derived. Owing to the strong nonlinear couplings in the PDE system, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional will be of standard type.
- ItemDistributed optimal control of a nonstandard nonlocal phase field system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenWe 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.
- ItemOptimal control of a phase field system of Caginalp type with fractional operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn their recent work ``Well-posedness, regularity and asymptotic analyses for a fractional phase field system'' (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables.
- ItemOptimal distributed control of a diffuse interface model of tumor growth(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Rocca, Elisabetta; Sprekels, JürgenIn 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.
- ItemDistributed optimal control of a nonstandard system of phase field equations : dedicated to Prof. Dr. Ingo Müller on the occasion of his 75th birthday(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, Jürgen; Müller, IngoWe 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.
- ItemAnalysis and optimal control theory for a phase field model of Caginalp type with thermal memory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Colli, Pierluigi; Signori, Andrea; Sprekels, JürgenA nonlinear extension of the Caginalp phase field system is considered that takes thermal memory into account. The resulting model, which is a first-order approximation of a thermodynamically consistent system, is inspired by the theories developed by Green and Naghdi. Two equations, resulting from phase dynamics and the universal balance law for internal energy, are written in terms of the phase variable (representing a non-conserved order parameter) and the so-called thermal displacement, i.e., a primitive with respect to time of temperature. Existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem. Then, an optimal control problem is investigated for a suitable cost functional, in which two data act as controls, namely, the distributed heat source and the initial temperature. Fréchet differentiability between suitable Banach spaces is shown for the control-to-state operator, and meaningful first-order necessary optimality conditions are derived in terms of variational inequalities involving the adjoint variables. Eventually, characterizations of the optimal controls are given.
- ItemOptimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn 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.