arXiv subjects

Jürgen Sprekels

Publications and source records attributed to Jürgen Sprekels.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic analyis of an optimal control problem for a phase field tumor growth model with hyperbolic relaxation of the chemical potential

In this paper, we study the optimal control of a phase field tumor growth model of Cahn--Hilliard type with possibly singular potential in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We investigate the asymptotic behavior of the control problem as the relaxation parameter $\,α\,$ approaches zero. It is shown that the corresponding state and adjoint state variables of the relaxed problem converge in a suitable sense to their counterparts for the unrelaxed regime where $α=0$. Also, weak limits of optimal controls for the relaxed problem can be shown to be optimal controls for the unrelaxed case. Moreover, every fixed optimal control $\,u^*\,$ for the unrelaxed system turns out to be the limit of a family $\{u^*_α\}_{α>0}$ of minimizers of suitable optimal control problems for the relaxed systems, where the original cost functional is replaced by the so-called ``adapted'' cost functional. It is also shown that the variational inequalities satisfied by $u^*_α$, which constitute the first-order necessary optimality conditions of the relaxed problems with adapted cost functional, converge to the corresponding variational inequality fulfilled by $u^*$ for the unrelaxed problem with the original (nonadapted) cost functional.

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arXiv2026

Optimal velocity control of a Brinkman-Cahn-Hilliard system with curvature effects

We address an optimal control problem governed by a system coupling a Brinkman-type momentum equation for the velocity field with a sixth-order Cahn-Hilliard equation for the phase variable, incorporating curvature effects in the free energy. The control acts as a distributed velocity control, allowing for the manipulation of the flow field and, consequently, the phase separation dynamics. We establish the existence of optimal controls, prove the Fréchet differentiability of the control-to-state operator, and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. We also discuss the aspect of sparsity. Beyond its analytical novelty, this work provides a rigorous control framework for Brinkman-Cahn-Hilliard systems incorporating a curvature regularization, offering a foundation for applications in microfluidic design and controlled pattern formation.

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arXiv2026

Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

In this paper, we study the optimal control of a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. Both the cases when the double-well potential governing the phase evolution is of either regular or logarithmic type are covered by the analysis. We show the Fréchet differentiability of the associated control-to-state operator in suitable Banach spaces and establish first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. The necessary optimality conditions are then used to derive sparsity results for the optimal controls.

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arXiv2026

On the hyperbolic relaxation of the chemical potential in a phase field tumor growth model

In this paper, we study a phase field model for a tumor growth model of Cahn--Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. We show that the resulting initial-boundary value problem is well posed and that its solutions depend continuously on two given functions: one appearing in the mass balance equation and one in the nutrient equation, representing, respectively, sources of drugs (e.g. chemotherapy) and antiangiogenic therapy. We also discuss regularity properties of the solutions. Moreover, in the case of a constant proliferation function, we rigorously analyze the asymptotic behavior as the coefficient of the inertial term tends to zero, establishing convergence to the corresponding viscous Cahn--Hilliard tumor growth model. Our results apply to a broad class of double-well potentials, including nonsmooth ones.

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arXiv2025

On Brinkman flows with curvature-induced phase separation in binary mixtures

The mathematical analysis of diffuse-interface models for multiphase flows has attracted significant attention due to their ability to capture complex interfacial dynamics, including curvature effects, within a unified, energetically consistent framework. In this work, we study a novel Brinkman-Cahn-Hilliard system, coupling a sixth-order phase-field evolution with a Brinkman-type momentum equation featuring variable shear viscosity. The Cahn-Hilliard equation includes a nonconservative source term accounting for mass exchange, and the velocity equation contains a non divergence-free forcing term. We establish the existence of weak solutions in a divergence-free variational framework, and, in the case of constant mobility and shear viscosity, prove uniqueness and continuous dependence on the forcing. Additionally, we analyze the Darcy limit, providing existence results for the corresponding reduced system.

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arXiv2025

On the optimal control of viscous Cahn-Hilliard systems with hyperbolic relaxation of the chemical potential

In this paper, we study an optimal control problem for a viscous Cahn--Hilliard system with zero Neumann boundary conditions in which a hyperbolic relaxation term involving the second time derivative of the chemical potential has been added to the first equation of the system. For the initial-boundary value problem of this system, results concerning well-posedness, continuous dependence and regularity are known. We show Fréchet differentiability of the associated control-to-state operator, study the associated adjoint state system, and derive first-order necessary optimality conditions. Concerning the nonlinearities driving the system, we can include the case of logarithmic potentials. In addition, we perform an asymptotic analysis of the optimal control problem as the relaxation coefficient approaches zero.

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arXiv2024

Hyperbolic relaxation of the chemical potential in the viscous Cahn-Hilliard equation

In this paper, we study a hyperbolic relaxation of the viscous Cahn--Hilliard system with zero Neumann boundary conditions. In fact, we consider a relaxation term involving the second time derivative of the chemical potential in the first equation of the system. We develop a well-posedness, continuous dependence and regularity theory for the initial-boundary value problem. Moreover, we investigate the asymptotic behavior of the system as the relaxation parameter tends to 0 and prove the convergence to the viscous Cahn--Hilliard system.

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arXiv2024

Optimality conditions for sparse optimal control of viscous Cahn-Hilliard systems with logarithmic potential

In this paper we study the optimal control of a parabolic initial-boundary value problem of viscous Cahn-Hilliard type with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. It is assumed that the nonlinear functions driving the physical processes within the spatial domain are double-well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the L1-norm, which leads to sparsity of optimal controls. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls. In the approach to second-order sufficient conditions, the main novelty of this paper, we adapt a technique introduced by E. Casas, C. Ryll and F. Tröltzsch in the paper [SIAM J. Control Optim. 53 (2015), 2168-2202]. In this paper, we show that this method can also be successfully applied to systems of viscous Cahn-Hilliard type with logarithmic nonlinearity. Since the Cahn-Hilliard system corresponds to a fourth-order partial differential equation in contrast to the second-order systems investigated before, additional technical difficulties have to be overcome.

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arXiv2024

Solvability and optimal control of a multi-species Cahn-Hilliard-Keller-Segel tumor growth model

This paper investigates an optimal control problem associated with a two-dimensional multi-species Cahn-Hilliard-Keller-Segel tumor growth model, which incorporates complex biological processes such as species diffusion, chemotaxis, angiogenesis, and nutrient consumption, resulting in a highly nonlinear system of nonlinear partial differential equations. The modeling derivation and corresponding analysis have been addressed in a previous contribution. Building on this foundation, the scope of this study involves investigating a distributed control problem with the goal of optimizing a tracking-type cost functional. This latter aims to minimize the deviation of tumor cell location from desired target configurations while penalizing the costs associated with implementing control measures, akin to introducing a suitable medication. Under appropriate mathematical assumptions, we demonstrate that sufficiently regular solutions exhibit continuous dependence on the control variable. Furthermore, we establish the existence of optimal controls and characterize the first-order necessary optimality conditions through a suitable variational inequality.

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arXiv2024

Second-order optimality conditions for the sparse optimal control of nonviscous Cahn-Hilliard systems

In this paper we study the optimal control of an initial-boundary value problem for the classical nonviscous Cahn-Hilliard system with zero Neumann boundary conditions. Phase field systems of this type govern the evolution of diffusive phase transition processes with conserved order parameter. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a sparsity-enhancing nondifferentiable term like the L1-norm. For such cases, we establish first-order necessary and second-order sufficient optimality conditions for locally optimal controls, where in the approach to second-order sufficient conditions we employ a technique introduced by E. Casas, C. Ryll and F. Tröltzsch in the paper [SIAM J. Control Optim. 53 (2015), 2168-2202]. The main novelty of this paper is that this method, which has recently been successfully applied to systems of viscous Cahn-Hilliard type, can be adapted also to the classical nonviscous case. Since in the case without viscosity the solutions to the state and adjoint systems turn out to be considerably less regular than in the viscous case, numerous additional technical difficulties have to be overcome, and additional conditions have to be imposed. In particular, we have to restrict ourselves to the case when the nonlinearity driving the phase separation is regular, while in the presence of a viscosity term also nonlinearities of logarithmic type turn could be admitted. In addition, the implicit function theorem, which was employed to establish the needed differentiability properties of the control-to-state operator in the viscous case, does not apply in our situation and has to be substituted by other arguments.

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arXiv2024

Curvature effects in pattern formation: well-posedness and optimal control of a sixth-order Cahn-Hilliard equation

This work investigates the well-posedness and optimal control of a sixth-order Cahn-Hilliard equation, a higher-order variant of the celebrated and well-established Cahn-Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improve the understanding of the mathematical properties and control aspects of the sixth-order Cahn-Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.

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arXiv2023

On a Cahn-Hilliard system with source term and thermal memory

A nonisothermal phase field system of Cahn-Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

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arXiv2023

Nutrient control for a viscous Cahn-Hilliard-Keller-Segel model

In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn-Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller-Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.

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arXiv2023

Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the $L^1$-norm in order to enhance the occurrence of sparsity effects in the optimal control, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before.

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arXiv2023

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

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 Fréchet derivative of the control-to-state operator and carry out a thorough and detailed investigation about the related properties.

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arXiv2023

Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions

In this paper we study the optimal control of a parabolic initial-boundary value problem of Allen--Cahn type with dynamic boundary conditions. Phase field systems of this type govern the evolution of coupled diffuse phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear function driving the physical processes within the bulk and on the surface are double well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the $L^1$-norm leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls.

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arXiv2023

Optimal temperature distribution for a nonisothermal Cahn-Hilliard system in two dimensions with source term and double obstacle potential

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails further mathematical difficulties because the mass conservation of the order parameter is no longer satisfied. We study the case that the double-well potential driving the evolution of the phase transition is given by the nondifferentiable double obstacle potential, thereby complementing recent results obtained for the differentiable cases of regular and logarithmic potentials. We derive first-order necessary optimality conditions for the control problem. The analysis is carried out by employing the so-called deep quench approximation in which the nondifferentiable double obstacle potential is approximated by a family of logarithmic potentials for which meaningful first-order necessary optimality conditions are available. Since the results for the logarithmic potentials crucially depend on the validity of the so-called strict separation property which is only available in the spatially two-dimensional situation, our whole analysis is restricted to the two-dimensional case.

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arXiv2023

Optimal temperature distribution for a nonisothermal Cahn-Hilliard system with source term

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn-Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem.

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