Jürgen Sprekels’s research while affiliated with Weierstrass Institute for Applied Analysis and Stochastics and other places

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Publications (232)


Optimality Conditions for Sparse Optimal Control of Viscous Cahn–Hilliard Systems with Logarithmic Potential
  • Article
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October 2024

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49 Reads

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2 Citations

Applied Mathematics & Optimization

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Jürgen Sprekels

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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 L1L1L^1-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 Casas et al. in the paper (SIAM J Control Optim 53:2168–2202, 2015). 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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Hyperbolic relaxation of the chemical potential in the viscous Cahn-Hilliard equation

August 2024

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51 Reads

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.


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

July 2024

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62 Reads

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.



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

June 2024

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54 Reads

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. Tr\"oltzsch 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.



Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential

November 2023

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31 Reads

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4 Citations

ESAIM Control Optimisation and Calculus of Variations

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 L1L^1--norm in order to enhance the occurrence of sparsity effects in the optimal controls, 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. 2020 Mathematics Subject Classification 35K57, 37N25, 49J50, 49J52, 49K20, 49K40.


Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

August 2023

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55 Reads

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3 Citations

Applied Mathematics & Optimization

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. Crucial for the whole analysis to work is the boundedness property stating that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions.


Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

July 2023

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81 Reads

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6 Citations

We analyze a phase field model for tumor growth consisting of a Cahn–Hilliard–Brinkman system, ruling the evolution of the tumor mass, coupled with an advection-reaction-diffusion equation for a chemical species acting as a nutrient. The main novelty of the paper concerns the discussion of the existence of weak solutions to the system covering all the meaningful cases for the nonlinear potentials; in particular, the typical choices given by the regular, the logarithmic, and the double obstacle potentials are admitted in our treatise. Compared to previous results related to similar models, we suggest, instead of the classical no-flux condition, a Dirichlet boundary condition for the chemical potential appearing in the Cahn–Hilliard-type equation. Besides, abstract growth conditions for the source terms that may depend on the solution variables are postulated.


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

June 2023

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49 Reads

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 L1L^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.


Citations (55)


... is well known and was already investigated in a number of papers (see [8,9,[11][12][13][14][15][20][21][22][23] to quote some recent contributions), while to our knowledge an inertial term like α∂ tt µ in (1.1) is not common and possibly deserves to be examined. In this class of problems, the unknown functions ϕ and µ ususally stand for the order parameter, which can represent a scaled density of one of the involved phases, and the chemical potential associated with the phase separation process, respectively. ...

Reference:

Hyperbolic relaxation of the chemical potential in the viscous Cahn-Hilliard equation
Optimality Conditions for Sparse Optimal Control of Viscous Cahn–Hilliard Systems with Logarithmic Potential

Applied Mathematics & Optimization

... quite a number of works have been dedicated to the study of cases in which the Cahn-Hilliard system is coupled to other systems; in this connection, we quote Cahn-Hilliard-Navier-Stokes models (see [32,42,44,45,60]) and the Cahn-Hilliard-Oono (see [15,36]), Cahn-Hilliard-Darcy (see [1,58]), Cahn-Hilliard-Brinkman (see [30]), Cahn-Hilliard-Keller-Segel (see [37]), and Cahn-Hilliard with curvature effects (see [16]) systems. None of the papers cited above is concerned with the aspect of sparsity, i.e., the possibility that any locally optimal control may vanish in subregions of positive measure of the space-time cylinder Q that are controlled by the sparsity parameter κ. ...

Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation
  • Citing Article
  • July 2024

SIAM Journal on Mathematical Analysis

... For a more detailed physical overview, the reader may see the papers [11,13]. In this respect, κ 1 and κ 2 in (1.3) stand for prescribed positive coefficients related to the heat flux, which is here assumed in the Green-Naghdi form (see [19][20][21]26]) ...

On a Cahn–Hilliard system with source term and thermal memory
  • Citing Article
  • March 2024

Nonlinear Analysis

... This method was originally introduced for a class of semilinear second-order parabolic problems with smooth nonlinearities. In the recent papers [56,57] two of the present authors have demonstrated that it can be adapted correspondingly to the sparse optimal control of Allen-Cahn systems with dynamic boundary conditions and to a large class of systems modeling tumor growth, Applied Mathematics & Optimization (2024) 90:47 Applied Mathematics & Optimization (2024) 90:47 ...

Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential

ESAIM Control Optimisation and Calculus of Variations

... As already said, the other functionals from (1.2) were already investigated in [5]. Further contributions which address more complicated state equations are [8,19]. ...

Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions
  • Citing Article
  • January 2023

Discrete and Continuous Dynamical Systems - S

... Fig. 2). 10 Further, we can observe at t = 0.005 the placement of the 16 actuators, as we chose M = 2 11 in this simulation. The divergence lim M →+∞ α H M = +∞ is shown in [34,Sect. ...

Optimal Temperature Distribution for a Nonisothermal Cahn–Hilliard System with Source Term

Applied Mathematics & Optimization

... The mass and momentum equations are complemented with mass and momentum exchange terms between the phases, and appropriate constitutive laws to close the model equations. Recent theoretical and numerical studies of Cahn-Hilliard type tumour growth models, which describe tumour progression in avascular early stages, can be found e.g. in [2,3,15,17,19,22,25,26,[30][31][32], while complete models which describe tumour progression in vascular late stages can be found e.g. in [4,46]. In this paper, we consider the following tumour growth model, based on the one introduced in [35]: ...

Cahn–Hilliard–Brinkman model for tumor growth with possibly singular potentials

... quite a number of works have been dedicated to the study of cases in which the Cahn-Hilliard system is coupled to other systems; in this connection, we quote Cahn-Hilliard-Navier-Stokes models (see [32,42,44,45,60]) and the Cahn-Hilliard-Oono (see [15,36]), Cahn-Hilliard-Darcy (see [1,58]), Cahn-Hilliard-Brinkman (see [30]), Cahn-Hilliard-Keller-Segel (see [37]), and Cahn-Hilliard with curvature effects (see [16]) systems. None of the papers cited above is concerned with the aspect of sparsity, i.e., the possibility that any locally optimal control may vanish in subregions of positive measure of the space-time cylinder Q that are controlled by the sparsity parameter κ. ...

Nutrient control for a viscous Cahn–Hilliard–Keller–Segel model with logistic source describing tumor growth
  • Citing Article
  • January 2023

Discrete and Continuous Dynamical Systems - S

... There are numerous papers studying optimal control problems for problems with dynamic boundary conditions. Without claiming to be complete, we cite here the works [1,13,23] for the Allen-Cahn equation and [11,12,16,17,18,19,20,21,29] for systems of Cahn-Hilliard type. ...

A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions

... with c 2 > 0. However, the double obstacle case is not included in the subsequent analysis, although we expect that, with similar techniques as those employed in [10], it is possible to extend the analysis also to this kind of nonregular potentials. ...

Optimal control of a nonconserved phase field model of Caginalp type with thermal memory and double obstacle potential
  • Citing Article
  • January 2023

Discrete and Continuous Dynamical Systems - S