Fredi Tröltzsch

• Technische Universität Berlin

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 pr...

For a nonlinear ordinary differential equation with time delay, the differentiation of the solution with respect to the delay is investigated. Special emphasis is laid on the second-order derivative. The results are applied to an associated optimization problem for the time delay. A first- and second-order sensitivity analysis is performed includin...

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 occurrenc...

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...

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 assume...

In this work, we investigate the applicability of unstructured space-time methods to the numerical solution of inverse problems, considering as our model problem the classical inverse problem of the reconstruction of the initial temperature in the heat equation from an observation of the temperature 𝑢𝛿 𝑇 ∈ 𝐿2 (Ω) at a finite time horizon:

Optimal control problems for the linear heat equation with final observation and pointwise constraints on the control are considered, where the control depends only on the time. It is shown that to each finite number of given switching points, there is a final target such that the optimal objective value is positive, the optimal control is bang ban...

A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder sta...

We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the L1-norm of the control that accounts for its spatio-temporal sparsity...

Optimal control problems of partial differential equations are studied. Though the focus lies on elliptic partial differential equations, similar methods can be used for the analysis of control problems associated to evolution equations. In the first section, the state equation is semilinear. The existence of solutions is studied, first and second...

This work is devoted to the reconstruction of the initial temperature in the backward heat equation using the space-time finite element method on fully unstructured space-time simplicial meshes proposed by Steinbach (2015). Such a severely ill-posed problem is tackled by the standard Tikhonov regularization method. This leads to a related optimal c...

In this article we study an optimal control problem subject to the Fokker-Planck equation ∂ t ρ − ν ∆ ρ − div( ρB [ u ]) = 0 The control variable u is time-dependent and possibly multidimensional, and the function B depends on the space variable and the control. The cost functional is of tracking type and includes a quadratic regularization term on...

An optimal control problem for a semilinear heat equation with distributed control is discussed, where two-sided pointwise box constraints on the control and two-sided pointwise mixed control-state constraints are given. The objective functional is the sum of a standard quadratic tracking type part and a multiple of the $L^1$-norm of the control th...

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The...

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [12]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The...

We analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{-1}(\Omega))$. By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the contr...

This work presents and analyzes space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of parabolic optimal control problems. Using Babu\v{s}ka's theorem, we show well-posedness of the first-order optimality systems for a typical model problem with linear state equations, but without control...

We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the $L^1$-norm of the control that accounts for its spatio-temporal spars...

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented that stabilize the system in a periodic state with a given period and given boundary traces. We consider the system both with boundary feedback laws of Pyragas type and...

In this article we study an optimal control problem subject to the Fokker-Planck equation \[ \partial_t \rho - \nu \Delta \rho - {\rm div } \big(\rho B[u]\big) = 0. \] The control variable $u$ is time-dependent and possibly multidimensional, and the function $B$ depends on the space variable and the control. The cost functional is of tracking type...

In this article we study an optimal control problem subject to the Fokker-Planck equation \[ \partial_t \rho - \nu \Delta \rho - {\rm div } \big(\rho B[u]\big) = 0. \] The control variable $u$ is time-dependent and possibly multidimensional, and the function $B$ depends on the space variable and the control. The cost functional is of tracking type...

In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new indust...

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a nondifferentiable term with the measure norm of the control. Existence, uniqueness, and regularity of the solution of the state equation, as well as differentia...

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the soluti...

The optimal control of a system of nonlinear reaction–diffusion equations is considered that covers several important equations of mathematical physics. In particular equations are covered that develop traveling wave fronts, spiral waves, scroll rings, or propagating spot solutions. Well-posedness of the system and differentiability of the control-...

A class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the soluti...

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a non-differentiable term with the measure-norm of the control. Existence, uniqueness and regularity of the solution of the state equation, as well as differentia...

A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism. The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the control function is a transient voltage and the aim of the control is the best approximation of desired electric and ma...

An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak Gâteaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and an associa...

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $\nu$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is disc...

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $\nu$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is disc...

We consider a particular model for electromagnetic fields in the context of optimal control. Special emphasis is laid on a non-standard H-based formulation of the equations of low-frequency electromagnetism in multiply connected conductors. By this technique, the low-frequency Maxwell equations can be solved with reduced computational complexity. W...

The ability to control a desired dynamics or pattern in reaction-diffusion systems has attracted considerable attention over the last decades and it is still a fundamental problem in applied nonlinear science. Besides traveling waves, moving localized spots -- also called dissipative solitons -- represent yet another important class of self-organiz...

Traveling localized spots represent an important class of self-organized two-dimensional patterns in reaction-diffusion systems. We study open-loop control intended to guide a stable spot along a desired trajectory with desired velocity. Simultaneously, the spot's concentration profile does not change under control. For a given protocol of motion,...

A class of optimal control problems for electromagnetic fields is considered. Special emphasis is laid on a non-standard H-based formulation of the equations of low-frequency electromagnetism in multiply connected conductors. By this technique, the low-frequency Maxwell equations can be solved with reduced computational complexity. While the magnet...

A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schl\"ogl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optima...

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schlögl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control to...

Second-order sufficient optimality conditions are considered for a simplified class of semilinear parabolic equations with quadratic objective functional including distributed and terminal observation. Main emphasis is laid on problems where the objective functional does not include a Tikhonov regularization term. Here, standard second-order condit...

This work deals with the position control of selected patterns in reaction-diffusion systems. Exemplarily, the Schl\"{o}gl and FitzHugh-Nagumo model are discussed using three different approaches. First, an analytical solution is proposed. Second, the standard optimal control procedure is applied. The third approach extends standard optimal control...

A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schl\"ogl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optima...

Optimal sparse control problems are considered for the FitzHugh-Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a the...

We investigate optimal sparse control problems for reaction diffusion equations with non-monotonous cubic non-linearities. In particular, we consider the Schlöl equation as well as the FitzHugh-Nagumo system. In these models, the solutions form pattern of traveling wave fronts or spiral waves. To control them turns out to be very challenging and co...

We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the number of constraints. In a first strategy, we apply uniformly refined meshes, whereas in a second more heuristic strategy we use ad...

The Schlögl system is governed by a nonlinear reaction-diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable. In t...

Two optimal control problems for instationary magnetization processes are considered in 3D spatial domains that include electrically conducting and nonconducting regions. The magnetic fields are generated by induction coils. In the first model, the induction coil is considered as part of the conducting region and the electrical current is taken as...

Some optimal control problems for linear and nonlinear ordinary differential equations related to the optimal switching between different magnetic fields are considered. The main aim is to move an electrical initial current by a controllable voltage in shortest time to a desired terminal current and to hold it afterwards. Necessary optimality condi...

Optimal control problems are considered for transient magnetization processes arising from electromagnetic flow measurement. The magnetic fields are generated by an induction coil and are defined in 3D spatial domains that include electrically conducting and nonconducting regions. Taking the electrical voltage in the coil as control, the state equa...

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L 1 -norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. S...

A mathematical model for instationary magnetization processes is considered, where the underlying spatial domain includes electrically conducting and nonconducting regions. The model accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil. By a theorem of Showalter...

If \(f: \mathbb{R}^{n} \to \mathbb{R}\) is twice continuously differentiable, f′(u)=0 and f″(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order sufficient optimality condition to the case \(f: U \to \mathbb{R}\), where U is an infinite-dimensional linear normed space. The read...

Recently, inertial mircofluidics has emerged as a promising tool to manipulate complex liquids with possible biomedical applications, for example, to particle separation. Indeed, in experiments different particle types were separated based on their sizes (A.J. Mach, D. Di Carlo, Biotechnol. Bioeng. 107, 302 (2010)). In this article we use a theoret...

We investigate the problem of sparse optimal controls for the so-called Schlögl model and the FitzHugh-Nagumo system. In these reaction-diffusion equations, traveling wave fronts occur that can be controlled in different ways. The L1-norm of the distributed control is included in the objective functional so that optimal controls exhibit effects of...

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation. The well-posedness of the optimal control problem and the regularity of...

In this paper we present a derivative-free optimization algorithm for finite minimax problems. The algorithm calculates an approximate gradient for each of the active functions of the finite max function and uses these to generate an approximate subdifferential. ...

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally...

Book Front Matter of AICT 391

This paper is concerned with a PDE-constrained optimization problem of induction heating, where the state equations consist of 3D time-dependent heat equations coupled with 3D time-harmonic eddy current equations. The control parameters are given by finite real numbers representing applied alternating voltages which enter the eddy current equations...

An abstract optimization problem of minimizing a functional on a convex subset of a Banach space is considered. We discuss natural assumptions on the functional that permit establishing sufficient second-order optimality conditions with minimal gap with respect to the associated necessary ones. Though the two-norm discrepancy is taken into account,...

A theorem on error estimates for smooth nonlinear programming prob-lems in Banach spaces is proved that can be used to derive optimal error estimates for optimal control problems. This theorem is applied to a class of optimal control problems for quasilinear elliptic equations. The state equation is approximated by a finite element scheme, while di...

A class of one-dimensional parabolic optimal boundary control problems is considered. The discussion includes Neumann, Robin, and Dirichlet boundary conditions. The reachability of a given target state in final time is discussed under box constraints on the control. As a mathematical tool, related exponential moment problems are investigated. Moreo...

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: Let an arbitrary control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second order sufficient optimality condition for the (unknown) locally optimal con...

The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a...

In this work, boundary control problems governed by a system of semilinear parabolic PDEs with pointwise control constraints are considered. This class of problems is related to applications in the chemical catalysis. After discussing existence and uniqueness of the state equation with both linear and nonlinear boundary conditions, the existence of...

Solutions to optimization problems with pde constraints inherit special properties; the associated state solves the pde which in the optimization problem takes the role of a equality constraint, and this state together with the associated control solves an optimization problem, i.e. together with multipliers satisfies first and second order necessa...

Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results...

In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One...

We investigate two control problems related to the aerodynamic optimization of flows around airfoils in high-lift configurations. The first task is the steady state maximization of lift subject to restrictions on the drag. This leads to a boundary control problem for the 2D stationary Navier-Stokes equations with constrained control functions belon...

Discretizations of optimal control problems for elliptic equations by finite element methods are considered. The problems are subject to constraints on the control and may also contain pointwise state constraints. Some techniques are surveyed to estimate the distance between the exact optimal control and the associated optimal control of the discre...

We derive a priori error estimates for linear-quadratic elliptic optimal control problems with pointwise state constraints in a compact subdomain of the spatial domain Ω for a class of problems with finite-dimensional control space. The problem formulation leads to a class of semi-infinite programming problems, whose constraints are implicitly give...

In this paper we derive a priori error estimates for linear-quadratic elliptic optimal control problems with finite dimensional control space and state constraints in the whole domain, which can be written as semi-infinite optimization problems. Numerical experiments are con-ducted to underline our theory.

The finite element approximation of optimal control problems for semilin- ear elliptic partial dierential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the do- main. Under the standard linear independency condition on the active gradients and a strong second-...

The main focus of this paper is on an a-posteriori analysis for the method of proper orthogonal decomposition (POD) applied to optimal control problems governed by parabolic and elliptic PDEs. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the POD model, is from the (unknown) exact one. Numeric...

A state-constrained optimal boundary control problem gov-erned by a linear elliptic equation is considered. In order to obtain the optimality conditions for the solutions to the model problem, a Slater as-sumption has to be made that restricts the theory to the two-dimensional case. This difficulty is overcome by a source representation of the cont...

In this paper we study Lavrentiev-type regularization concepts for linear-quadratic parabolic control problems with pointwise state constraints. In the first part, we apply classical Lavrentiev regularization to a problem with distributed control, whereas in the second part, a Lavrentiev-type regularization method based on the adjoint operator is a...

A class of optimal control problems for quasilinear elliptic equations is considered, where the coefficients of the elliptic differential operator depend on the state function. First- and second-order optimality conditions are discussed for an associated control-constrained optimal control problem. Main emphasis is laid on second-order sufficient o...

In den vorangegangenen Abschnitten haben wir nur lineare partielle Differentialgleichungen behandelt und damit viele wichtige Anwendungen ausgeschlossen. Jetzt sind auch nichtlineare Gleichungen zugelassen, wie zum Beispiel in der Aufgabe

Elliptische Differentialgleichungen beschreiben zeitlich stationäre physikalische Vorgänge wie z.B. Wärmeleitprozesse mit einer Temperaturverteilung im Gleichgewichtszustand. Liegt diese Stationarität nicht vor, dann kommt als weiterer physikalischer Parameter die Zeit ins Spiel. Wir betrachten als Modellfall die Aufgabe der optimalen zeitlich inst...

Wir beginnen mit der Bereitstellung einiger Begriffe der Funktionalanalysis. Dabei gehen wir nach dem Prinzip vor, immer nur das abzuhandeln, was zum Verst¨andnis des jeweils folgenden Abschnitts unbedingt nötig ist. Die entsprechenden Aussagen werden hier nicht bewiesen. Dazu verweisen wir auf Standardwerke der Funktionalanalysis wie Alt [6], Heus...

Die mathematische Theorie der optimalen Steuerung hat sich im Zusammenhang mit Berechnungen für die Raumfahrt schnell zu einem wichtigen und eigenständigen Gebiet der angewandten Mathematik entwickelt. Die Bewegungsgleichungen von Luft- und Raumfahrzeugen werden durch Systeme von gewöhnlichen Differentialgleichungen beschrieben und Aspekte der Opti...

Das Arbeiten mit Sobolewräumen wird wesentlich durch Einbettungs- und Spursätze bestimmt, die wir hier bereitstellen. Dabei halten wir uns an Adams [2].

Die in den vorangegangenen Abschnitten mehrfach erprobte formale Lagrangetechnik beruht auf einer mathematisch exakten Grundlage. Hier werden Grundzüge dieser Theorie skizziert, so weit sie zum Verständnis der Behandlung von Aufgaben mit Zustandsbeschr änkungen nötig sind. Beweise und weitere Sätze sind in Büchern zur Optimierung in allgemeinen Räu...

In diesem Kapitel behandeln wir analog zu Aufgaben der Optimalsteuerung semilinearer elliptischer Gleichungen den parabolischen Fall. Die Theorie der Existenz und Regularität von Lösungen parabolischer Gleichungen unterscheidet sich in mancher Hinsicht von der für elliptische Aufgaben. Die optimierungstheoretischen Aspekte weisen aber viele Paralle...