Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

Gegenstand dieser Arbeit ist die numerische Analysis von Optimierungsproblemen mit partiellen Differentialgleichungen (PDEs), deren Zustand oder Steuerung punktweisen Ungleichungsbeschränkungen unterliegt. Wir interessieren uns insbesondere für nichtkonvexe Probleme mit semilinearer Zustandsgleichung. Es liegt in der Natur derartiger Probleme, dass eine Lösung oftmals nur numerisch gefunden werden kann. Man interessiert sich deshalb für den Fehler zwischen einer (lokalen) Lösung des kontinuierlichen Problems und einer zugehörigen (lokalen) diskreten Lösung. Eine umfassende Diskussion des kontinuierliche Problems ist dabei eine Grundvoraussetzung. Insbesondere bei vorhanden punktweisen Zustandsbeschränkungen treten spezifische Schwierigkeiten sowohl analytischer als auch numerischer Art auf, denen man entweder direkt oder mit Hilfe von Regularisierungsansätzen begegnen kann. Mit dieser Dissertation leisten wir auf verschiedene Weise neue Beiträge zur Diskussion von Optimalsteuerungsproblemen mit punktweisen Zustandsschranken aber auch reinen Kontrollschranken. Nach kurzer Einführung in die Thematik und Bereitstellung gewisser Grundlagen beschäftigen wir uns in Kapitel 3 mit einem elliptischen semiinfiniten Optimalsteuerungsproblem. Bekannte Resultate zu notwendigen und hinreichenden Optimalitätsbedingungen, die vergleichsweise hohe Regularität von Lösungen elliptischer PDEs und die Endlichdimensionalität des Steuerungsraumes lassen eine direkte Diskussion von a priori Diskretisierungsfehlerabschätzungen für dieses Problem zu. Unser Hauptergebnis in diesem Kapitel ist eine a priori Fehlerschranke der Ordnung O(h^2|ln h|) für lokale Lösungen einer Finite-Elemente-Diskretisierung des Optimalsteuerungsproblems mit Gitterweite h in einem zweidimensionalen Ortsgebiet. Dazu stellen wir gewisse Annahmen an die Struktur der aktiven Menge. In Kapitel 4 betrachten wir ein parabolisches Optimalsteuerungsproblem mit punktweise beschräkten Steuerungsfunktionen, semilinearer Zustandsgleichung und punktweisen Zustandsbeschränkungen im gesamten Orts-Zeit-Gebiet. Im Gegensatz zu dem in Kapitel 3 diskutierten elliptischen Problem sind hier u.a. hinreichende Bedingungen zweiter Ordnung nur für eindimensionale Ortsgebiete verfügbar. Auch läßt die Verwendung von Steuerungsfunktionen an Stelle endlich vieler Parameter keine sinnvollen a priori Annahmen an die Struktur der aktiven Menge zu. Wir regularisieren daher das Problem mit der auf Meyer, Rösch und Tröltzsch zurückgehenden Lavrentievregularisierung, und können so unter anderem auf bekannte Resultate zurückgreifen, die eine höhere Regularität der Lagrangeschen Multiplikatoren sichern und eine tiefergehende Analysis ermöglichen. Wir beweisen ein Konvergenzresultat für lokale Lösungen des regularisierten Problems und weisen die lokale Eindeutigkeit regularisierter Lösungen nach. In Kapitel 5 untersuchen wir die Finite-Element-Diskretisierung eines kontrollbeschränkten parabolischen Optimalsteuerungsproblems mit semilinearer Zustandsgleichung. Wir beweisen Fehlerordnungen für diskrete lokale Lösungen in der L^2-Norm. Dabei erweitern wir Resultate, die für linear-quadratische Probleme bekannt sind, auf den nichtkonvexen Fall. Es müssen insbesondere Beschränktheitsresultate in der L^infty-Norm der semidiskreten und diskreten Zustände gezeigt werden, die unabhängig von den Diskretisierungsparametern gelten. Außerdem erfordert die Diskussion lokal optimaler Lösungen, dass Konvergenzresultate und quadratische Wachstumsbedingungen in denselben Normen betrachtet werden.

In this paper, we give some new global regularity criteria for three-dimensional incompressible magnetohydrodynamics (MHD) equations. More precisely, we provide some sufficient conditions in terms of the derivatives of the velocity or pressure, for the global regularity of strong solutions to 3D incompressible MHD equations in the whole space, as well as for periodic boundary conditions. Moreover, the regularity criterion involving three of the nine components of the velocity gradient tensor is also obtained. The main results generalize the recent work by Cao and Wu (2010 Two regularity criteria for the 3D MHD equations J. Diff. Eqns 248 2263–74) and the analysis in part is based on the works by Cao C and Titi E (2008 Regularity criteria for the three-dimensional Navier–Stokes equations Indiana Univ. Math. J. 57 2643–61; 2011 Gobal regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor Arch. Rational Mech. Anal. 202 919–32) for 3D incompressible Navier–Stokes equations.

In this paper, based on the stabilization technique, the Oseen iterative method and the two-level finite element algorithm are combined to numerically solve the stationary incompressible magnetohydrodynamic (MHD) equations. For the low regularity of the magnetic field, when dealing with the magnetic field sub-problem, the Lagrange multiplier technique is used. The stabilized method is applied to approximate the flow field sub-problem to circumvent the inf-sup condition restrictions. One- and two-level stabilized finite element algorithms are presented, and their stability and convergence analysis is given. The two-level method uses the Oseen iteration to solve the nonlinear MHD equations on a coarse grid of size H, and then employs the linearized correction on a fine grid with grid size h. The error analysis shows that when the grid sizes satisfy , the two-level stabilization method has the same convergence order as the one-level one. However, the former saves more computational cost than the latter one. Finally, through some numerical experiments, it has been verified that our proposed method is effective. The two-level stabilized method takes less than half the time of the one-level one when using the second class Nédélec element to approximate magnetic field, and even takes almost a third of the computing time of the one-level one when adopting the first class Nédélec element.

The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles.

The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is very challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible resistive MHD equations using a stable nodal (Q2/Q1) discretization for the hydrodynamics and a stable edge-node discretization of a reduced form of the Maxwell equations. This paper presents new approximate block factorization preconditioners for this system which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods. These preconditioners include a new augmentation-based approximation for the magnetic induction saddle point system as well as efficient approximations of the Schur complements that arise from the complex coupling between the Navier--Stokes equations and the Maxwell equations.

The large time L1-behavior of the strong solution (including the first and second order spacial derivatives) to the incompressible magneto-hydrodynamic (MHD) equations is given in a half-space. The main tool employed in this article is a new weighted estimate for the Stokes flow in L1(R+n), such a study is of independent interest.

A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

In the paper be considered the problem of optimal control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. Particular attention is given to the connection of the differential-differential system with the evolutionary differential system and the search conditions in which the properties of the differential system are preserved. This connection establishes a universal method of semi- digitization by temporal variable for differential system, providing an effective tool in finding conditions of uniqueness solvability and continuity on the initial data for the differential- differential system. A priori estimates of the norms of a weak solution of differential- differential system give an opportunity to establish not only the solvability of this system but also the existence of a weak solution of the evolutionary differential system. For the differential-difference system analysis of the optimal control problem is presented, containing natural in that cases a additional study of the problem with a time lag. This essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — defining ratios that determine optimal control or the set optimal controls. The work shows courses to transfer the results in case of analysis of optimal control problems in the class of functions with bearer in network-like domains. The transition from an evolutionary differential system to a differential-difference system was a natural step in the study of applied problems of the theory of the transfer of solid mediums. The obtained results underlie the analysis of optimal control problems for differential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.

Discrete-time hybrid control in Borel spaces: Average cost optimality criterion

Previous article Next article The Reduction of Certain Control Problems to an “Ordinary Differential” TypeJ. WargaJ. Wargahttps://doi.org/10.1137/1010036PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails Stationarity conditions in optimal control problems class related to dynamical objects group control Cross Ref On the First- and Second-Order Conditions in Optimal Control Problems Related to Autonomous Objects Group lifecycle Cross Ref Optimal Control Problems with Time Delays: Constancy of the HamiltonianRichard B. Vinter25 July 2019 | SIAM Journal on Control and Optimization, Vol. 57, No. 4AbstractPDF (525 KB)On obtaining the stationarity conditions in an optimal control problem for a trajectory with a smooth contact with the phase boundary Cross Ref On one Method to Obtain Stationarity Conditions for an Optimal Control Problem Trajectory With a Smooth Boundary ContactIFAC-PapersOnLine, Vol. 51, No. 32 Cross Ref The Maximum Principle for Optimal Control Problems with Time DelaysA. Boccia and R. B. Vinter19 September 2017 | SIAM Journal on Control and Optimization, Vol. 55, No. 5AbstractPDF (407 KB)Problems with relaxed shifted controls and the pointwise maximum principleNonlinear Analysis: Theory, Methods & Applications, Vol. 75, No. 3 Cross Ref The Instability of Optimal Control Problems to Time DelayAlexey S. Matveev26 July 2006 | SIAM Journal on Control and Optimization, Vol. 43, No. 5AbstractPDF (340 KB)On the Characterization of Properly Relaxed Delayed ControlsJournal of Mathematical Analysis and Applications, Vol. 209, No. 1 Cross Ref Approximation of strongly relaxed minimizers with ordinary delayed controlsApplied Mathematics & Optimization, Vol. 32, No. 1 Cross Ref Jack Warga: In Appreciation14 July 2006 | SIAM Journal on Control and Optimization, Vol. 33, No. 2AbstractPDF (809 KB)Minimizing approximate original solutions for commensurate delayed controlsApplied Mathematics Letters, Vol. 7, No. 5 Cross Ref How to Properly Relax Delayed Controls Cross Ref A proper relaxation of shifted and delayed controlsJournal of Mathematical Analysis and Applications, Vol. 169, No. 2 Cross Ref Optimal controls and their discontinuities in quadratic problems of delay systemsIEEE Transactions on Automatic Control, Vol. 30, No. 7 Cross Ref Optimal Controls with PseudodelaysJ. Warga18 July 2006 | SIAM Journal on Control, Vol. 12, No. 2AbstractPDF (1227 KB)A conservative bound on the estimation error covariance matrix in the presence of correlated driving noise and correlated discrete measurement noiseJournal of Mathematical Analysis and Applications, Vol. 43, No. 2 Cross Ref References Cross Ref Volume 10, Issue 2| 1968SIAM Review History Submitted:18 October 1967Published online:18 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1010036Article page range:pp. 219-222ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

This paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled Schrodinger equations. The first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space. (The minimal time control problems are also called the first type of optimal time control problems.) The second one is an equivalence of three optimal control problems: optimal target control problems, optimal norm control problems, and the second type of optimal time control problems.

This paper studies the optimum design of beam networks modeled with Timoshenko beams. To account for multiple load cases, an auxiliary optimal control problem is introduced. Optimal distributed control problems for Timoshenko beam networks are solved through the associated optimality system, where the shape functional of the network is defined by the optimal value of the control cost. For control problems exhibiting the turnpike property, the optimum network design is carried out using the steady-state beam model and the corresponding steady-state control problem. A domain decomposition method is adopted to handle topological changes, while the Steklov–Poincaré operator is used to reformulate the beam network model as an interface problem on subdomain boundaries. This approach is applicable under additional assumptions on the network loading. Consequently, the topological derivative of the Steklov–Poincaré operator is incorporated into the optimality system of the control problem, enabling sensitivity analysis with respect to topological changes. The topological derivative of the cost functional with respect to the size of small cycles is derived and computed. Finally, numerical experiments are presented to illustrate and corroborate the analytical results.

We study the stability of the Couette flow $$(y,0,0)^T$$ in the presence of a uniform magnetic field $$\alpha (\sigma , 0, 1)$$ on $${{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {T}}}$$ using the 3D incompressible magnetohydrodynamics (MHD) equations. We consider the inviscid, ideal conductor limit $$\mathbf{Re} ^{-1}$$, $$\mathbf{R }_m^{-1} \ll 1$$ and prove that for strong and suitably oriented background fields the Couette flow is asymptotically stable to perturbations small in the Sobolev space $$H^N$$. More precisely, we show that if $$\mathbf{Re} ^{-1} = \mathbf{R }_m^{-1} \in (0,1]$$, $$\alpha > 0$$ and $$N > 0$$ are sufficiently large, $$\sigma \in {{\mathbb {R}}}{\setminus } {\mathbb {Q}}$$ satisfies a generic Diophantine condition, and the initial perturbations $$u_{\text{ in }}$$ and $$b_{\text{ in }}$$ to the Couette flow and magnetic field, respectively, satisfy $$\Vert u_{\text{ in }}\Vert _{H^N} + \Vert b_{\text{ in }}\Vert _{H^N} = \epsilon \ll \mathbf{Re} ^{-1}$$, then the resulting solution to the 3D MHD equations is global in time and the perturbations $$u(t,x+yt,y,z)$$ and $$b(t,x+yt,y,z)$$ remain $${\mathcal {O}}(\mathbf{Re} ^{-1})$$ in $$H^{N'}$$ for some $$1 \ll N'(\sigma ) < N$$. Our proof establishes enhanced dissipation estimates describing the decay of the x-dependent modes on the timescale $$t \sim \mathbf{Re} ^{1/3}$$, as well as inviscid damping of the velocity and magnetic field with a rate that agrees with the prediction of the linear stability analysis. In the Navier–Stokes case, high regularity control on the perturbation in a coordinate system adapted to the mixing of the Couette flow is known only under the stronger assumption $$\epsilon \ll \mathbf{Re} ^{-3/2}$$ (Bedrossian et al. in Ann. Math. 185(2): 541–608, 2017). The improvement in the MHD setting is possible because the magnetic field induces time oscillations that partially suppress the lift-up effect, which is the primary transient growth mechanism for the Navier–Stokes equations linearized around Couette flow.

Method-of-lines discretizations are demanding test problems for stiff integration methods. However, for PDE problems with known analytic solution, the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To overcome these drawbacks, we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numerically observed convergence orders for selected implicit Runge–Kutta and Peer two-step methods of classical order four, which are suitable for optimal control problems.