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An inverse breaking line identification problem formulated as an optimal control problem with a suitable PDE constraint is studied. The constraint is a boundary value problem describing the anti-plane equilibrium of an elastic body with a stress-free breaking line under the action of a traction force at the boundary. The behavior of the displacement is observed on a subset of the boundary, and the optimal breaking line is identified by minimizing the L 2 {L^{2}} -distance between the displacement and the observation. Then the optimal control problem is solved by shape optimization techniques via a Lagrangian approach. Several numerical experiments are carried out to show its performance in diverse situations. [ABSTRACT FROM AUTHOR]

A theoretical framework and numerical techniques to solve optimal control problems with a spatial trace term in the terminal cost and governed by regularized nonlinear hyperbolic conservation laws are provided. Depending on the spatial dimension, the set at which the optimum of the trace term is reached under the action of the control function can be a point, a curve or a hypersurface. The set is determined by geometric parameters. Theoretically the lack of a convenient functional framework in the context of optimal control for hyperbolic systems leads us to consider a parabolic regularization for the state equation, in order to derive optimality conditions. For deriving these conditions, we use a change of variables encoding the sensitivity with respect to the geometric parameters. As illustration, we consider the shallow-water equations with the objective of maximizing the height of the wave at the final time, a wave whose location and shape are optimized via the geometric parameters. Numerical results are obtained in 1D and 2D, using finite difference schemes, combined with an immersed boundary method for iterating the geometric parameters. [ABSTRACT FROM AUTHOR]

Nonsmooth nonconvex optimization problems involving the ℓp quasi-norm, p∈(0,1], of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction. [ABSTRACT FROM AUTHOR]

An approach to optimal actuator design based on shape and topology optimization techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology. [ABSTRACT FROM AUTHOR]

Optimal control approaches have proved useful in designing RF pulses for large tip-angle applications. A typical challenge for optimal control design is the inclusion of constraints resulting from physiological or technical limitations that assure the realizability of the optimized pulses. In this paper, we show how to treat such inequality constraints, in particular, amplitude constraints on the B1 field, the slice-selective gradient, and its slew rate, as well as constraints on the slice profile accuracy. For the latter, a pointwise profile error and additional phase constraints are prescribed. Here, a penalization method is introduced that corresponds to a higher order tracking instead of the common quadratic tracking. The order is driven to infinity in the course of the optimization. We jointly optimize for the RF and slice-selective gradient waveform. The amplitude constraints on these control variables are treated efficiently by semismooth Newton or quasi-Newton methods. The method is flexible, adapting to many optimization goals. As an application, we reduce the power of refocusing pulses, which is important for spin echo-based applications with a short echo spacing. Here, the optimization method is tested in numerical experiments for reducing the pulse power of simultaneous multislice refocusing pulses. The results are validated by phantom and in-vivo experiments. [ABSTRACT FROM PUBLISHER]

Sufficient conditions for the existence of a solution to an abstract optimization problem in Banach spaces are given, which do not rely on convexity, regularity properties or a straightforward coerciveness assumption. Applications to sparsity-constrained optimization and to problems from mechanics are provided. [ABSTRACT FROM AUTHOR]

Lp optimal control with p ∈ [0, 1) is investigated. The difficulty of natural lack of convexity and thus of weak lower semicontinuity is addressed by introducing appropriately chosen regularization terms. Existence results and necessary optimality conditions are obtained, and convergence of a monotone scheme is proved. Special attention is given to the particular case of optimal control problems with quadratic tracking and regularized L0 control costs are given. A maximum principle is derived and existence of controls, in some cases relaxed controls, is proved, and an estimate on the consequences of relaxation are estimated. [ABSTRACT FROM AUTHOR]

Sparsity optimization for linear least squares problems formulated as non-smooth regularization problems are considered in infinite-dimensional sequence spaces ℓp with p ∈ [0, 1]. Necessary optimality conditions in the format of a complementarity system are obtained. A monotonically convergent scheme is developed for the case p ∈ (0, 1]. For the case p = 0 a primal dual active set strategy based on the Lagrange multiplier rule is proposed and analyzed for special cases. [ABSTRACT FROM AUTHOR]

Time optimal control of the wave equation is analyzed on the basis of a regularized formulation which is considered as a bilevel optimization problem. For the lower level problems, which are constrained optimal control problems for the wave equation, a detailed sensitivity analysis is carried out. Further a semismooth Newton method is analyzed and proved to converge locally superlinearly. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

Portfolio optimization problems on a finite time horizon under proportional transaction costs are considered. The objective is to maximize the expected utility of the terminal wealth. The ensuing non-smooth time-dependent Hamilton-Jacobi-Bellman equation is solved by regularization and the application of a semi-smooth Newton method. Discretization in space is carried out by finite differences or finite elements. Computational results for one and two risky assets are provided. [ABSTRACT FROM AUTHOR]

Abstract: For the monodomain approximation of the bidomain equations, a weak solution concept is proposed. We analyze it for the FitzHugh–Nagumo and the Rogers–McCulloch ionic models, obtaining existence and uniqueness theorems. Subsequently, we investigate optimal control problems subject to the monodomain equations. After proving the existence of global minimizers, the system of the first-order necessary optimality conditions is rigorously characterized. For the adjoint system, we prove an existence and regularity theorem as well. [Copyright &y& Elsevier]

Lagrange multiplier rules for abstract optimization problems with mixed smooth and convex terms in the cost, with smooth equality constrained and convex inequality constraints, are presented. The typical case for the equality constraints that the theory is meant for is given by differential equations. Applications are given to L¹-minimum norm control problems, L∞-norm minimization, and a class of optimal control problems with distributed state constraints and nonsmooth cost. [ABSTRACT FROM AUTHOR]

general framework for calculating shape derivatives for domain optimization problems with partial differential equations as constraints is presented. The first order approximation of the cost with respect to the geometry perturbation is arranged in an efficient manner that allows the computation of the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state variable is only required to be Lipschitz continuous with respect to geometry perturbations. Application to shape optimization with the Navier-Stokes equations as PDE constraint is given. [ABSTRACT FROM AUTHOR]

This work addresses an optimal control approach for a model problem in cardiac electrophysiology with the goal of extinction of a reentry phenomenon. After the introduction of the mathematical model, the derivation of the optimality system, the description of its discretization and a numerical feasibility study in a parallel environment are provided. [ABSTRACT FROM AUTHOR]

The bidomain equations, a continuum approximation of cardiac tissue based on the idea of a functional syncytium, are widely accepted as one of the most complete descriptions of cardiac bioelectric activity at the tissue and organ level. Numerous studies employed bidomain simulations to investigate the formation of cardiac arrhythmias and their therapeutical treatment. They consist of a linear elliptic partial differential equation and a non-linear parabolic partial differential equation of reaction-diffusion type, where the reaction term is described by a set of ordinary differential equations. The monodomain equations, although not explicitly accounting for current flow in the extracellular domain and its feedback onto the electrical activity inside the tissue, are popular since they approximate, under many circumstances of practical interest, the bidomain equations quite well at a much lower computational expense, owing to the fact that the elliptic equation can be eliminated when assuming that conductivity tensors of intracellular and extracellular space are related to each other. Optimal control problems suggest themselves quite naturally for this important class of modelling problems and the present paper is a first attempt in this direction. Specifically, we present an optimal control formulation for the monodomain equations with an extra-cellular current, I, as the control variable. I must be determined such that electrical activity in the tissue is damped in an optimal manner. The derivation of the optimality system is given and a method for its numerical realization is proposed. The solution of the optimization problem is based on a non-linear conjugate gradient (NCG) method. The main goals of this work are to demonstrate that optimal control techniques can be employed successfully to this class of highly nonlinear models and that the influence of I judiciously applied can in fact serve as a successful control for the dampening of propagating wavefronts. [ABSTRACT FROM AUTHOR]

Abstract: In this work adaptive and high resolution numerical discretization techniques are demonstrated for solving optimal control of the monodomain equations in cardiac electrophysiology. A monodomain model, which is a well established model for describing the wave propagation of the action potential in the cardiac tissue, will be employed for the numerical experiments. The optimal control problem is considered as a PDE constrained optimization problem. We present an optimal control formulation for the monodomain equations with an extra-cellular current as the control variable which must be determined in such a way that excitations of the transmembrane voltage are damped in an optimal manner. The focus of this work is on the development and implementation of an efficient numerical technique to solve an optimal control problem related to a reaction–diffusions system arising in cardiac electrophysiology. Specifically a Newton-type method for the monodomain model is developed. The numerical treatment is enhanced by using a second order time stepping method and adaptive grid refinement techniques. The numerical results clearly show that super-linear convergence is achieved in practice. [ABSTRACT FROM AUTHOR]

Abstract: Optimal control of parabolic variational inequalities is studied in the case where the spatial domain is not necessarily bounded. First, strong and weak solutions concepts for the variational inequality are proposed and existence results are obtained by a monotone and a finite difference technique. An optimal control problem with the control appearing in the coefficient of the leading term is investigated and a first order optimality system in a Lagrangian framework is derived. [Copyright &y& Elsevier]

A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given. [ABSTRACT FROM AUTHOR]

Optimal control of variational inequalities with the controls given by the obstacles is considered. Existence optimal solutions are proved for obstacles with H1 regularity and first-order optimality conditions are derived which, under additional assumptions, are also sufficient. A numerical algorithm is proposed and its practical feasibility is investigated. [ABSTRACT FROM AUTHOR]

Sufficient conditions for global convergence of the primal-dual active set strategy for finite and infinite dimensional quadratic, as well as nonlinear optimization, problems with affine equality and inequality constraints are presented. These conditions involve diagonal dominance and cone preserving properties of the operator defining the cost functional. Globalization strategies are also provided, and specific sufficient conditions for the primal-dual active set step to have a descent property are given. [ABSTRACT FROM AUTHOR]