Solving Nonlinear Inverse Problems Based on the Regularized Modified Gauss–Newton Method

Doklady Mathematics - A nonlinear operator equation is investigated in the case when the Hadamard correctness conditions are violated. A two-stage method is proposed for constructing a stable...     

Modified Legendre–Gauss–Radau Collocation Method for Optimal Control Problems with Nonsmooth Solutions

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre–Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh interval. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Furthermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.

     

A Primal-Dual Active Set Method for Bilaterally Control Constrained Optimal Control of the Navier–Stokes Equations

Abstract

This paper is devoted to the analysis and numerical solution of distributed optimal control of the Navier–Stokes equations in presence of bilateral pointwise control constraints. The analysis of the problem involves the proof of existence of an optimal solution, as well as the presentation of necessary and sufficient conditions for optimality. For the numerical solution of the problem we apply a primal-dual active set strategy and show global and local convergence properties of the method. Finally, some numerical experiments, which illustrate the performance of the method, are discussed.     

Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

There are very few results about analytic solutions of problems of optimal control with minimal L norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L norm that steers the system to the target.We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.     

Semi-Smooth Newton Methods for Optimal Control of the Dynamical Lamé System with Control Constraints

Optimal control problems governed by the dynamical Lamé system with additional constraints on the controls are analyzed. Different types of control action are considered: distributed, Neumann boundary and Dirichlet boundary control. To treat the inequality control constraints semi-smooth Newton methods are applied and their convergence is analyzed. Although semi-smooth Newton methods are widely studied in the context of PDE-constrained optimization little has been done in the context of the dynamical Lamé system. The novelty of the article is the proof that in case of distributed and Neumann boundary control the Newton method converges superlinearly. In case of Dirichlet control superlinear convergence is shown for a strongly damped Lamé system. The results are an extension of [14 A. Kröner , K. Kunisch , and B. Vexler ( 2011 ). Semi-smooth Newton methods for optimal control of the wave equation with control constraints . SIAM J. Control Optim. 49 : 830 – 858 .[Crossref], [Web of Science ®] , [Google Scholar]], where optimal control problems of the classical wave equation are considered. The control problems are discretized by finite elements and numerical examples are presented.     

Optimal L ∞(L 2)-Error Estimates for the DG Method Applied to Nonlinear Convection–Diffusion Problems with Nonlinear Diffusion

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ^∞(L ²)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular.     

Hölder Continuity and Optimal Control for Nonsmooth Elliptic Problems

The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.     

Dissipative Control Systems and Disturbance Attenuation for Nonlinear H ∞ Problems

We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of an epicontingent solution, or a viscosity supersolution to a partial differential equation called Isaacs' equation. Links between supersolutions and epicontingent solutions to Isaacs' equation are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H ^∞ problem from contingent formulation of Isaacs' equation. Accepted 20 January 1998     

Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

Finite Difference Method for an Optimal Control Problem for a Nonlinear Time-dependent Schrödinger Equation

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coe?cient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved.     

Optimal Control for non–isothermal crystallization of polymers

In this paper an optimal control problem for polymer crystallization is investigated. The crystallization is described by a non–isothermal Avrami–Kolmogorov model. The boundary temperature serves as control variable. The cost functional takes the spatial variation of the crystallinity and the final degree of crystallization into account. The resulting boundary control problem for a parabolic equation coupled with two ordinary differential equations is treated by an adjoint variable approach. A steepest descent algorithm is used to solve the problem numerically. Simulations illustrate the applicability and performance of the optimization algorithm. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Grönwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.     

A Homotopy Interior point Method for Nonlinear Complementarity Problems

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Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods

In this paper, algorithms of solving an inverse source problem for systems of production–destruction equations are considered. Numerical schemes that are consistent to satisfy Lagrange’s identity for solving direct and adjoint problems are constructed. With the help of adjoint equations, a sensitivity operator with a discrete analog is constructed. It links perturbations of the measured values with those of the sought-for model parameters. This operator transforms the inverse problem to a quasilinear system of equations and allows applying Newton–Kantorovich methods to it. A numerical comparison of gradient algorithms based on consistent and inconsistent numerical schemes and a Newton–Kantorovich algorithm applied to solving an inverse source problem for a nonlinear Lorenz model is done.     

RESEARCH ANNOUNCEMENTS──Convergence Analysis of Quasi－Newton Methods for Nonlinear Programming Problems

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Approximation and Optimal Control of Dissipative Solutions to the Ericksen–Leslie System

Abstract

We analyze the Ericksen–Leslie system equipped with the Oseen–Frank energy in three space dimensions. Recently, the author introduced the concept of dissipative solutions. These solutions show several advantages in comparison to the earlier introduced measure-valued solutions. In this article, we argue that dissipative solutions can be numerically approximated by a relatively simple scheme, which fulfills the norm-restriction on the director in every step. We introduce a semi-discrete scheme and derive an approximated version of the relative-energy inequality for solutions of this scheme. Passing to the limit in the semi-discretization, we attain dissipative solutions. Additionally, we introduce an optimal control scheme, showing the existence of an optimal control and a possible approximation strategy. We prove that the cost functional is lower semi-continuous with respect to the convergence of this approximation and argue that an optimal control is attained in the case that there exists a solution admitting additional regularity.