Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

Abstract

In this paper semi-smooth Newton methods for optimal control problems governed by the dynamical Lamé system are considered and their convergence behavior with respect to superlinear convergence is analyzed. Techniques from Kröner, Kunisch, Vexler (2011), where semi-smooth Newton methods for optimal control of the classical wave equation are considered, are transferred to control of the dynamical Lamé system. Three different types of control actions are examined: distributed control, Neumann boundary control and Dirichlet boundary control. The problems are discretized by finite elements and numerical examples are presented. 2 U , subject to y = S(u), y ∈ Y, u ∈ U ad ⊂ U ω with control space U ω , state space Y and α > 0. The control-to-state operator S : U ω → Y is assumed to be affine-linear, the functional G : Y → R to be quadratic. The control and state space and the operators are defined in more detail in the next section. The choice of the control-to-state operator incorporates distributed as well as Neumann and Dirichlet boundary control problems of the dynamical Lamé system which we will consider later. The set of admissible controls is defined by U ad = { u ∈ U ω | u a ≤ u ≤ u b } for given u a , u b ∈ U ω. To specify the control-to-state operator we introduce the dynamical Lamé system. Let Ω ⊂ R d , d = 1, 2, 3, be a bounded domain with C 2-boundary (bounded

Norm-minimal Neumann boundary control of the wave equation

“…Only if ν(T ) ≤ |u a |, an admissible control exists. In [10], the corresponding problems of distributed optimal control subject to box constraints for the control are considered.…”

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

“…In Luo et al, 16 the finite volume element method is applied to the hyperbolic optimal control problems and a priori error estimation is derived. The semi-smooth Newton methods are investigated in Kroner et al, 19 where the control is restricted by pointwise lower and upper bounds. In Li et al, 3 a new central finite difference scheme and in Kunisch and Reiterer, 18 a Gautschi time-stepping scheme is proposed for optimal control problem of wave equations.…”

“…In Li et al, 3 a new central finite difference scheme and in Kunisch and Reiterer, 18 a Gautschi time-stepping scheme is proposed for optimal control problem of wave equations. The semi-smooth Newton methods are investigated in Kroner et al, 19 where the control is restricted by pointwise lower and upper bounds. For state constrained optimal distributed control of the wave equation, the Lavrentiev regularization is considered in Gugat et al 20 In this work, we propose another method on the basis of the shooting method and proper orthogonal decomposition method for solving the distributed optimal control problem (1).…”

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

“…Some more examples of the optimization problem governed by semilinear elliptic equations include, the optimal control of current in a cathodic protection system, the optimal control problem in radiation and scattering, and so forth. As far as numerical approximation of these problems is concerned, finite element method (FEM) has been extensively used (see e.g., ). Most of these publications analyzed the convergence behavior of control error in L 2 ‐norm and established the convergence order for piecewise constant discretization of control.…”