AMS Subj. Classification: 49J15, 49M15
The control problem of minimal time transition between two stationary points are
formulated in a framework of an indirect numerical method. The problem is regularized and
the monotone behavior of the regularisation procedure is investigated. Semi-smooth Newton
method applied on the regularized problems converge superlinearly and usually produce a very
accurate solution. Differently from other methods, this one does not need a-priory knowledge of
the...
We consider optimal control problems for the bidomain equations of cardiac electrophysiology together with two-variable ionic models, the Rogers–McCulloch model. After ensuring the existence of global minimizers, we provide a rigorous proof for the system of first-order necessary optimality conditions. The proof is based on a stability estimate for the primal equations and an existence theorem for weak solutions of the adjoint system.
Semi-smooth Newton methods for elliptic equations with gradient constraints are investigated. The one- and multi-dimensional cases are treated separately. Numerical examples illustrate the approach and as well as structural features of the solution.
The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented lagrangians allows to apply an infinite-dimensional...
Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an estimate for the penalized...
In this paper sufficient second order optimality conditions for optimal control problems
subject to stationary variational inequalities of obstacle type are derived. Since
optimality conditions for such problems always involve measures as Lagrange multipliers,
which impede the use of efficient Newton type methods, a family of regularized problems is
introduced. Second order sufficient optimality conditions are derived for the regularized
problems...
The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier–Stokes equations, semilinear wave equations and reaction diffusion systems are given.
Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with...
An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived.
Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained....
Output least squares stability for the diffusion coefficient in an elliptic equation in dimension two is analyzed. This guarantees Lipschitz stability of the solution of the least squares formulation with respect to perturbations in the data independently of their attainability. The analysis shows the influence of the flow direction on the parameter to be estimated. A scale analysis for multi-scale resolution of the unknown parameter is provided.
Semi-smooth Newton methods are analyzed for the Signorini problem. A proper regularization is introduced which guarantees that the semi-smooth Newton method is superlinearly convergent for each regularized problem. Utilizing a shift motivated by an augmented Lagrangian framework, to the regularization term, the solution to each regularized problem is feasible. Convergence of the regularized problems is shown and a report on numerical experiments is given.
Lagrangian and augmented Lagrangian methods for nondifferentiable
optimization problems that arise from the total bounded variation formulation
of image restoration problems are analyzed. Conditional convergence of the
Uzawa algorithm and unconditional convergence of the first order augmented
Lagrangian schemes are discussed. A Newton type method based on an active
set strategy defined by means of the dual variables is developed and
analyzed. Numerical examples for blocky signals and images perturbed
by...
Output least squares stability for the diffusion coefficient in an elliptic equation in dimension
two is analyzed. This guarantees Lipschitz stability of the solution of the least squares
formulation with respect to perturbations in the data independently of their attainability.
The analysis shows the influence of the flow direction on the parameter to be estimated.
A scale analysis for multi-scale resolution of the unknown parameter is provided.
The receding horizon control strategy for
dynamical systems posed in infinite dimensional spaces is analysed. Its
stabilising property is verified provided control
Lyapunov functionals are used as terminal penalty functions.
For closed loop dissipative systems the terminal penalty can
be chosen as quadratic functional. Applications to the Navier–Stokes
equations, semilinear wave equations and reaction diffusion systems are given.
The construction of reduced order models for dynamical systems using
proper orthogonal decomposition (POD) is based on the information
contained in so-called snapshots. These provide the spatial
distribution of the dynamical system at discrete time instances.
This work is devoted to optimizing the choice of these time
instances in such a manner that the error between the POD-solution
and the trajectory of the dynamical system is minimized. First and
second order optimality systems are given. Numerical...
In this paper sufficient second order optimality conditions for optimal control problems
subject to stationary variational inequalities of obstacle type are derived. Since
optimality conditions for such problems always involve measures as Lagrange multipliers,
which impede the use of efficient Newton type methods, a family of regularized problems is
introduced. Second order sufficient optimality conditions are derived for the regularized
problems...
The 2D-Signorini contact problem with Tresca and Coulomb friction
is discussed in infinite-dimensional Hilbert spaces. First, the
problem with given friction (Tresca friction) is considered. It
leads to a constraint non-differentiable minimization problem. By
means of the Fenchel duality theorem this problem can be transformed
into a constrained minimization involving a smooth functional. A
regularization technique for the dual problem motivated by augmented
Lagrangians allows to apply an...
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