Walter Alt

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L 1-norm and with order 1/2 w.r.t. the L 2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

An implicit discretization scheme for linear-quadratic control problems with bang–bang solutions

We analyse an implicit discretization scheme for a class of linear-quadratic optimal control problems. First, we show convergence of order O(h) for the optimal values of the objective function, where h is the mesh size. Under the additional assumption that the optimal control has bang–bang structure we show that the discrete and the continuous controls coincide except on a set of measure O(√h).

Lipschitz stability for elliptic optimal control problems with mixed control-state constraints

A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established.

Mesh-Independence of the Lagrange–Newton Method for Nonlinear Optimal Control Problems and their Discretizations

In a recent paper we proved a mesh-independence principle for Newtons method applied to stable and consistent discretizations of generalized equations. In this paper we introduce a new consistency condition which is easier to check in applications. Using this new condition we show that the mesh-independence principle holds for the Lagrange–Newton method applied to nonlinear optimal control problems with mixed control-state constraints and their discretizations by Eulers method or Ritz type methods.

Regularization and implicit Euler discretization of linear-quadratic optimal control problems with bang-bang solutions

We analyze the implicit Euler discretization for a class of convex linear-quadratic optimal control problems with control appearing linearly. Constraints are defined by lower and upper bounds for the controls, and the cost functional may depend on a regularization parameterźź. Without any structural assumption on the optimal control we prove convergence of orderź1 w.r.t. the mesh size for the discrete optimal values. Under the additional assumption that the optimal control is of bang-bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters. By applying this result to the implicit Euler discretization we improve existing error estimates for discretizations based on the explicit Euler method. Numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang-bang controls.

Improved Error Estimate for an Implicit Discretization Scheme for Linear-Quadratic Control Problems with Bang-Bang Solutions

We analyze an implicit discretization scheme for a class of linear-quadratic optimal control problems without mixed state-control terms. Under the assumption that the optimal control has bang-bang structure we show convergence of the discrete approximation and improve existing error estimates to order \(\mathcal {O}(h)\).

Unrestringierte Optimierungsprobleme: Verfahren

Wie in Kapitel 2 betrachten wir das unrestringierte Optimierungsproblem

Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions

\left( {PU} \right)\,\mathop {\min }\limits_{x \in {R^n}} \,f\left( x \right)