Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L1-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing L2-regularization parameter for the control. The associated convergence rate is estimated.This author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711

Characterization of maximum hands-off control

Maximum hands-off control aims to maximize the length of time over which zero actuator values are applied to a system when executing specified control tasks. To tackle such problems, recent literature has investigated optimal control problems which penalize the size of the support of the control function and thereby lead to desired sparsity properties. This article gives the exact set of necessary conditions for a maximum hands-off optimal control problem using an $L_0$-(semi)norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an $L_0$ cost leads to a sparse control, whereas an $L_1$-relaxation in singular problems leads to a non-sparse solution.Comment: 6 page

Regularization in fractional order Sobolev spaces for a parameter identification problem

In this work we aim the identification of an unknown parameter function in the main part of an elliptic partial differential equation. It is a well known fact, that identification problems are in general ill-posed. Our idea is to apply a Tichonov-type regularization in fractional order Sobolev spaces. For such a problem, we derive existence of solutions and first order necessary conditions. Under a size condition for the regularization parameters, corresponding to the fractional order of differentiation, we are able to derive a second order sufficient condition as well. Fractional order Sobolev norm are challenging to implement. We therefore prove their equivalence to a multilevel based operator norm, for s ∈ [0,3/2), which we can implement. This operator norm and the second order sufficient condition enable us to show superlinear convergence of an SQP-method. In the end we present a numerical example.In dieser Arbeit beabsichtigen wir unbekannte Parameterfunktionen im Hauptteil von elliptischen partiellen Differentialgleichungen zu identifizieren. Es ist eine allgemein bekannte Tatsache, dass solche Identifikationsprobleme im Allgemeinen schlecht gestellte Probleme darstellen. Daher ist unsere Idee, das Problem mit einem Tichonovterm in Sobolevräumen von reellwertiger Ordnung zu regularisieren. Für dieses Problem leiten wir Existenz von Lösungen und notwendige Optimalitätsbedingungen erster Ordnung her. Setzen wir eine Bedingung an den Regularisierungsparameter s, der der Ordnung des Sobolevraums entspricht, voraus, können wir auch hinreichende Optimalitätsbedingungen zweiter Ordnung herleiten. Normen zu Sobolevräumen reellwertiger Ordnung sind sehr schwierig zu implementieren. Daher führen wir eine Multilevel basierte Operatornorm ein, deren Äquivalenz zu Sobolevnormen wir für s∈[0,3/2) beweisen. Diese sind einfacher zu implementieren. Die Operatornorm und die hinreichende Optimalitätsbedingung sind Zutaten mit deren Hilfe wir superlineare Konvergenz eines SQP-Verfahrens zeigen. Am Ende stellen wir ein numerisches Beispiel vor

Optimal trajectory tracking

This thesis investigates optimal trajectory tracking of nonlinear dynamical systems with affine controls. The control task is to enforce the system state to follow a prescribed desired trajectory as closely as possible. The concept of so-called exactly realizable trajectories is proposed. For exactly realizable desired trajectories exists a control signal which enforces the state to exactly follow the desired trajectory. For a given affine control system, these trajectories are characterized by the so-called constraint equation. This approach does not only yield an explicit expression for the control signal in terms of the desired trajectory, but also identifies a particularly simple class of nonlinear control systems. Based on that insight, the regularization parameter is used as the small parameter for a perturbation expansion. This results in a reinterpretation of affine optimal control problems with small regularization term as singularly perturbed differential equations. The small parameter originates from the formulation of the control problem and does not involve simplifying assumptions about the system dynamics. Combining this approach with the linearizing assumption, approximate and partly linear equations for the optimal trajectory tracking of arbitrary desired trajectories are derived. For vanishing regularization parameter, the state trajectory becomes discontinuous and the control signal diverges. On the other hand, the analytical treatment becomes exact and the solutions are exclusively governed by linear differential equations. Thus, the possibility of linear structures underlying nonlinear optimal control is revealed. This fact enables the derivation of exact analytical solutions to an entire class of nonlinear trajectory tracking problems with affine controls. This class comprises mechanical control systems in one spatial dimension and the FitzHugh-Nagumo model.Comment: 240 pages, 36 figures, PhD thesi