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Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated. The controls are subject to a cost functional which promotes sparsity in time. The focus is put on deriving first order optimality conditions. This ...

In this paper, we consider an optimal control problem for the two-dimensional evolutionary Navier--Stokes system. Looking for sparsity, we take controls as functions of time taking values in a space of Borel measures. The cost functional does not involve ...

In this paper, we investigate optimal control problems subject to a semilinear elliptic partial differential equation. The cost functional contains a term that measures the size of the support of the control, which is the so-called $L^0$-norm. We provide ...

An optimal control problem for a semilinear elliptic equation is discussed, where the control appears nonlinearly in the state equation but is not included in the objective functional. The existence of optimal controls is proved by a measurable selection ...

In this paper, we consider an optimal control problem for the two-dimensional stationary Navier--Stokes system. Looking for sparsity, we take Borel measures as controls. We prove the well-posedness of the control problem and derive necessary and ...

In this paper, we analyze optimal control problems of semilinear parabolic equations, where the controls are distributed and depend only on time. Box constraints for the controls are imposed and the cost functional does not involve the control itself, ...

We study a control problem governed by a semilinear parabolic equation. The control is a measure that acts as the kernel of a possibly nonlocal time delay term and the functional includes a nondifferentiable term with the measure norm of the control. ...

We consider bilinear optimal control problems whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth ...

We provide sufficient optimality conditions for optimal control problems with bang-bang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with ...

Stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional. This formulation allows that the optimal control completely shuts ...

Optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed. This choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of ...

We study a Dirichlet optimal control problem for a quasi-linear monotone elliptic equation, the so-called weighted $p$-Laplace problem. The coefficient of the $p$-Laplacian, the weight $u$, we take as a control in $BV(\Omega)\cap L^\infty(\Omega)$. In this ...

The velocity tracking problem for the evolutionary Navier--Stokes equations in three dimensions is studied. The controls are of distributed type and are submitted to bound constraints. The classical cost functional is modified so that a full analysis of the ...

Optimal sparse control problems are considered for the FitzHugh--Nagumo system including the so-called Schlögl model. The nondifferentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the $L^1$-norm of the ...

Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient ...

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 $L^1$-norm of the control as part of ...

Optimal control problems in measure spaces lead to controls that have small support, which is desirable, e.g., in the context of optimal actuator placement. For problems governed by parabolic partial differential equations, well-posedness is guaranteed in the ...

In this paper, we derive some sufficient second order optimality conditions for control problems of partial differential equations (PDEs) when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it. ...

Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions. Optimality conditions are derived and some of the structural properties of ...

In this paper, we study the approximation of a Dirichlet control problem governed by an elliptic equation defined on a curved domain $\Omega$. To solve this problem numerically, it is usually necessary to approximate $\Omega$ by a (typically polygonal) ...