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The minimization of drag functional for the stationary, isothermal, compressible Navier-Stokes equations (N-S-E) in three spatial dimensions is considered. In order to establish the existence of an optimal shape the general result [26] on compactness of families of generalized solutions to N-S-E is applied. The family of generalized solutions to N-S-E is constructed over a family of admissible domains $$ \mathcal{U}_{ad} $$. Any admissible domain Ω = B\S contains an obstacle S, e.g., a wing profile. Compactness properties of the family of admissible domains are imposed. It turns out that we require the compactness of the family of admissible domains with respect to the Hausdorff metrics as well as in the sense of Kuratowski-Mosco. The analysis is performed for the range of adiabatic ratio γ > 1 in the pressure law p(ρ) = grγ and it is based on the technique proposed in [24] for the discretized N-S-E. [ABSTRACT FROM AUTHOR]

This paper deals with the control of a moving dynamical domain in which a non cylindrical dynamical boundary value problem is considered. We consider weak Eulerian evolution of domains through the convection of a measurable set by (non necessarily smooth) vector field V. We introduce the concept of tubes by "product space" and we show a closure result leading to existence results for a variational shape principle. We illustrate this by new results: heat equation and wave equation in moving domains with various boundary conditions and also the geodesic characterisation for two Eulerian shape metrics leading to the Euler equation through the transverse field considerations. We consider the non linear Hamilton-Jacobi like equation associated with level set parametrization of the moving domain and give new existence result of possible topological change in finite time in the solution. [ABSTRACT FROM AUTHOR]

We investigate optimal control problems with vector-valued controls. As model problem serve the optimal distributed control of the instationary Navier-Stokes equations. We study pointwise convex control constraints, which is a constraint of the form u(x, t) ∈ U(x, t) that has to hold on the domain Q. Here, U is an set-valued mapping that is assumed to be measurable with convex and closed images. We establish first-order necessary as well as second-order sufficient optimality conditions. And we prove regularity results for locally optimal controls. [ABSTRACT FROM AUTHOR]

Quantum control is traditionally expressed through bilinear models and their associated Lie algebra controllability criteria. But, the first order approximation are not always sufficient and higher order developments are used in recent works. Motivated by these applications, we give in this paper a criterion that applies to situations where the evolution operator is expressed as sum of possibly non-linear real functionals of the same control that multiplies some time independent (coupling) operators. [ABSTRACT FROM AUTHOR]

The feedback stabilization of the Navier-Stokes equations around an unstable stationary solution is related to the feedback stabilization of the Oseen equations (the linearized Navier-Stokes equations about the unstable stationary solution). In this paper we investigate the regularizing properties of feedback operators corresponding to a family of optimal control problems for the Oseen equations. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of determining parameters in nonlinear partial differential equations of hyperbolic type from boundary measurements. In order to investigate the qualitative behavior of this class of identification problems, we analyze the model problem of identifying c in the nonlinear wave equation dtt − (c(dx)dx)x = 0 and discuss stability and identifiability for this problem. Moreover, we derive applicability of these results to material parameter identification in piezoelectricity and provide numerical reconstruction results. [ABSTRACT FROM AUTHOR]

We consider a sequential quadratic programming (SQP) method for the solution of an optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions arise from conductive-radiative heat transfer in non-convex domains. After stating firstand second-order optimality conditions, we introduce the SQP algorithm that uses an active set method to solve the linear quadratic subproblems arising in each step. The corresponding optimality systems are discretized by linear finite elements, using a partly exact summarized midpoint rule for the discretization of the nonlocal radiation interface conditions. The paper ends with some numerical results demonstrating the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper discusses semi-smooth Newton methods for solving nonlinear non-smooth equations in Banach spaces. Such investigations are motivated by complementarity problems, variational inequalities and optimal control problems with control or state constraints, for example. The function F(x) for which we desire to find a root is typically Lipschitz continuous but not C1 regular. The primal-dual active set strategy for the optimization with the inequality constraints is formulated as a semi-smooth Newton method. Sufficient conditions for global convergence assuming diagonal dominance are established. Globalization strategies are also discussed assuming that the merit function F(x)2 has appropriate descent directions. [ABSTRACT FROM AUTHOR]

We discuss optimal and model predictive control techniques applied to the Boussinesq approximation of the Navier-Stokes system. We focus on mathematical modeling, discuss possible control scenarios, and provide a concise description of the numerical implementation. Furthermore, several numerical examples are provided. [ABSTRACT FROM AUTHOR]

Efficient numerical techniques for the solution of constrained optimal control problems for the nonlinear heat equation are considered. The nonlinearity in the governing equation is due to the boundary conditions which cover the Boltzmann radiation boundary condition. With respect to numerical algorithms, variants of semismooth Newton methods are proposed which allow a convergence analysis in function space. For the latter aspect the concept of generalized (Newton, or slant) differentiability is invoked. The paper ends with a comparison of the proposed algorithms among each other and with a sequential quadratic programming method. [ABSTRACT FROM AUTHOR]

The present article is concerned with boundary control stabilization of semi-linear parabolic systems which are unstable if uncontrolled. A particular emphasis is put on Dirichlet control in that context. We investigate two different numerical approaches to solve these problems. The first approach relies on the extension method proposed by A.V. Fursikov where the considered partial differential equations are first solved on an extended domain with suitable initial value leading a stable solution. The needed control is then de-fined as an appropriate trace of this solution. The second approach relies on the formulation of the stabilization problem as an optimization problem with constraints based on partial differential equations. We address the numerical issues related to both class of approaches toward a comparison of their specific stabilization properties. The considered methodology is applied to the solution of test parabolic problems assuming linear and nonlinear models. [ABSTRACT FROM AUTHOR]

In active flood hazard mitigation, lateral flow withdrawal is used to reduce the impact of flood waves in rivers. Through emergency side channels, lateral outflow is generated. The optimal outflow controls the flood in such a way that the cost of the created damage is minimized. The flow is governed by a networked system of nonlinear hyperbolic partial differential equations, coupled by algebraic node conditions. Two types of integrals appear in the objective function of the corresponding optimization problem: Boundary integrals (for example, to measure the amount of water that flows out of the system into the floodplain) and distributed integrals. For the evaluation of the derivative of the objective function, we introduce an adjoint backwards system. For the numerical solution we consider a discretized system with a consistent discretization of the continuous adjoint system, in the sense that the discrete adjoint system yields the derivatives of the discretized objective function. Numerical examples are included. [ABSTRACT FROM AUTHOR]

We develop an adaptive finite element method for a class of distributed optimal control problems with control constraints. The method is based on a residual-type a posteriori error estimator and incorporates data oscillations. The analysis is carried out for conforming P1 approximations of the state and the co-state and elementwise constant approximations of the control and the co-control. We prove convergence of the error in the state, the co-state, the control, and the co-control. Under some additional non-degeneracy assumptions on the continuous and the discrete problems, we then show that an error reduction property holds true at least asymptotically. The analysis uses the reliability and the discrete local efficiency of the a posteriori estimator as well as quasi-orthogonality properties as essential tools. Numerical results illustrate the performance of the adaptive algorithm. [ABSTRACT FROM AUTHOR]

We review recent results on the boundary and interior feedback stabilization of Navier-Stokes equations, d = 2, 3, and provide new ones. [ABSTRACT FROM AUTHOR]

We consider a boundary stabilization problem for the plate equation in a square. The feedback law gives the bending moment on a part of the boundary as function of the velocity field of the plate. The main result of the paper asserts that the obtained closed loop system is exponentially stable if and only if the controlled part of the boundary contains a vertical and a horizontal part of non-zero length (the geometric optics condition introduced by Bardos, Lebeau and Rauch in [2] for the wave equation is thus not necessary in this case). The proof of the main result uses the methodology introduced in Ammari and Tucsnak [1], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system. The second essential ingredient of the proof is an observability inequality recently proved by Ramdani, Takahashi, Tenenbaum and Tucsnak [7] [ABSTRACT FROM AUTHOR]