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Existence and uniqueness of solutions to the Navier-Stokes equations in dimension two with forces in the space Lq((0, T); W−1,p(Ω)) for p and q in appropriate parameter ranges are proven. The case of spatially measured-valued forces is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions for any 1 < p, q < ∞. [ABSTRACT FROM AUTHOR]

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 directly the control but we assume some constraints on them. We prove the well-posedness of the control problem and derive necessary and sufficient conditions for local optimality of the controls. [ABSTRACT FROM AUTHOR]

Existence and uniqueness of solutions to the Navier-Stokes equations in dimension two with forces in the space Lq((0, T); W−1,p(Ω)) for p and q in appropriate parameter ranges are proven. The case of spatially measured-valued forces is included. For the associated Stokes equation the well-posedness results are verified in arbitrary dimensions for any 1 < p, q < ∞. [ABSTRACT FROM AUTHOR]

The approximation of the value function associated to a stabilization problem formulated as optimal control problem for the Navier-Stokes equations in dimension three by means of solutions to generalized Lyapunov equations is proposed and analyzed. The specificity, that the value function is not differentiable on the state space must be overcome. For this purpose a new class of generalized Lyapunov equations is introduced. Existence of unique solutions to these equations is demonstrated. They provide the basis for feedback operators, which approximate the value function, the optimal states and controls, up to arbitrary order. [ABSTRACT FROM AUTHOR]

The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated. [ABSTRACT FROM AUTHOR]

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 sufficient conditions for local optimality of the controls. Finally, under a second order condition, we prove rates of the optimal states with respect to small perturbations in the data of the control problem. [ABSTRACT FROM AUTHOR]

A receding horizon framework for stabilization of a class of infinite-dimensional controlled systems is presented. No terminal costs or constraints are used to ensure asymptotic stability of the controlled system. The key assumption is a stabilizability assumption, which can be guaranteed, for example, for the Burgers equations with periodic and with homogeneous Neumann boundary conditions. Numerical experiments validate the theoretical results. Comparisons to the case with terminal penalties acting as control Lyapunov functions are included. [ABSTRACT FROM AUTHOR]

Feedback control for the monodomain equations is studied. The dynamics of interest are governed by a coupled PDE-ODE reaction diffusion system with nonmonotone nonlinearity of FitzHugh-Nagumo type. A localized distributed control is used to locally stabilize the nonlinear system. This is achieved by a Riccati-based feedback law, determined by the linearized system. It is shown that the Riccati equation corresponding to the PDE variable suffices for exponential stabilization of the linearized PDE-ODE system. The theoretical findings are underlined by several numerical examples. [ABSTRACT FROM AUTHOR]

general framework for calculating shape derivatives for domain optimization problems with partial differential equations as constraints is presented. The first order approximation of the cost with respect to the geometry perturbation is arranged in an efficient manner that allows the computation of the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In doing so, the state variable is only required to be Lipschitz continuous with respect to geometry perturbations. Application to shape optimization with the Navier-Stokes equations as PDE constraint is given. [ABSTRACT FROM AUTHOR]

A general framework for calculating shape derivatives for optimization problems with partial differential equations as constraints is presented. The proposed technique allows to obtain the shape derivative of the cost without the necessity to involve the shape derivative of the state variable. In fact, the state variable is only required to be Lipschitz continuous with respect to the geometry perturbations. Applications to inverse interface problems, and shape optimization for elliptic systems and the Navier-Stokes equations are given. [ABSTRACT FROM AUTHOR]

A reduced-order method for optimal control problems in infinite dimensions based on approximate inertial manifolds is developed. Convergence of the cost, optimal controls, and optimal states of the finite dimensional, reduced-order, optimal control problems to the original optimal control problem is analyzed. Special attention is given to the particular case when the dynamics are described by the Navier-Stokes equations in dimension two. [ABSTRACT FROM AUTHOR]

Second order methods for open loop optimal control problems governed by the twodimensional instationary Navier-Stokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasi-Newton methods as well as various variants of SQP methods are developed for applications to optimal flow control, and their complexity in terms of system solves is discussed. Local convergence and rate of convergence are proved. A numerical example illustrates the feasibility of solving optimal control problems for twodimensional instationary Navier Stokes equations by second order numerical methods in a standard workstation environment. [ABSTRACT FROM AUTHOR]