Neumann Boundary Control Research Articles

Rapid exponential stabilization for a linear Korteweg-de Vries equation

We consider a control system for a Korteweg-de Vries equation with homogeneous Dirichlet boundary conditions and Neumann boundary control. We address the rapid exponential stabilization problem. More precisely, we build some feedback laws forcing the solutions of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates. We also perform some numerical computations in order to illustrate this theoretical result.

Local A Posteriori Error Estimates for Convex Boundary Control Problems

In this paper, we derive some improved a posteriori error estimates for finite element approximation of Neumann boundary control problems. We first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems. We then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications.

Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain

It is known that the linear Korteweg–de Vries (KdV) equation with homogeneous Dirichlet boundary conditions and Neumann boundary control is not controllable for some critical spatial domains. In this paper, we prove in these critical cases, that the nonlinear KdV equation is locally controllable around the origin provided that the time of control is large enough. It is done by performing a power series expansion of the solution and studying the cascade system resulting of this expansion.

On saturation effects in the Neumann boundary control of elliptic optimal control problems

AbstractA Neumann boundary control problem for a linear‐quadratic elliptic optimal control problem in a polygonal domain is investigated. The main goal is to show an optimal approximation order for discretized problems after a postprocessing process. It turns out that two saturation processes occur: The regularity of the boundary data of the adjoint state is limited if the largest angle of the polygon is at least 2π /3. Moreover, piecewise linear finite elements cannot guarantee the optimal order, if the largest angle of the polygon is greater than π /2. We will derive error estimates of order hσ with σ ∈ [3/2, 2] depending on the largest angle and properties of the finite elements. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Neumann boundary control of semilinear wave equation

The initial state (x,x) ∈ H(Ω) × L(Ω) is fixed. The importance of the problem with control on the boundary appears in optimal control theory see e.g. [1]. Our aim is to extend the type of nonlinearity in (1). To underline that we simply add a new function f to well known type nonlinearity Φ. We shall study (1) by variational method, i.e. we shall consider (1) as the Euler—Lagrange equation of the action functional:

Error estimates for the numerical approximation of Neumann control problems

We continue the discussion of error estimates for the numerical analysis of Neumann boundary control problems we started in Casas et al. (Comput. Optim. Appl. 31:193-219, 2005). In that paper piecewise constant functions were used to approximate the control and a convergence of order O(h) was obtained. Here, we use continuous piecewise linear functions to discretize the control and obtain the rates of convergence in L2(Γ). Error estimates in the uniform norm are also obtained. We also discuss the approach suggested by Hinze (Comput. Optim. Appl. 30:45-61, 2005) as well as the improvement of the error estimates by making an extra assumption over the set of points corresponding to the active control constraints. Finally, numerical evidence of our estimates is provided.

The role of an L^2(\Omga­)-energy estimate in the theories of uniform stabilization and exact controllability for Schrödinger equations with Neumann boundary control

 The present paper deals with (linear) Schrödinger equations, of very general form, which are deffined on a bounded domain ­ \Omega \subset Rn. With focus on these dynamics, we shall then discuss and analyze the specific and foundational topic of a-priori energy identities, with the goal of deriving control-theoretic implications. These will include the issue of optimal regularity, as well as the problems of exact controllability (by open loop controls) and of uniform stabilization (by closed loop feedback controls).

Exact Controllability for the Time Dependent Transport Equation

Exact Controllability for the Time Dependent Transport Equation

Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems

In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear-quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.

Adjoint concepts for the optimal control of Burgers equation

Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradient-based optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.

Uniform controllability of semidiscrete approximations of parabolic control systems

Controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model. In the present paper, under the main assumptions that the discretized semigroup is uniformly analytic, and that the control operator is mildly unbounded, we prove that the semidiscrete approximation models are uniformly controllable. Moreover, we provide a computationally efficient way to compute the approximation controls. An example of application is implemented for the one- and two-dimensional heat equation with Neumann boundary control.

Optimal boundary control of hyperbolic equations with pointwise state constraints

In this paper, we consider dynamic optimization problems for hyperbolic systems with boundary controls and pointwise state constraints. In contrast to parabolic dynamics, such systems have not been sufficiently studied in the literature. The reason is the lack of regularity in the case of hyperbolic dynamics. We present necessary optimality conditions for both Neumann and Dirichlet boundary control problems and discuss differences and relationships between them.

Adaptive Wavelet Methods for Linear-Quadratic Elliptic Control Problems: Convergence Rates

We propose an adaptive algorithm based on wavelets for the fast numerical solution of control problems governed by elliptic boundary value problems with distributed or Neumann boundary control. A quadratic cost functional that may involve fractional Sobolev norms of the state and the control is to be minimized subject to linear constraints in weak form. Placing the problem into the framework of (biorthogonal) wavelets allows us to formulate the functional and the constraints equivalently in terms of $\ell_2$-norms of wavelet expansion coefficients and constraints in the form of an $\ell_2$ automorphism. The resulting first order necessary conditions are then derived as a (still infinite) system in $\ell_2$. Applying the machinery developed in [A. Cohen, W. Dahmen, and R. DeVore, Math. Comp., 70 (2001), pp. 27--75; A. Cohen, W. Dahmen, and R. DeVore, Found. Comput. Math., 2 (2002), pp. 203--245], we propose an adaptive method which can be interpreted as an inexact gradient descent method, where in each iteration step the primal and the adjoint system need to be solved up to a prescribed accuracy. Convergence of the adaptive algorithm is proved. In addition, we show that the adaptive algorithm is asymptotically optimal, that is, the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

Neumann Boundary Control of Hyperbolic Equations with Pointwise State Constraints

We consider optimal control problems for hyperbolic equations with controls in Neumann boundary conditions with pointwise constraints on the control and state functions. Focusing on the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the form of a pointwise Pontryagin maximum principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.

Well-Posedness of Boundary Control Systems

Continuity of the input/output map for boundary control systems is shown through the system transfer function. Our approach transforms the question of continuity of the input/output map of a boundary control system to uniform boundedness of the solution to a related elliptic problem. This is shown for a class of boundary control systems with Dirichlet, Neumann, or Robin boundary control.

Mesh Adaptation for Stationary Flow Control

We consider local mesh adaptation for stationary flow control problems. The state equation is given by the incompressible Navier-Stokes equations with Neumann boundary control. As an example, the functional to be minimized is the drag coefficient of an immersed body. We use finite elements on locally refined meshes to discretize the first order necessary conditions based on the Lagrangian. An a posteriori error estimate is derived which is directly related to the control problem. It is used to successively enrich the finite element space until the computed solution satisfies prescribed accuracy requirements.

A Posteriori Error Estimates for Convex Boundary Control Problems

In this paper, we present an a posteriori error analysis for the finite element approximation of convex optimal Neumann boundary control problems. We derive a posteriori error estimates for both the state and the control approximation, first on polygonal domains and then on Lipschitz piecewise C2 domains. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximation schemes for the control problems. Explicit estimates are shown for some model problems that frequently appear in applications.

On global stabilization of Burgers’ equation by boundary control

Although often referred to as a one-dimensional “cartoon” of Navier–Stokes equation because it does not exhibit turbulence, the Burgers equation is a natural first step towards developing methods for control of flows. Recent references include Burns and Kang [Nonlinear Dynamics 2 (1991) 235–262], Choi et al. [J. Fluid Mech. 253 (1993) 509–543], Ito and Kang [SIAM J. Control Optim. 32 (1994) 831–854], Ito and Yan [J. Math. Anal. Appl. 227 (1998) 271–299], Byrnes et al. [J. Dynam. Control Systems 4 (1998) 457–519] and Van Ly et al. [Numer. Funct. Anal. Optim. 18 (1997) 143–188]. While these papers have achieved tremendous progress in local stabilization and global analysis of attractors, the problem of global asymptotic stabilization has remained open. This problem is non-trivial because for large initial conditions the quadratic (convective) term – which is negligible in a linear/local analysis – dominates the dynamics. We derive nonlinear boundary control laws that achieve global asymptotic stability. We consider both the viscous and the inviscid Burgers’ equation, using both Neumann and Dirichlet boundary control. We also study the case where the viscosity parameter is uncertain, as well as the case of stochastic Burgers’ equation. For some of the control laws that would require the measurement in the interior of the domain, we develop the observer-based versions.

On The Observability Inequalities for Exact Controllability of Wave Equations With Variable Coefficients

This paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part, subject to Dirichlet or Neumann boundary controls. The observability inequalities are established by the Riemannian geometry method under some geometric condition for the Dirichlet problem and for the Neumann problem, respectively. Next, a number of nontrivial examples are presented to verify the observability inequality. In particular, a counterexample is given without boundary exact controllability, where the control is exerted on the whole boundary.

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

We introduce a penalized Neumann boundary control approach for solving an optimal Dirichlet boundary control problem associated with the two- or three-dimensional steady-state Navier--Stokes equations. We prove the convergence of the solutions of the penalized Neumann control problem, the suboptimality of the limit, and the optimality of the limit under further restrictions on the data. We describe the numerical algorithm for solving the penalized Neumann control problem and report some numerical results.