Optimal Boundary Control Problem Research Articles

Improved boundary regularity for a Stokes-Lamé system

<p style='text-indent:20px;'>This paper recalls a partial differential equations system, which is the linearization of a recognized fluid-elasticity interaction three-dimensional model. A collection of regularity results for the traces of the fluid variable on the interface between the body and the fluid is established, in the case a suitable boundary dissipation is present. These regularity estimates are geared toward ensuring the well-posedness of the Riccati equations which arise from the associated optimal boundary control problems on a finite as well as infinite time horizon. The theory of operator semigroups and interpolation provide the main tools.</p>

On the Tykhonov Well-Posedness of an Antiplane Shear Problem

We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by $\cP$. We associated to problem $\cP$ a boundary optimal control problem, denoted by $\cQ$. For Problem $\cP$ we introduce the concept of well-posedness and for Problem $\cQ$ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem $\cP$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem $\cQ$ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results.

Optimal boundary control problem for ill‐posed elliptic equation in domains with rugous boundary. Existence result and optimality conditions

SummaryThe main purpose is the study of optimal control problem in a domain with rough boundary for the mixed Dirichlet‐Neumann boundary value problem for the strongly nonlinear elliptic equation with exponential nonlinearity. A density of surface traction u acting on a part of rough boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable functional class in which we look for solutions, we prove the consistency of the original optimal control problem and show that it admits a unique optimal solution. Then we derive a first‐order optimality system assuming the optimal solution is slightly more regular.

A Finite Element Method for Elliptic Dirichlet Boundary Control Problems

Abstract We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in H 1 / 2 ⁢ ( Γ ) {H^{1/2}(\Gamma)} . To avoid computing the latter norm numerically, we realize it using the H 1 ⁢ ( Ω ) {H^{1}(\Omega)} norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the H 1 {H^{1}} and L 2 {L_{2}} norm are proven. We also consider and analyze the case of control constrained problems.

The Continuous Classical Boundary Optimal Control Vector Governing by Triple Linear Partial Differential Equations of Parabolic Type

In this paper, the continuous classical boundary optimal control problem (CCBOCP) for triple linear partial differential equations of parabolic type (TLPDEPAR) with initial and boundary conditions (ICs & BCs) is studied. The Galerkin method (GM) is used to prove the existence and uniqueness theorem of the state vector solution (SVS) for given continuous classical boundary control vector (CCBCV). The proof of the existence theorem of a continuous classical boundary optimal control vector (CCBOCV) associated with the TLPDEPAR is proved. The derivation of the Fréchet derivative (FrD) for the cost function (CoF) is obtained. At the end, the theorem of the necessary conditions for optimality (NCsThOP) of this problem is stated and proved.

Solvability for classical continuous boundary optimal control problem dominating by triple hyperbolic equations

This work concerns with the study of the continuous classical boundary optimal control problem (CCBUOCP) dominating by triple linear hyperbolic (TLH) partial differential equations(TLHPDEQS). The existing theorem for a unique state vector solution(SVES) for the TLH boundary value problem (TLHBVP) as well as for its adjoint triple linear equations (ATLEQ) is proved using the method of Galerkin (MG) when the continuous classical boundary control vector (CCBUCV) is known. The existing theorem of a continuous classical boundary optimal control vector (CCBUOCV) dominating by the TLHBVP is proved. The directional derivative (DIDE) for the cost functional (CFu) is derived. Finally the theorem for necessity conditions for optimality (THNCO) of the problem is proved.

Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems

We consider a steady-state heat conduction problem in a multidimensional bounded domain $$\Omega $$ for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion $$\Gamma _1$$ of the boundary and a constant heat flux q in the remaining portion $$\Gamma _2$$ of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary $$\Gamma _1$$ with heat transfer coefficient $$\alpha $$ and external temperature b. We obtain explicitly, for a rectangular domain in $${\mathbb {R}}^{2}$$ , an annulus in $${\mathbb {R}}^{2}$$ and a spherical shell in $${\mathbb {R}}^{3}$$ , the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on $$\Gamma _1$$ converge, when $$\alpha \rightarrow \infty $$ , to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on $$\Gamma _1$$ . Also, we analyze the order of convergence in each case, which turns out to be $$1/\alpha $$ being new for these kind of elliptic optimal control problems.

On Tikhonov regularization of optimal Neumann boundary control problem for an ill-posed strongly nonlinear elliptic equation with an exponential type of non-linearity

We discuss the existence of solutions to an optimal control problem for the mixed Dirichlet-Neumann boundary value problem for strongly non-linear elliptic equations with an exponential type of nonlinearity. A density of surface traction $u$ acting on a part of boundary of open domain is taken as a boundary control. The optimal control problem is to minimize the discrepancy between a given distribution $y_d\in L^2(\Omega)$ and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this, we make use of a variant of the classical Tikhonov regularization. We eliminate the differential constraints between control and state and allow such pairs run freely in their respective sets of feasibility by introducing some additional variable which plays the role of ``defect". We show that this special residual function can be determined in a unique way. We introduce a special family of regularized optimization problems and show that each of these problem is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we establish sufficient conditions of the existence of optimal solutions to the given class of nonlinear Dirichlet BVP and propose the way for their approximation.

Optimal control of a fractional Sturm–Liouville problem on a star graph

ABSTRACT This paper is devoted to elliptic fractional boundary value problems of Sturm–Liouville type on an interval and on a general star graph. We first give some existence and uniqueness results on an open bounded real interval, prove the existence of solutions to a quadratic boundary optimal control problem and provide a characterization via optimality conditions. We then investigate the analogous problems for a fractional Sturm–Liouville problem on a star graph with mixed Dirichlet and Neumann boundary controls.

On optimal boundary control of Ericksen–Leslie system in dimension two

In this paper, we consider the boundary value problem of a simplified Ericksen-Leslie system in dimension two with non-slip boundary condition for the velocity field $u$ and time-dependent boundary condition for the director field $d$ of unit length. For such a system, we first establish the existence of a global weak solution that is smooth away from finitely many singular times, then establish the existence of a unique global strong solution that is smooth for $t>0$ under the assumption that the image of boundary data is contained in the hemisphere $\mathbb S^2_+$. Finally, we apply these theorems to the problem of optimal boundary control of the simplified Ericksen-Leslie system and show both the existence and a necessary condition of an optimal boundary control.

Global superconvergence analysis of a nonconforming FEM for Neumann boundary OCPs with elliptic equations

In this paper, a nonconforming finite element method (FEM) is proposed for the Neumann type boundary optimal control problems (OCPs) governed by elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection through the adjoint state. Some superclose behaviors are derived by full use of the distinguish characters of this element, such as the interpolation operator equals to the Ritz projection, and the consistency error is higher than its interpolation error in the broken energy norm. After that, the global superconvergence results are obtained by employing the so-called post-interpolation technique. Finally, some numerical results are provided to verify the theoretical analysis.

Optimal boundary control for steady motions of a self-propelled body in a Navier-Stokes liquid

Consider a rigid body 𝒮 ⊂ ℝ3 immersed in an infinitely extended Navier-Stokes liquid and the motion of the body-fluid interaction system described from a reference frame attached to 𝒮. We are interested in steady motions of this coupled system, where the region occupied by the fluid is the exterior domain Ω = ℝ3 \ 𝒮. This paper deals with the problem of using boundary controls v*, acting on the whole ∂Ω or just on a portion Γ of ∂Ω, to generate a self-propelled motion of 𝒮 with a target velocity V (x) := ξ + ω × x and to minimize the drag about 𝒮. Firstly, an appropriate drag functional is derived from the energy equation of the fluid and the problem is formulated as an optimal boundary control problem. Then the minimization problem is solved for localized controls, such that supp v*⊂ Γ, and for tangential controls, i.e, v*⋅ n|∂Ω = 0, where n is the outward unit normal to ∂Ω. We prove the existence of optimal solutions, justify the Gâteaux derivative of the control-to-state map, establish the well-posedness of the corresponding adjoint equations and, finally, derive the first order optimality conditions. The results are obtained under smallness restrictions on the objectives |ξ| and |ω| and on the boundary controls.

Mathematical modeling in task of optimal managing by savings in the middle class

Consumption and saving balance issues in the middle class, as the most economically active cluster of society, are the subject of extensive expert discussion and require systematic government regulation. The present paper deals with mathematical model of middle class differentiation by savings which dynamics is described by initial boundary value problem with a parabolic equation. This study aims to investigate a case of savings regulation by changing of non-savers share. The paper presents formulation this problem as an optimal boundary control problem for distributed system of savings. Based on the Lagrange principle, the necessary conditions for the solvability of the problem are derived in the form of an optimization system. The optimal control law establishing the relationship between the non-savers share and the structure of the middle class in terms of savings is obtained. The paper also considers an approach to the numerical implementation of the optimal control model in the Comsol Multiphysics simulation software. An example of model calculating for a region of Russia based on real data is given.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations

<p style='text-indent:20px;'>This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.</p>

A policy iteration algorithm for the American put option and free boundary control problems

We develop a solution method for American put options that directly employs the policy iteration principle of dynamic programming. The method iteratively improves exercise policies, obtains monotonically increasing value functions and converges quadratically under reasonable assumptions. We present a numerical implementation that exhibits these features. The same principle is also applied to obtain a monotonically improving policy iteration scheme for general free boundary optimal control problems.

Asymptotic analysis of a boundary optimal control problem on a general branched structure

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar‐type (non‐smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch.

Meshless methods for solving Dirichlet boundary optimal control problems governed by elliptic PDEs

In this paper, two meshless schemes are proposed for solving Dirichlet boundary optimal control problems governed by elliptic equations. The first scheme uses radial basis function collocation method (RBF-CM) for both state equation and adjoint state equation, while the second scheme employs the method of fundamental solution (MFS) for the state equation when it has a zero source term, and RBF-CM for the adjoint state equation. Numerical examples are provided to validate the efficiency of the proposed schemes.

Optimal valve closure operations for pressure suppression in fluid transport pipelines

When a valve is suddenly closed in fluid transport pipelines, a pressure surge or shock is created along the pipeline due to the momentum change. This phenomenon, called hydraulic shock, can cause major damage to the pipelines. In this paper, we introduce a hyperbolic partial differential equation &#x0028 PDE &#x0029 system to describe the fluid flow in the pipeline and propose an optimal boundary control problem for pressure suppression during the valve closure. The boundary control in this system is related to the valve actuation located at the pipeline terminus through a valve closing model. To solve this optimal boundary control problem, we use the method of lines and orthogonal collocation to obtain a spatial-temporal discretization model based on the original pipeline transmission PDE system. Then, the optimal boundary control problem is reduced to a nonlinear programming &#x0028 NLP &#x0029 problem that can be solved using nonlinear optimization techniques such as sequential quadratic programming &#x0028 SQP &#x0029. Finally, we conclude the paper with simulation results demonstrating that the full parameterization &#x0028 FP &#x0029 method eliminates pressure shock effectively and costs less computation time compared with the control vector parameterization &#x0028 CVP &#x0029 method.

On optimal boundary control problem for a strongly degenerate elliptic equation

In this paper we study an optimal control problem for a linear boundary value problem with strongly degenerate coefficient in the main part of the elliptic operator and with the Neumann boundary control. Given a cost functional, the objective is to provide the well-posedness analysis of the corresponding optimal control problem, prove existence of the optimal solutions and propose the scheme for their approximation.