State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

Optimal control problems for an extensible beam equation

Optimal control problems are studied for extensible beam equations. The Gâteaux differentiability of solution mapping on control variables is proved, and the various types of necessary optimality conditions corresponding to the distributive and terminal value observations are established.     

Regularized state-constrained boundary optimal control of the Navier-Stokes equations

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau-Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.     

An initial-boundary value problem for the Maxwell equations

In this paper, by using methods from complex analysis and quaternionic analysis, we investigate an initial-boundary value problem for the Maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases. In addition, by using a similar method, we also discuss an initial-boundary value problem for a hyperbolic complex system of first order equations in R³.     

Optimal control problems for the equation of motion of membrane with strong viscosity

Optimal control problems are studied for the equation of membrane with strong viscosity. The Gâteaux differentiability of solution mapping on control variables is proved and the various types of necessary optimality conditions corresponding to the distributive and terminal values observations are established.     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

Fréchet differentiability of solution mappings for semilinear second order evolution equations

In this paper we establish the strong Fréchet differentiability of maps from the set of initial values and forcing terms into the set of solutions for semilinear second order evolution equations. Also, under the weaker conditions of nonlinear terms we establish the strong Gâteaux differentiability of the solution maps. An application of results to semilinear strongly damped wave equations is given.     

Analysis of multiscale finite element method for the stationary Navier-Stokes equations

In this work, a multiscale finite element method is proposed for the stationary incompressible Navier-Stokes equations. And the inf-sup stability of the method for the P₁/P₁ triangular element is established. The optimal error estimates are obtained.     

Local existence of solution to free boundary value problem for compressible Navier-Stokes equations

This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local (in time) existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.     

A trace regularity result for thermoelastic equations with application to optimal boundary control

We consider a mixed problem for a Kirchoff thermoelastic plate model with clamped boundary conditions. We establish a sharp regularity result for the outer normal derivative of the thermal velocity on the boundary. The proof, based upon interpolation techniques, benefits from the exceptional regularity of traces of solutions to the elastic Kirchoff equation. This result, which complements recent results obtained by the second and third authors, is critical in the study of optimal control problems associated with the thermoelastic system when subject to thermal boundary control. Indeed, the present regularity estimate can be interpreted as a suitable control-theoretic property of the corresponding abstract dynamics, which is crucial to guarantee well-posedness for the associated differential Riccati equations.     

Incompressible limits of the Navier-Stokes equations for all time

We study the asymptotic behaviors of the regular solutions to the compressible Navier-Stokes equations for “well-prepared” initial data for all time as the Mach number tends to zero, by deriving a differential inequality with certain decay property. The estimates obtained in this paper are uniform both in time and Mach number.     

Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation

In this paper, we investigate the large-time behavior of solutions to an outflow problem for compressible Navier-Stokes equations. In 2003, Kawashima, Nishibata and Zhu [S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003) 483-500] showed there exists a boundary layer (i.e., stationary solution) to the outflow problem and the boundary layer is nonlinearly stable under small initial perturbation. In the present paper, we show that not only the boundary layer above but also the superposition of a boundary layer and a rarefaction wave are stable under large initial perturbation. The proofs are given by an elementary energy method.     

A note on the non-formation of vacuum states for compressible Navier-Stokes equations

We give a refinement of Lemma 2.2 in [D. Hoff, J.A. Smoller, Non-formation of vacuum states for compressible Navier-Stokes equations, Comm. Math. Phys. 216 (2001) 255-276] and complete the proof of non-formation of vacuum states for one-dimensional compressible Navier-Stokes equation given there.     

Free boundary value problem for the one-dimensional compressible Navier-Stokes equations with density-dependent viscosity

We study the free boundary value problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in this paper. Under certain assumptions imposed on the initial data, we show that there exists a unique global strong solution, the interface separating the flow and vacuum state propagates along particle path and expands outwards at an algebraic time-rate, the flow density is strictly positive from blow for any finite time and decays pointwise to zero also at an algebraic time-rate as the time tends to infinity.     

Hybrid functions approach for optimal control of systems described by integro-differential equations

In this paper, a new numerical method for solving the optimal control of a class of systems described by integro-differential equations with quadratic performance index is presented. This optimization problem plays an important role in describing the dynamics of an elastic aircraft with allowance for non-steady flow past its profile. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the solution of optimization problem to a nonlinear programming one to which existing well-developed algorithms may be applied. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Asymptotic analysis of an interior optimal control problem governed by Stokes equations

This article introduces an interior optimal control problem (OCP) in a two-dimensional domain with a highly oscillatory boundary governed by the stationary Stokes equations. We consider the periodic controls in the oscillating region of the domain and use the unfolding operators to characterize the optimal controls. We establish the convergences of optimal control, state, and pressure in a suitable space to the ones of the limit system in a fixed domain.     

Existence of an optimal control for a coupled FBSDE with a non degenerate diffusion coefficient

We consider a controlled system driven by a coupled forward–backward stochastic differential equation with a non degenerate diffusion matrix. The cost functional is defined by the solution of the controlled backward stochastic differential equation, at the initial time. Our goal is to find an optimal control which minimizes the cost functional. The method consists to construct a sequence of approximating controlled systems for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we establish the existence of a relaxed optimal control to the initial problem. The existence of a strict control follows from the Filippov convexity condition.     

Analysis and finite element approximation of an optimal control problem for the Oseen viscoelastic fluid flow

In this article we study a boundary control problem for an Oseen-type model of viscoelastic fluid flow. The existence of a unique optimal solution is proved and an optimality system is derived by the first-order necessary condition. We investigate finite element approximations to a solution of the optimality system, and a solution algorithm for the system based on the gradient method.