Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

The construction of solutions to pseudoparabolic equations in noncylindrical domains

We consider the first initial boundary value problem for pseudoparabolic equations in a domain with time dependent boundaries. Sufficient conditions on the coefficients of the equation and the boundary of the region are given in order that this initial boundary value problem has a unique solution.     

On the Maxwell system under impedance boundary conditions with memory

We consider an initial boundary value problem for the system of the Maxwell equations in a bounded domain with smooth boundary on a finite time interval with new boundary conditions with memory. In appropriate function spaces, we define and study the nonselfadjoint operator that is generated by the Maxwell operator under a boundary condition with memory. Using the operator method, we prove an existence and uniqueness theorem for a solution to the initial boundary value problem.     

Solving Linear Boundary Value Problems Via Non-commutative Gröbner Bases

A new approach for symbolically solving linear boundary value problems is presented. Rather than using general-purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique Moore—Penrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of non-commutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green’s function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed.     

The planar Dirichlet problem for the Stokes equations

The Dirichlet problem for the Stokes equations is studied in a planar domain. We construct a solution of this problem in form of appropriate potentials and determine the unknown source densities via integral equation systems on the boundary of the domain. The solution is given explicitly in the form of a series. As a consequence we determine a solution of the Dirichlet problem for a compressible Stokes system and a solution of a boundary value problem on a domain with cracks. Copyright © 2011 John Wiley & Sons, Ltd.     

On a possibility of generalization of the Hopf lemma to the case of the Navier-Stokes system with nonzero flows

We examine the Hopf lemma (Leray inequality) which is used in proving the existence of a solution to a nonhomogeneous boundary value problem for the stationary Navier-Stokes equations of an incompressible fluid in a bounded domain. We study a possibility of generalization of a weakened variant of the lemma to the case of nonzero flows through the connected components of the boundary of the domain.     

A boundary value problem for a class of mixed-type equations in an unbounded domain

We consider a boundary value problem for an equation of the mixed type with a singular coefficient in an unbounded domain. The uniqueness of the solution of the problem is proved with the use of the extremum principle. In the proof of the existence of a solution of the problem, we use the method of integral equations.     

Estimates for the difference between exact and approximate solutions of parabolic equations on the basis of Poincaré inequalities for traces of functions on the boundary

We study a method for the derivation of majorants for the distance between the exact solution of an initial–boundary value reaction–convection–diffusion problem of the parabolic type and an arbitrary function in the corresponding energy class. We obtain an estimate (for the deviation from the exact solution) of a new type with the use of a maximally broad set of admissible fluxes. In the definition of this set, the requirement of pointwise continuity of normal components of the dual variable (which was a necessary condition in earlier-obtained estimates) is replaced by the requirement of continuity in the weak (integral) sense. This result can be achieved with the use of the domain decomposition and special embedding inequalities for functions with zero mean on part of the boundary or for functions with the zero mean over the entire domain.     

Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.     

Asymptotics of a Solution of a Three-Dimensional Nonlinear Wave Equation near a Butterfly Catastrophe Point

We consider an optimal distributed control problem in a planar convex domain with smooth boundary and a small parameter at the highest derivatives of an elliptic operator. The zero Dirichlet condition is given on the boundary of the domain, and the control is included additively in the inhomogeneity. The set of admissible controls is the unit ball in the corresponding space of square integrable functions. Solutions of the obtained boundary value problems are considered in the generalized sense as elements of a Hilbert space. The optimality criterion is the sum of the squared norm of the deviation of the state from a given state and the squared norm of the control with a coefficient. This structure of the optimality criterion makes it possible to strengthen, if necessary, the role of either the first or the second term of the criterion. In the first case, it is more important to achieve the desired state, while, in the second case, it is preferable to minimize the resource consumption. We study in detail the asymptotics of the problem generated by the sum of the Laplace operator with a small coefficient and a first-order differential operator. A feature of the problem is the presence of the characteristics of the limit operator which touch the boundary of the domain. We obtain a complete asymptotic expansion of the solution of the problem in powers of the small parameter in the case where the optimal control is an interior point of the set of admissible controls.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.     

An embedding domain approach for a class of 2-d shape optimization problems: mathematical analysis

This contribution combines a shape optimization approach to free boundary value problems of Bernoulli type with an embedding domain technique. A theoretical framework is developed which allows to prove continuous dependence of the primal and dual variables in the resulting saddle point problems with respect to the domain. This ensures the existence of a solution of a related shape optimization problem in a sufficiently large class of admissible domains.     

Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density

We consider the boundary value problem for the stationary Navier-Stokes equations describing an inhomogeneous incompressible fluid in a two dimensional bounded domain. We show the existence of a weak solution with boundary values for the density prescribed in L^￥L^{\infty}.     

Construction and Structure Properties of Solutions of a Periodic Boundary Value Problem for a Generalization of the Matrix Lyapunov and Riccati Equations

We obtain constructive sufficient conditions for the unique solvability of a periodic boundary value problem for a matrix differential equation that generalizes the Lyapunov and Riccati equations, develop an algorithm for constructing the solution of this equation, estimate the domain where the solution is localized, and study the structural properties of the solution.     

Boundary value problem for a system of fractional partial differential equations

We prove the unique solvability of a boundary value problem for a system of fractional partial differential equations in a rectangular domain and construct the solution in closed form.     

Asymptotic behavior of weak solutions for the relativistic Vlasov-Maxwell equations with large light speed

We study here the behavior of time periodic weak solutions for the relativistic Vlasov-Maxwell boundary value problem in a three-dimensional bounded domain with strictly star-shaped boundary when the light speed becomes infinite. We prove the convergence toward a time periodic weak solution for the classical Vlasov-Poisson equations.     

Regularity of mixed boundary value problems in ℝ3 and boundary element methods on graded meshes

The solution of the three-dimensional mixed boundary value problem for the Laplacian in a polyhedral domain has special singular forms at corners and edges. A ‘tensor-product’ decomposition of those singular forms along the edges is derived. We present a strongly elliptic system of boundary integral equations which is equivalent to the mixed boundary value problem. Regularity results for the solution of this system of integral equations are given which allow us to analyse the influence of graded meshes on the rate of convergence of the corresponding boundary element Galerkin solutions. We show that it suffices to refine the mesh only towards the edges of the surfaces to regain the optimal rate of convergence.     

Polynomial and nonpolynomial spline methods for fractional sub-diffusion equations

We consider one-dimensional fractional sub-diffusion equations on an unbounded domain. For a problem of this type for which an exact or approximate artificial boundary condition is available we reduce it to an initial-boundary value problem on a bounded domain. We then analyze the numerical solution of the problem by polynomial and nonpolynomial spline methods. The consistency and the Von Neumann stability analysis of these methods are also discussed. Numerical experiments clarify the effectiveness and order of accuracy of the proposed methods.     

二阶非线性椭圆型方程带位移的边值问题

In this paper, we consider the boundary value problem with the shift for nonlinear uniformly elliptic equations of second order in a multiply connected domain. For this sake, we propose a modified boundary value problem for nonlinear elliptic systems of first order equations, and give a priori estimates of solutions for the modified boundary value problem. Afterwards we prove by using the Schauder fixedpoint theorem that this boundary value problem with some conditions has a solution. The result obtained is the generlization of the corresponding theorem on the Poincare boundary value problem.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.