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.     

Parameter identification problems for an extensible beam equation

Parameter identification problems are studied for extensible beam equations. The problem is formulated by minimizations of quadratic cost functionals by distributive measurements. The existence of optimal parameters and necessary conditions for the parameters are proved.     

Free boundary problem for the equation of one-dimensional motion of compressible gas with density-dependent viscosity

We consider a free boundary problem for the equation of the one-dimensional isentropic motion with density-dependent viscosity μ =b _ϱ ^β, whereb and β are positive constants. We prove that there exists an unique weak solution globally in time, provided that β<1/3.

---

Sunto Si considera un problema di frontiera libera per l’equazione del moto unidimensionale isoentropico con viscosità dipendente dalla densità secondo la legge μ =b _ϱ ^β, doveb e β sono costanti positive. Si dimostra che esiste un’unica soluzione debole globale nel tempo, purché β<1/3.

     

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.     

Global existence of strong solutions of the Navier-Stokes equations for isentropic compressible fluids with density-dependent viscosity

This paper is concerned with global strong solutions of the isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient in one-dimensional bounded intervals. Precisely, the viscosity coefficient μ=μ(ρ) and the pressure P is proportional to ργ with γ>1. The important point in this paper is that the initial density may vanish in an open subset. We also show that the strong solution obtained above is unique provided that the initial data satisfies additional regularity and a compatible condition. Compared with former results obtained by Hyunseok Kim in [H. Kim, Global existence of strong solutions of the Navier-Stokes equations for one-dimensional isentropic compressible fluids, available at: http://com2mac.postech.ac.kr/papers/2001/01-38.pdf], we deal with density-dependent viscosity coefficient.     

Semiconcavity of the value function for the Bolza control problem

In this paper we investigate the regularity of the value function of the Bolza control problem. We propose sufficient conditions for the value function to be semiconcave or locally Lipschitz when the controls are unbounded.     

Some kind of Pareto stationarity for multiobjective problems with equilibrium constraints

ABSTRACT

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that the generalized Guignard constraint qualification is the weakest constraint qualification for the strong Pareto Kuhn-Tucker optimality. Furthermore, under certain convexity or generalized convexity assumptions, we show that the strong Pareto Kuhn-Tucker optimality conditions are also sufficient for several popular locally Pareto-type optimality conditions for MOPEC.     

Mathematical analysis of absorbing boundary conditions for the wave equation: the corner problem

Our goal in this work is to establish the existence and the uniqueness of a smooth solution to what we call in this paper the corner problem, that is to say, the wave equation together with absorbing conditions at two orthogonal boundaries. First we set the existence of a very smooth solution to this initial boundary value problem. Then we show the decay in time of energies of high order--higher than the order of the boundary conditions. This result shows that the corner problem is strongly well-posed in spaces smaller than in the half-plane case. Finally, specific corner conditions are derived to select the smooth solution among less regular solutions. These conditions are required to derive complete numerical schemes.

     

On boundary-value problem for hyperbolic-type equation of the third order

We formulate some boundary-value problems for a linear third-order equation with hyperbolic operator in the main part and study the unique solvability. Under certain conditions to given functions, using the Riemann method, we obtain an integral representation of solutions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 591–603, October–December, 2007.     

A remark on the Cauchy problem of 1D compressible Navier-Stokes equations with density-dependent viscosity coefficients

This paper is concerned with the Cauchy problems of one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefcients.By assumingρ0∈L1（R）,we will prove the existence of weak solutions to the Cauchy problems forθ〉0.This will improve results in Jiu and Xin’s paper（Kinet.Relat.Models,1（2）：313–330（2008））in whichθ〉12is required.In addition,We will study the large time asymptotic behavior of such weak solutions.     

Approximation of solutions for an initial and terminal value problem for the forced Duffing equation with non-viscous damping

An algorithm to find explicit approximate solutions of an initial and terminal value problem for the forced Duffing equation with non-viscous damping is accomplished via a generalized quasilinearization method. In fact, we obtain a monotone sequence of approximate solutions converging uniformly and quadratically to a unique solution of the problem.     

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over‐specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45–52) applied to the Cauchy problem for the two‐dimensional modified Helmholtz equation. The two mixed, well‐posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross‐validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two‐dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On a nonlocal problem for the Laplace equation in the unit ball with fractional boundary conditions

In this paper, we investigate the correct solvability for the Laplace equation with a nonlocal boundary condition in the unit ball. The considered boundary operator is of fractional order. This problem is a generalization of the well‐known Bitsadze–Samarskii problem. Copyright © 2015 John Wiley & Sons, Ltd.     

具有一个细焦点和一个粗焦点的二次系统极限环的个数

Abstract. It is proved that the quadratic system with a weak focus and a strong focus has atmost one limit cycle around the strong focus, and as the weak focus is a 2nd -order (or 3rd-order ) weak focus the quadratic system has at most two (one) limit cycles which have (1,1)-distribution ((0,1)-distribution).     

A boundary value problem for the spherically symmetric motion of a pressureless gas with a temperature‐dependent viscosity

We consider an initial‐boundary value problem for the equations of spherically symmetric motion of a pressureless gas with temperature‐dependent viscosity µ(θ) and conductivity κ(θ). We prove that this problem admits a unique weak solution, assuming Belov's functional relation between µ(θ) and κ(θ) and we give the behaviour of the solution for large times. Copyright © 2009 John Wiley & Sons, Ltd.     

Nondifferentiable fractional programming with Hanson-Mond classes of functions

A class of nonlinear fractional programming problems is considered in which the numerator of the objective function involves a finite sum of square roots of quadratic forms. Inequality and equality constraints are considered. Necessary and sufficient optimality conditions and weak, strong, and converse duality theorems are established in the framework of the Hanson-Mond classes of functions.The author wishes to thank the unknown referee for the helpful comments that improved the quality of this paper.     

Optimal control of a system governed by a parabolic equation with an infinite number of variables

A distributed control problem for a parabolic operator with an infinite number of variables is considered. The performance index is more general than the quadratic one an has an integral form. Making use of the Dubovicki-Milutin theorem, necessary and sufficient conditions of optimality are derived for the Dirichlet problem.     

The initial boundary value problem for quasi-linear wave equation with viscous damping

In this paper, the existence and uniqueness of the local generalized solution and the local classical solution for the initial boundary value problem of the quasi-linear wave equation with viscous damping are proved. The nonexistence of the global solution for this problem is discussed by an ordinary differential inequality. Finally, an example is given.     

Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.