Constructive observability inequalities for weak generalized solutions of the wave equation with elastic restraint

Problems with one-sided boundary controls and a homogeneous Robin boundary condition set on the uncontrolled end are considered in the class of strong generalized solutions of the variable coefficient wave equation. In the adjoint class of weak generalized solutions of the dual problems with one-sided observations, new constructive observability inequalities are obtained that differ from previously known ones by an optimal threshold time. It is shown that, in the considered functional classes, the estimated constants degenerate as the time interval length approaches the threshold. Numerical illustrations are given showing that the stability of approximate solutions to control problems can be substantially enhanced by taking into account a priori information contained in the resulting observability inequalities.     

Threshold optimization in observability inequality for the wave equation with homogeneous Robin-type boundary condition

For the wave equation with variable coefficients, problems with one-side boundary controls of three basic types and a boundary condition of the third kind at the uncontrolled end are considered. For dual problems with one-side boundary observations in the classes of strong generalized solutions, new constructive observability inequalities are obtained that are superior to the earlier known ones in two respects. First, inequalities with an optimal value of the controllability-observability threshold are derived, and second, the value of the final evaluation constant is bounded away from zero on time intervals whose length is close to the critical length. This opens up a possibility of constructing stable approximate solutions to the indicated classes of dual control and observation problems on time intervals not only of an arbitrary supercritical but also of precisely critical length.     

The optimal exchange algorithm and comparisons with the generalized remes algorithm † ‡

This paper is concerned with the general problem of the determination of the best uniform approximation of a given function. A special case is the calculation of the minimax solution of an overdetermined linear system. Single point exchange algorithms produce successive approximate solutions for such problems. An example is furnished by the generalized Remes algorithm, which includes both the original Remes algorithm and the Stiefel algorithm as special cases. The optimal exchange algorithm is similar, but it has the important feature that every exchange is optimal in a certain desirable sense. It is proved that eventually the optimal exchange and generalized Remes algorithms coincide. However, early exchanges in the latter algorithm can be far from optimal and quite inefficient. A combination of the optimal exchange and generalized Remes algorithms is suggested as a reasonable strategy.     

System of generalized resolvent equations with corresponding system of variational inclusions

The purpose of this paper is to introduce a new system of generalized resolvent equations with corresponding system of variational inclusions in uniformly smooth Banach spaces. We establish an equivalence relation between system of generalized resolvent equations and system of variational inclusions. The iterative algorithms for finding the approximate solutions of system of generalized resolvent equations are proposed. The convergence of approximate solutions of system of generalized resolvent equations obtained by the proposed iterative algorithm is also studied.      

Numerical Methods for Optimal Dividend Payment and Investment Strategies of Markov-Modulated Jump Diffusion Models with Regular and Singular Controls

This work focuses on numerical methods for finding optimal dividend payment and investment policies to maximize the present value of the cumulative dividend payment until ruin; the surplus is modeled by a regime-switching jump diffusion process subject to both regular and singular controls. Using the dynamic programming principle, the optimal value function obeys a coupled system of nonlinear integro-differential quasi-variational inequalities. Since the closed-form solutions are virtually impossible to obtain, we use Markov chain approximation techniques to approximate the value function and optimal controls. Convergence of the approximation algorithms are proved. Examples are presented to illustrate the applicability of the numerical methods.     

Regularizing algorithms with optimal and extra-optimal quality

The notion of a special quality for approximate solutions to ill-posed inverse problems is introduced. A posteriori estimates of the quality are studied for different regularizing algorithms. Examples of typical quality functionals are provided, which arise in solving linear and nonlinear inverse problems. A technique and a numerical algorithm for calculating a posteriori quality estimates for approximate solutions of general nonlinear inverse problems are developed. New notions of optimal and extra-optimal quality of a regularizing algorithm are introduced. The theory of regularizing algorithms with optimal and extra-optimal quality is presented, which includes an investigation of optimal properties for estimation functions of the quality. Examples of regularizing algorithms with extra-optimal quality of solutions are given, as well as examples of regularizing algorithms without such a property. Results of numerical experiments illustrate a posteriori quality estimation.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

关于GMP和GMT问题的最优解

进一步研究模糊推理的非模糊形式，在几个重要的逻辑系统中形式地讨论GMP(广义取式)和GMT(广义拒取式)问题的最优解。结果表明，GMP和GMT问题的三I解和一种新的三I解都是某种意义下的最优解。还讨论所给算法的还原性问题。     

Estimates for normal solutions of problems with zone controls in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> for the wave equation

For the wave equation with variable coefficients and homogeneous boundary conditions of the first kind, we consider problems with regular zone controls and dual zone observation problems. For weak generalized solutions of the observation problem on sufficiently large time intervals, we obtain constructive estimates that imply the well-posed solvability of the observation operator. These estimates contain information that permits one to construct stable approximate solutions of both problems with the use of a variational method suggested earlier by the author for linear equations with nonuniformly perturbed operators.     

Numerical construction of Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game

Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.     

Observability inequality for a wave equation with elastic fastening in the case of critical time interval

Problems with one-sided boundary controls of three basic types and homogeneous boundary third conditions on uncontrollable ends are considered for a wave equation in classes of strong generalized solutions in time intervals of strict critical length. New constructive observability inequalities are obtained for dual problems in adjoint classes of weak generalized solutions.     

Generalized Exact Boundary Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls

This paper deals with the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls in the framework of weak solutions. A necessary and sufficient condition for the generalized exact boundary synchronization is obtained, and some results for its generalized exactly synchronizable states are given.     

Existence and iterative algorithm of solutions for a class of bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP are suggested and analyzed. The strong convergence of the iterative sequences generated by the algorithms are proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Finite-difference approximation of dirichlet observation problems for weak solutions to the wave equation subject to Robin boundary conditions

For the wave equation with variable coefficients subject to Neumann and Robin boundary conditions, two mutually dual problems are considered: the Dirichlet observation problem with weak generalized solutions and the control problem with strong generalized solutions. Both problems are approximated by finite differences preserving the duality relation. The convergence of the approximate solutions is established in the norms of the corresponding dual spaces.     

Global solutions for initial-boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial-boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial-boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.     

Utilizing expected improvement and generalized data envelopment analysis in multi-objective genetic algorithms

Meta-heuristic methods such as genetic algorithms (GA) and particle swarm optimization (PSO) have been extended to multi-objective optimization problems, and have been observed to be useful for finding good approximate Pareto optimal solutions. In order to improve the convergence and the diversity in the search of solutions using meta-heuristic methods, this paper suggests a new method to make offspring by utilizing the expected improvement (EI) and generalized data envelopment analysis (GDEA). In addition, the effectiveness of the proposed method will be investigated through several numerical examples in comparison with the conventional multi-objective GA and PSO methods.     

Existence and algorithms for bilevel generalized mixed equilibrium problems in Banach spaces

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) to compute the approximate solutions of the generalized mixed equilibrium problems (GMEP) and bilevel generalized mixed equilibrium problems (BGMEP) involving set-valued mappings is introduced. By using a minimax inequality, the existence and uniqueness of solutions of the AGMEP is proved under quite mild conditions. By using auxiliary principle technique, new iterative algorithm to compute the approximate solutions of the GMEP and the BGMEP is suggested and analyzed. Strong convergence of the iterative sequences generated by the proposed algorithms is proved under quite mild assumptions. These results are new and generalize some recent results in this field.     

Approximate solution of nonlinear problems of optimal control of oscillatory processes using the method of canonical separation of motions : PMM vol.40, n 6, 1976, pp. 1024–1033

An asymptotic method of solving certain problems of optimal control of motion of the standard type systems with rotating phase is developed. It is assumed that the controls enter only the small perturbing terms, and that the fixed time interval over which the process is being considered is long enough to ensure that the slow variables change essentially. Assuming also that the system and the controls satisfy the necessary requirements of smoothness, the method of canonical averaging [1] is used to construct a scheme for deriving a simplified boundary value problem of the maximum principle. The structure of the set of solutions of the boundary value problem is investigated and a scheme for choosing the optimal solution with the given degree of accuracy in the small parameter is worked out. The validity of the approximate method of solving the boundary value problem is proved. The method suggested in [2] for constructing a solution in the first approximation for similar problems of optimal control is developed.     

广义非线性集值混合拟变分包含的扰动近似点算法

本文研究一类广义非线性集值混合拟变分包含,概括了尚明生等人引入与研究过的熟知的广义集值变分包含类成特例.运用预解算子的技巧,建立了广义非线性集值混合拟变分包含与不动点问题之间的等价性,其中,预解算子JρA(·,x)是具有常数1/(1+cρ)的Lipschitz连续算子.本文还建立了几个扰动迭代算法,并提供了由算法生成的逼近解的收敛判据,所得算法与结果改进与推广了尚明生等人的相应算法与结果.     

Best proximity points: approximation and optimization

A best proximity point theorem explores the existence of an optimal approximate solution, known as a best proximity point, to the equations of the form Tx = x where T is a non-self mapping. The purpose of this article is to establish some best proximity point theorems for non-self non-expansive mappings, non-self Kannan- type mappings and non-self Chatterjea-type mappings, thereby producing optimal approximate solutions to some fixed point equations. Also, algorithms for determining such optimal approximate solutions are furnished in some cases.