Control problems of an age-dependent predator-prey system

This paper is concerned with optimal harvesting problems for a system consisting oftwo populations with age-structure and interaction of predator-prey. Existence and uniquenessof non-negative solutions to the system and the continuous dependence of solutions on controlvariables are investigated. Existence of optimal policy is discussed, optimality conditions arederived by means of normal cone and adjoint system techniques.     

Application of ODT to constant volume autoignition problems

The One-Dimensional Turbulence (ODT) model is applied to a constant volume configuration by means of a periodic, one-dimensional domain subject to randomized ensemble members with initial inhomogeneous temperature fields and homogeneous mass fraction profiles. The multidimensional turbulent interactions in the flow are modeled by the separate implementation of turbulent advection and the diffusion-reaction processes, neglecting the mean advection of the system. On one hand, turbulent advection is modeled by means of the eddy events defined within the framework of ODT; on the other hand, the diffusion-reaction system is solved by means of the Zero-Mach limit conservation equations discretized with a 1D Finite Volume Method (FVM). The treatment is specialized in this work to constant volume systems. Due to the inherent stiffness of the diffusion-reaction system, an operator splitting approach is also included in the formulation. Results for n-Heptane chemistry comprising the temporal evolution of the heat release rate, pressure and normalized density-weighted displacement speed are shown and compared to DNS results from Yoo et al. [Combust. Flame 158 (2011) 1727-1741], in terms of individual ensemble members and mean ensemble behavior. The results show that it is possible to obtain reasonably good results in comparison to the DNS if an appropriate set of initial conditions is used. Furthermore, it is shown that the model uncertainty is negligible in comparison to the ensemble standard deviation introduced by randomized initial conditions. Overall, this work introduces the framework for constant volume autoignition in ODT and shows its efficiency for complex chemistry simulations. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Structure of extremals of autonomous convex variational problems

In this paper we study the structure of approximate solutions of autonomous variational problems with convex integrands. We are interested in a turnpike property of the extremals which is independent of the length of the interval, for all sufficiently large intervals. To have this property means, roughly speaking, that the approximate solutions of the variational problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions.     

On a class of constrained functional minimization problems and their numerical solution

We investigate a class of functional minimization problems with constraints. By means of variational principles, optimal control theory, and numerical methods for nonlinear equations, numerical methods and the corresponding computer software are established to solve the problems. These tools can be used in fitting curves with arbitrary smoothness, different boundary conditions, and constraints. For special boundary conditions, analytical expressions of the curves are derived. Numerical examples are given to demonstrate the effectiveness of the algorithms by the means of curve fitting.     

Positive solutions of fractional differential equation boundary value problems at resonance

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.     

Numerical solutions to continuous linear programming problems

Summary A method for finding approximate solutions for continuous linear programming problems is suggested. The required conditions to be met are: a) the matrix associated with the integrals in the constraints is constant; b) all functions involved are of bounded variation; c) the matrices involved satisfy certain boundedness conditions, and d) there exist feasible solutions.The approximations converge almost everywhere to an optimal solution. The optimal solution is shown to be a function of bounded variation.The method is illustrated by means of a numerical example. Here, the approximate solution reveals also the structure of the exact (analytical) solution and makes its construction possible.     

Periodic perturbations of a class of resonant problems

We study the existence of solution for nonlinear problems at resonance under Dirichlet boundary conditions. We deal with PDE's as well as systems of ODE's. The nonlinear terms considered are periodic functions: in particular, the problem is strongly resonant at infinity. By means of variational methods, we prove nondegeneracy under some hypotheses on the nonlinearities. Received: 31 October 2003, Accepted: 12 July 2004, Published online: 8 February 2005 Mathematics Subject Classification (2000): 34B15, 35B34, 35J20 The authors have been supported by the Ministry of Science and Technology of Spain (BFM2002-02649), and by J. Andalucía (FQM 116)     

Blow-up in parabolic problems under Robin boundary conditions

By means of a first-order differential inequality technique, sufficient conditions are determined which imply that blow-up of the solution does occur or does not occur for the semilinear heat equation under Robin boundary conditions. In addition, a lower bound on blow-up time is obtained when blow-up does occur.     

Lower bounds for blow-up time in parabolic problems under Neumann conditions

We consider an initial boundary value problem for the semilinear heat equation under homogeneous Neumann boundary conditions in which the solution may blow up in finite time. A lower bound for the blow-up time is determined by means of a differential inequality argument when blow up occurs. Under alternative conditions on the nonlinearity, some additional bounds for blow-up time are also determined.     

Graph algorithm for partially-linear problems obtained by absolute value functions

The increasing interest in partially-linear models is caused by their role in economics, technology and mathematics. Every continuous partially-linear function with a finite number of linear parts can be formulated by means of linear function and absolute value functions. If we presebt every absolote value function as vertex of a graph we obtain the graph interpretaion of this optimization problem. The Partially-linear problems are often not only nondifferentiable but nonconvex as well,which makes their solving by standard optimization methods difficult. In this work we give the necessary and sufficient conditions for local optimality. An algorithm for solving such problems, based on the constructive optimization methods is suggested.     

Double control problems of age-distributed population dynamics

This paper is concerned with some optimal control problems of an age-structured population model controlled by fecundity and culling rates. The first-order necessary optimality conditions of the following four problems are deduced by means of a Dubovitskii–Milyutin functional analysis approach: a problem with fixed horizon and terminal constraint set, a free horizon counterpart, a differential game and a mini–max problem with a non-differentiable objective functional.     

Complexity of vector optimization problems on graphs

The paper is dedicated to the computation complexity of multi-objective optimization problems on graphs. The classes of multi-objective problems with polynomial complexity or being polynomially reduced to be NP-hard are marked out. The unsolvability of a series of combinatorial multi-objective problems has been set up by means of linear convolution algorithm. The sufficient conditions under which these algorithms are statistically efficient have also been obtained.     

Differentiable difference approximations for nonlinear initial-value problems.I

This paper deals with the approximation of nonlinear initial-value problems by difference methiods. in the Present part I The basic definitions and concepts are presented and equivalence theorems for stability and continuous convergenc are proved. Here, The differentiability condition (d) and the boundedness condition (Bp) are of fundamental siginificance. the latter is one of the equivalent characterizations of stability. The equivalence theorems for stability and continuous convergence include characterizations by means of locally uniform two-sided Lipschitz conditions and tow-sided discertization error estimates. At the end of part I a generalized of the concept of stable convergence of Dahlquist [2] and Torng [16] is proved to series of equivalent conditions convergence. in part II the above results will yields a series of equivalent conditions for the concepts of weak stability and conditions convergence of certain order. Moreover, further convergence concepts for semi-homogeneous methods will be studied, and hyperbolic and parawbolic example will be treated     

Existence theorems for boundary value problems in difference equations

In this research, we study linear difference equations with constant coefficients subject to boundary conditions. Necessary and/or sufficient conditions for the existence of a unique solution will be established. The proofs of the existence and uniqueness theorems are established by means of special types of determinants called Mosaic Vandermonde determinants.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Duality for location problems with unbounded unit balls

Given an optimization problem with a composite of a convex and componentwise increasing function with a convex vector function as objective function, by means of the conjugacy approach based on the perturbation theory, we determine a dual to it. Necessary and sufficient optimality conditions are derived using strong duality. Furthermore, as special case of this problem, we consider a location problem, where the “distances” are measured by gauges of closed convex sets. We prove that the geometric characterization of the set of optimal solutions for this location problem given by Hinojosa and Puerto in a recently published paper can be obtained via the presented dual problem. Finally, the Weber and the minmax location problems with gauges are given as applications.     

On some optimum problems of vibration theory

Some optimum problems of vibration theory and the variational approach to their solution are discussed. The results of construction of optimum solutions obtained by means of necessary conditions of the calculus of variations are presented.     

On optimality conditions of relaxed non-convex variational problems

Non-convex variational problems in many situations lack a classical solution. Still they can be solved in a generalized sense, e.g., they can be relaxed by means of Young measures. Various sets of optimality conditions of the relaxed non-convex variational problems can be introduced. For example, the so-called “variations” of Young measures lead to a set of optimality conditions, or the Weierstrass maximum principle can be the base of another set of optimality conditions. Moreover the second order necessary and sufficient optimality conditions can be derived from the geometry of the relaxed problem. In this article the sets of optimality conditions are compared. Illustrative examples are included.     

Solvability of initial-boundary value problems for bending of thermoelastic plates with mixed boundary conditions

Initial-boundary value problems for bending of a thermoelastic plate with transverse shear deformation are studied under the assumption that various parts of the boundary are subjected to different types of physical conditions. The unique solvability of these problems is established in spaces of distributions by means of a combination of the Laplace transform and variational methods.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.