Regularization of optimal design problems for bars and plates,part 1

In this paper, we consider a number of optimal design problems for elastic bars and plates. The material characteristics of rigidity of an elastic nonhomogeneous medium are taken as the control variables. A linear functional of the solutions to the equilibrium boundary-value problem is minimized under additional restrictions upon the control variables.Special variations of the control within a narrow strip provide a necessary condition for a strong local minimum (Weierstrass test). This necessary condition can be used for the detailed analysis of the following problems: bar of extremal torsional rigidity; optimal distribution of isotropic material with variable shear modulus within a plate; and optimal orientation of principal axes of elasticity in an orthotropic plate. For all of these cases, the stationary solutions violate the Weierstrass test and therefore are not optimal. This is because, in the course of the approximation of the optimal solution, the curve dividing zones occupied by materials with different rigidities displays rapid oscillations sweeping over a two-dimensional region. Within this region, the material behaves as a composite medium assembled of materials of the initial class.     

Optimization problems for an energy functional with mass constraint revisited

In this paper we report on a variational problem under a constraint on the mass which is motivated by the torsional rigidity and torsional creep. Following a device by Alt, Caffarelli and Friedman we treat instead a problem without constraint but with a penalty term. We will complete some of the results of [C. Bandle, A. Wagner, Optimization problems for weighted Sobolev constants, Calc. Var. Partial Differential Equations 29 (2007) 481-507] where the existence of a Lipschitz continuous minimizer has been established. In particular we prove qualitative properties of the optimal shape.     

Torsional rigidity of reinforced concrete bars with arbitrary sectional shape

In this paper, the torsional rigidity of arbitrarily shape bar made of different materials is studied on the basis of theory of elasticity and finite element approach. With additional boundary conditions for the common boundaries of different materials from the continuous conditions of deformation and traction across the interior boundary, the torsion function can be solved numerically from the second boundary-value problem of potential theory. The traction jump boundary conditions across the interior surfaces are enforced in the alternate finite element approach. Several examples are shown to check the computational approach proposed, and the approach, at last, is applied to calculate the torsional rigidity of reinforced concrete bar and some multiply connected cross sections such as tower leg section of the Tsing Ma Bridge and other engineering structures.     

Regularization of optimal design problems for bars and plates,part 2

For the problems of optimal design considered in Ref. 1, contradictions arising in the necessary conditions of optimality are eliminated by suitable extension of the initially given class of admissible materials. The extended class includes composites of some special (layered) microstructure. Elastic properties of such composites are described, and alternative (regularized) formulations of the optimal design problems are given. Necessary conditions of Weierstrass are shown to be satisfied, both for the case in which the strip of variations is small compared with the width of the layers and for the opposite case. Numerical results are given for the regularized problem of a bar of extremal torsional rigidity.The authors are indebted to Dr. N. A. Lavrov for performing numerical calculations.     

Isoperimetric Inequalities for a Wedge-Like Membrane

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya–Szeg?, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.     

Stability of isoperimetric type inequalities for some Monge–Ampère functionals

We investigate the stability of some inequalities of isoperimetric type related to Monge–Ampère functionals. In particular, firstly we prove the stability of a reverse Faber–Krahn inequality for the Monge–Ampère eigenvalue and its generalization. Then we give a stability result for the Brunn–Minkowski inequality and for a consequent Urysohn’s type inequality for the so-called \(n\) -torsional rigidity, a natural extension of the usual torsional rigidity.     

Sharp estimates for the anisotropic torsional rigidity and the principal frequency

In this paper we generalize some classical estimates involving the torsional rigidity and the principal frequency of a convex domain to a class of functionals related to some anisotropic nonlinear operators.     

A Computational Method for an Optimal Design Problem in Uniform Torsion Under Local Stress Constraints

In this paper, we devise a computational method for simulating optimal elastic materials made of two phases that maximize torsional stiffness under local stress constraints. Our method relies on a theoretical relaxation result and it is capable to find optimal layouts involving microstructure among the constituents.     

The 2‐dimensional rigidity of certain families of graphs

Laman's characterization of minimally rigid 2‐dimensional generic frameworks gives a matroid structure on the edge set of the underlying graph, as was first pointed out and exploited by L. Lovász and Y. Yemini. Global rigidity has only recently been characterized by a combination of two results due to T. Jordán and the first named author, and R. Connelly, respectively. We use these characterizations to investigate how graph theoretic properties such as transitivity, connectivity and regularity influence (2‐dimensional generic) rigidity and global rigidity and apply some of these results to reveal rigidity properties of random graphs. In particular, we characterize the globally rigid vertex transitive graphs, and show that a random d‐regular graph is asymptotically almost surely globally rigid for all d ≥ 4. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 154–166, 2007     

Optimal control problem of the uncertain second‐order circuit based on first hitting criteria

First hitting criteria of a system are to initially achieve some performance indeces of the target state set. This paper primarily investigates the optimal control problem of the uncertain second‐order circuit based on first hitting criteria. First, considering time efficiency and different from the ordinary expected utility criterion over an infinite time horizon, two first hitting criteria which are reliability index and reliable time criteria are innovatively proposed. Second, assuming the circuit output voltage as an uncertain variable when the historical data is lacking, we better model the real circuit system with the uncertain second‐order differential equation which is essentially the uncertain fractional‐order differential equation. Then, based on the first hitting time theorem of the uncertain fractional‐order differential equation, the distribution function of the first hitting time under the second‐order circuit system is proposed and the uncertain second‐order circuit optimal control model (reliability index and reliable time‐based model) is transformed into corresponding crisp optimal problem. Lastly, analytic expressions of the optimal control for the reliability index model are obtained. Meanwhile, sufficient condition and guidance for parameters for the optimal solution of the reliable time‐based model are derived, and corresponding numerical examples are also given to demonstrate the fluctuation of our optimal solution for different parameters.     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Minimization of the ground state of the mixture of two conducting materials in a small contrast regime

We consider the problem of distributing two conducting materials with a prescribed volume ratio in a given domain so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions. The gap between the two conductivities is assumed to be small (low contrast regime). For any geometrical configuration of the mixture, we provide a complete asymptotic expansion of the first eigenvalue. We then consider a relaxation approach to minimize the second‐order approximation with respect to the mixture. We present numerical simulations in dimensions two and three to illustrate optimal distributions and the advantage of using a second‐order method. Copyright © 2016 John Wiley & Sons, Ltd.     

OPTIMAL HARVESTING AND SPATIAL PATTERNS IN A SEMIARID VEGETATION SYSTEM

We consider an infinite time horizon spatially distributed optimal harvesting problem for a vegetation and soil water reaction diffusion system, with rainfall as the main external parameter. By Pontryagin's maximum principle, we derive the associated four‐component canonical system (CS), and numerically analyze this and hence the optimal control problem in two steps. First, we numerically compute a rather rich bifurcation structure of flat (spatially homogeneous) canonical steady states and patterned canonical steady states (FCSS and PCSS, respectively), in 1D and 2D. Then, we compute time‐dependent solutions of the CS that connect to some FCSS or PCSS. The method is efficient in dealing with nonunique canonical steady states, and thus also with multiple local maxima of the objective function. It turns out that over wide parameter regimes the FCSS, i.e., spatially uniform harvesting, are not optimal. Instead, controlling the system to a PCSS yields a higher profit. Moreover, compared to (a simple model of) private optimization, the social control gives a higher yield, and vegetation survives for much lower rainfall. In addition, the computation of the optimal (social) control gives an optimal tax to incorporate into the private optimization.     

On the characterization of the compact embedding of Sobolev spaces

For every positive regular Borel measure, possibly infinite valued, vanishing on all sets of p-capacity zero, we characterize the compactness of the embedding ${W^{1,p}({\mathbb R}^N)\cap L^p ({\mathbb R}^N,\mu)\hookrightarrow L^q({\mathbb R}^N)}$ in terms of the qualitative behavior of some characteristic PDE. This question is related to the well posedness of a class of geometric inequalities involving the torsional rigidity and the spectrum of the Dirichlet Laplacian introduced by Pólya and Szeg? (Isoperimetric inequalities in mathematical physics, Annals of mathematics studies, 1951). In particular, we prove that finite torsional rigidity of an arbitrary domain (possibly with infinite measure), implies the compactness of the resolvent of the Laplacian.     

Algorithms for graph partitioning on the planted partition model

The NP‐hard graph bisection problem is to partition the nodes of an undirected graph into two equal‐sized groups so as to minimize the number of edges that cross the partition. The more general graph l‐partition problem is to partition the nodes of an undirected graph into l equal‐sized groups so as to minimize the total number of edges that cross between groups. We present a simple, linear‐time algorithm for the graph l‐partition problem and we analyze it on a random “planted l‐partition” model. In this model, the n nodes of a graph are partitioned into l groups, each of size n/l; two nodes in the same group are connected by an edge with some probability p, and two nodes in different groups are connected by an edge with some probability r<p. We show that if p−r≥n^−1/2+ϵ for some constant ϵ, then the algorithm finds the optimal partition with probability 1− exp(−n^Θ(ε)). © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 116–140, 2001     

Symplectic rigidity,symplectic fixed points,and global perturbations of Hamiltonian systems

In this paper we study a generalized symplectic fixed‐point problem, first considered by J. Moser in [20], from the point of view of some relatively recently discovered symplectic rigidity phenomena. This problem has interesting applications concerning global perturbations of Hamiltonian systems. © 2007 Wiley Periodicals, Inc.     

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers

• Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs).
• These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states.
• The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts.
• Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators.

     

THE MITRA‐WAN FORESTRY MODEL: A DISCRETE‐TIME OPTIMAL CONTROL PROBLEM

In this paper, we present some new properties of the Mitra‐Wan forestry model written as a discrete‐time optimal control problem. For this problem, the set of stationary states is characterized. For the optimal long‐run management, we consider the following optimality criteria: average optimality, good control policies, bias optimality, and overtaking optimality. We establish relationships between these criteria and show that the value of average optimal policies is constant and equals the value in the optimal stationary state.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

Brunn-Minkowski inequalities for variational functionals and related problems

Recently, several inequalities of Brunn-Minkowski type have been proved for well-known functionals in the Calculus of Variations, e.g. the first eigenvalue of the Laplacian, the Newton capacity, the torsional rigidity and generalizations of these examples. In this paper, we add new contributions to this topic: in particular, we establish equality conditions in the case of the first eigenvalue of the Laplacian and of the torsional rigidity, and we prove a Brunn-Minkowski inequality for another class of variational functionals. Moreover, we describe the links between Brunn-Minkowski type inequalities and the resolution of Minkowski type problems.