Maximum Principle in the Optimal Design of Plates with Stratified Thickness

An optimal design problem for a plate governed by a linear, elliptic equation with bounded thickness varying only in a single prescribed direction and with unilateral isoperimetrical-type constraints is considered. Using Murat–Tartars homogenization theory for stratified plates and Young-measure relaxation theory, smoothness of the extended cost and constraint functionals is proved, and then the maximum principle necessary for an optimal relaxed design is derived.     

Optimal distribution of the internal null control for the one-dimensional heat equation

This paper addresses the problem of optimizing the distribution of the support of the internal null control of minimal L²-norm for the 1-D heat equation. A measure constraint is imposed on the support but no topological assumption such as the number of connected components. Therefore, the problem typically lacks of solution in the class of characteristic functions and needs of relaxation. We show that the relaxed formulation is obtained by replacing the set of characteristic functions by its convex envelope. The proof requires that the observability constant related to the control problem be uniform with respect to the support, property which is obtained by the control transmutation method. The optimality conditions of the relaxed problem as well as the case where the number of connected components is fixed a priori are also discussed. Several numerical experiments complete the study and suggest the ill-posedness of the problem in contrast to the wave situation.     

Optimal Shape Design Problems for a Class of Systems Described by Parabolic Hemivariational Inequality

Optimal shape design problems for systems governed by a parabolic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.     

Global Optimization Problems in Optimal Design of Experiments in Regression Models

In this paper we show that optimal design of experiments, a specific topic in statistics, constitutes a challenging application field for global optimization. This paper shows how various structures in optimal design of experiments problems determine the structure of corresponding challenging global optimization problems. Three different kinds of experimental designs are discussed: discrete designs, exact designs and replicationfree designs. Finding optimal designs for these three concepts involves different optimization problems.     

Optimal Shape Design Problems for a Class of Systems Described by Hemivariational Inequalities

Optimal shape design problems for systems governed by an elliptic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.     

An Extremal Eigenvalue Problem for a Two-Phase Conductor in a Ball

The pioneering works of Murat and Tartar (Topics in the mathematical modeling of composite materials. PNLDE 31. Birkhäuser, Basel, 1997) go a long way in showing, in general, that problems of optimal design may not admit solutions if microstructural designs are excluded from consideration. Therefore, assuming, tactilely, that the problem of minimizing the first eigenvalue of a two-phase conducting material with the conducting phases to be distributed in a fixed proportion in a given domain has no true solution in general domains, Cox and Lipton only study conditions for an optimal microstructural design (Cox and Lipton in Arch. Ration. Mech. Anal. 136:101–117, 1996). Although, the problem in one dimension has a solution (cf. Kre?n in AMS Transl. Ser. 2(1):163–187, 1955) and, in higher dimensions, the problem set in a ball can be deduced to have a radially symmetric solution (cf. Alvino et al. in Nonlinear Anal. TMA 13(2):185–220, 1989), these existence results have been regarded so far as being exceptional owing to complete symmetry. It is still not clear why the same problem in domains with partial symmetry should fail to have a solution which does not develop microstructure and respecting the symmetry of the domain. We hope to revive interest in this question by giving a new proof of the result in a ball using a simpler symmetrization result from Alvino and Trombetti (J. Math. Anal. Appl. 94:328–337, 1983).     

An Optimal Design Problem in Wave Propagation

We consider an optimal design problem in wave propagation proposed in Sigmund and Jensen (Roy. Soc. Lond. Philos. Trans. Ser. A 361:1001–1019, 2003) in the one-dimensional situation: Given two materials at our disposal with different elastic Young modulus and different density, the problem consists of finding the best distributions of the two initial materials in a rod in order to minimize the vibration energy in the structure under periodic loading of driving frequency Ω. We comment on relaxation and optimality conditions, and perform numerical simulations of the optimal configurations. We prove also the existence of classical solutions in certain cases.     

Optimal Control and Applications to Aerospace: Some Results and Challenges

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.     

关于变分包含问题和不动点问题的一些结果及应用

基于一个广义迭代算法,考虑了逼近一类拟变分包含问题解集与一族无限多个非扩张映象公共不动点集的某一公共元问题.在实Hilbert空间的框架下,证明了由次广义迭代算法产生的迭代序列强收敛到某一公共元.     

Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

A Hypergraph Framework for Optimal Model-Based Decomposition of Design Problems

Decomposition of large engineering system models is desirable sinceincreased model size reduces reliability and speed of numericalsolution algorithms. The article presents a methodology for optimalmodel-based decomposition (OMBD) of design problems, whether or notinitially cast as optimization problems. The overall model isrepresented by a hypergraph and is optimally partitioned into weaklyconnected subgraphs that satisfy decomposition constraints. Spectralgraph-partitioning methods together with iterative improvementtechniques are proposed for hypergraph partitioning. A known spectralK-partitioning formulation, which accounts for partition sizes andedge weights, is extended to graphs with also vertex weights. TheOMBD formulation is robust enough to account for computationaldemands and resources and strength of interdependencies between thecomputational modules contained in the model.     

Sufficiency for Optimal Control Problems Involving Parameters

The optimal control problem is extended to the case where the performance index, the differential constraints, and the prescribed final conditions contain parameters. The sufficient condition for a minimum is derived for nonsingular problems using the sweep method. As expected, it involves the finiteness of a matrix or the location of the conjugate point. The minimum-time navigation problem is solved as a fixed final time problem to illustrate the application of the theory.     

Optimal location of the support of the control for the 1-D wave equation: numerical investigations

We consider in this paper the homogeneous 1-D wave equation defined on Ω⊂ℝ. Using the Hilbert Uniqueness Method, one may define, for each subset ω⊂Ω, the exact control v _ω of minimal L ²(ω×(0,T))-norm which drives to rest the system at a time T>0 large enough. We address the question of the optimal position of ω which minimizes the functional . We express the shape derivative of J as an integral on ∂ ω×(0,T) independently of any adjoint solution. This expression leads to a descent direction for J and permits to define a gradient algorithm efficiently initialized by the topological derivative associated with J. The numerical approximation of the problem is discussed and numerical experiments are presented in the framework of the level set approach. We also investigate the well-posedness of the problem by considering a relaxed formulation.     

Relaxation Through Homogenization for Optimal Design Problems with Gradient Constraints

The problem of the relaxation of optimal design problems for multiphase composite structures in the presence of constraints on the gradient of the state variable is addressed. A relaxed formulation for the problem is given in the presence of either a finite or infinite number of constraints. The relaxed formulation is used to identify minimizing sequences of configurations of phases.     

Some Existence Results for Vector Quasivariational Inequalities Involving Multifunctions and Applications to Traffic Equilibrium Problems

Some existence results for vector quasivariational inequalities with multifunctions in Banach spaces are derived by employing the KKM-Fan theorem. In particular, we generalize a result by Lin, Yang and Yao, and avoid monotonicity assumptions. We also consider a new quasivariational inequality problem and propose notions of weak and strong equilibria while applying the results to traffic network problems.     

Optimal Control and Stabilization for Some Fisher-Like Models

This article concerns optimal control and stabilization for some Fisher-like models with control acting in a subdomain ω. We investigate the optimal position of ω for some optimal harvesting problems. First, we refer to a logistic model with diffusion. We remember the necessary optimality conditions, and then obtain an iterative method to improve the position of ω for the optimal harvesting effort (for a simplified model without logistic term). Next, we consider the null stabilization for a controlled Fisher model and obtain a descent method to improve the position of ω in order to get a faster stabilization to zero. Numerical tests illustrating the effect of the last method are given. We also studied the null stabilization for a prey-predator system and have reduced it to the study of the null stabilizability for a related Fisher model.     

Time-Periodic Fitzhugh-Nagumo Equation and the Optimal Control Problems

In this work, the authors considered the periodic optimal control problem of Fitzhugh-Nagumo equation. They firstly prove the existence of time-periodic solution to Fitzhugh-Nagumo equation. Then they show the existence of optimal solution to the optimal control problem, and finally the first order necessary condition is obtained by constructing an appropriate penalty function.     

On Global Optimality Conditions for Nonlinear Optimal Control Problems

Let a trajectory and control pair maximize globally the functional g(x(T)) in the basic optimal control problem. Then (evidently) any pair (x,u) from the level set of the functional g corresponding to the value g( (T)) is also globally optimal and satisfies the Pontryagin maximum principle. It is shown that this necessary condition for global optimality of turns out to be a sufficient one under the additional assumption of nondegeneracy of the maximum principle for every pair (x,u) from the above-mentioned level set. In particular, if the pair satisfies the Pontryagin maximum principle which is nondegenerate in the sense that for the Hamiltonian H, we have along the pair on [0,T], and if there is no another pair (x,u) such that g(x(T))=g( (T)), then is a global maximizer.     

Homogenization of Optimal Control Problems for Functional Differential Equations

The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence.     

Existence and Structure of Optimal Solutions of Infinite-Dimensional Control Problems

In this work we analyze the structure of optimal solutions for a class of infinite-dimensional control systems. We are concerned with the existence of an overtaking optimal trajectory over an infinite horizon. The existence result that we obtain extends the result of Carlson, Haurie, and Jabrane to a situation where the trajectories are not necessarily bounded. Also, we show that an optimal trajectory defined on an interval [0,τ] is contained in a small neighborhood of the optimal steady-state in the weak topology for all t ∈ [0,τ] \backslash E , where E \subset [0,τ] is a measurable set such that the Lebesgue measure of E does not exceed a constant which depends only on the neighborhood of the optimal steady-state and does not depend on τ . Accepted 26 July 2000. Online publication 13 November 2000.