Optimal design of structures with unspecified loading distribution

The problem of the optimal distribution of loading on a structure that corresponds to the minimum of the elastic compliance or the maximum of the safety factor for plastic collapse is considered. Optimality criteria are derived, and their applicability is illustrated in the case of beams. Besides the optimally varying cross section, also the support positions and the load distribution are determined from the optimal solution.     

Optimal design of elastic structures for multipurpose loading

The optimal design of elastic beams subjected to two alternative loading systems is considered for compliance constraints and a minimum-cross-section constraint. Sufficient conditions for optimality are established, and a technique for determining the optimal design is presented. Two examples are given. Generalizations to more than two loading systems and more complex structures are straightforward.     

Optimal control of impulsive hybrid systems

In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms.     

Optimal design of elastic columns subject to the general conservative behaviour of loading

Zusammenfassung Optimale Formen eines gedrückten elastischen Stabes werden bei allgemeinem Verhalten der Druckkraft nach dem Stabilitätsverlust untersucht. Als Zielfunktion wird das minimale Volumen angenommen unter der Nebenbedingung einer konstanten kritischen Belastung. Das konservative Verhalten der Kraft ist mit einer Beziehung verbunden, welche die allgemeine Euler-Lagrangesche Gleichung vereinfacht. Es werden räumlich sowie eben verjüngte Stäbe untersucht und die Resultate bei einigen Beispielen diskutiert.     

Optimal synthesis in the problem of impulsive correction of motion

A minimax problem of correcting the motion of a dynamic system acted upon by perturbing forces of restricted magnitude is considered. The corrective action is carried out in the form of impulsive control, with a restriction on the total magnitude of the impulses and their number. This formulation models the problem of the impulsive correction of motion of an aircraft acted upon by external perturbations. The problem represents, in fact, a differentially-impulsive game /1/ in which the player controlling the correction aims to secure for himself a guaranteed minimum of the terminal functional. Following the methods used in /2/ we construct, for the problem with isotropic dynamics, an optimal synthesis of the correction instances. The present paper touches on the work done in /3–5/.     

Optimal impulsive control of piecewise deterministic Markov processes

In this paper, we study the infinite-horizon expected discounted continuous-time optimal control problem for Piecewise Deterministic Markov Processes with both impulsive and gradual (also called continuous) controls. The set of admissible control strategies is supposed to be formed by policies possibly randomized and depending on the past-history of the process. We assume that the gradual control acts on the jump intensity and on the transition measure, but not on the flow. The so-called Hamilton–Jacobi–Bellman (HJB) equation associated to this optimization problem is analyzed. We provide sufficient conditions for the existence of a solution to the HJB equation and show that the solution is in fact unique and coincides with the value function of the control problem. Moreover, the existence of an optimal control strategy is proven having the property to be stationary and non-randomized.     

Optimal control of impulsive systems with nonlocal boundary conditions

In this paper we study an optimal control problem, where states of a control system are described by impulsive differential equations with nonlocal boundary conditions. With the help of the contraction principle we prove the existence and uniqueness of a solution to the corresponding boundary value problem with fixed admissible controls. We calculate the first and second variation of the functional. Using the variation of controls, we establish various necessary optimality conditions of the second order.     

Optimal design via variational principles: the three-dimensional case

The analysis of some three-dimensional optimal design problems leads us to study variational principles under curl-free and divergence-free constraints simultaneously. We explicitly exploit the relationship between curl-free and div-free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials. Our discussion takes place in the three-dimensional situation. This is a first contribution in the sense that we only deal with the most basic issues.     

Failure of water-filled cylindrical glass-reinforced epoxy shells under internal impulsive loading

Conclusions 1. The explosive failure of water-filled, geometrically similar glass-reinforced epoxy shells, whose dimensions increase by a factor of 1.5–4.4, is not associated with an energetic scale effect. This is attributable to the imperfect similarity of the shells, since although their dimensions changed, the diameter of the glass reinforcing fibers remained the same.2. The relative mass of explosive required to fracture the shells is approximately 0.4%. With respect to this index, glass-reinforced epoxy is comparable to structural steels 20 and 17Mn1Si at R₀160 mm.3. The circumferential deformation of the shells at failure is about 4% and does not depend on the thickness of the shell wall, the dimensions of geometrically similar shells, or the initial strain rate on the interval 0.21 · 10³–1.2 · 10³ sec^–1. Thin-walled shells begin to fail from the outside surface, thick-walled shells from the inside surface.4. The modulus of elasticity of the GRE is 2.4 · 10⁵ kgf/cm² and does not depend on the strain rate on the interval 10^–3–1.5 · 10³ sec^–1. The material deforms elastically up to failure.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 283–289, March–April, 1978.     

Optimal impulsive control of compartment models,II: Algorithm

The optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. Under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of an ordinary finite-dimensional convex programming problem. An iterative algorithm is given for the situation in which the critical points cannot all be determineda priori.This work was supported in part by the National Science Foundation under Grant No. MPS-74-13332.     

Optimal design and sensitivity of large scale plane frames for the case of uncertainty

This paper deals with the optimal design and the robustness of large scale plane frames in dependence of their height. Using the first collapse theorem, the necessary and sufficient survival conditions of an elasto-plastic structure consist of the yield condition and the equilibrium condition. The basis for our consideration is provided by a plane n-storey frame which will be increased successively, and which is affected by applied random forces and moments. Taking into account these random applied loads, we get a stochastic structural optimization problem which cannot be solved using the traditional methods. Instead of that, an appropriate (deterministic) substitute problem is formulated. First, the recourse problem will be formulated in general and in the standard form of stochastic linear programming (SLP), and after the formulation of the stochastic optimization problem, the Recourse Problem based on Discretization (RPD) is introduced as a representative of substitute problems. The resulting (large scale) linear program (LP) can be solved efficiently by means of usual LP-solvers. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of impulsive Volterra equations with variable impulse times

We obtain necessary conditions of optimality for impulsive Volterra integral equations with switching and impulsive controls, with variable impulse time-instants. The present work continues and complements our previous work on impulsive Volterra control with fixed impulse times.     

Optimal control of impulsive switched systems with minimum subsystem durations

This paper presents a new computational approach for solving optimal control problems governed by impulsive switched systems. Such systems consist of multiple subsystems operating in succession, with possible instantaneous state jumps occurring when the system switches from one subsystem to another. The control variables are the subsystem durations and a set of system parameters influencing the state jumps. In contrast with most other papers on the control of impulsive switched systems, we do not require every potential subsystem to be active during the time horizon (it may be optimal to delete certain subsystems, especially when the optimal number of switches is unknown). However, any active subsystem must be active for a minimum non-negligible duration of time. This restriction leads to a disjoint feasible region for the subsystem durations. The problem of choosing the subsystem durations and the system parameters to minimize a given cost function is a non-standard optimal control problem that cannot be solved using conventional techniques. By combining a time-scaling transformation and an exact penalty method, we develop a computational algorithm for solving this problem. We then demonstrate the effectiveness of this algorithm by considering a numerical example on the optimization of shrimp harvesting operations.     

On the construction of primitive elements in field extensions

The subject of this article are a priori constructions of primitive elements in field extensions. Consider the totality of all separable polynomialsf of degreen over a fieldK with rootsx ₁,...,x _n and prescribed Galois groupG. A vector (b ₁,...,b _n )K ⁿ is called stably primitive (forG), if, for each suchf,b ₁ x ₁+...+b _n x _n generates the splitting field off. We develop representation theoretical devices to investigate the set of stably primitive vectors geometrically. A fundamental observation is that is either very large or very small (or even empty). These two cases are illustrated by various examples. Moreover, criteria are given to decide which case holds. For a number of groups where is recognized to be small we show.     

Canonical construction of finite elements

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, and . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for and . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: , and can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar - and -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

     

Optimal impulsive control of compartment models,I: Qualitative aspects

Optimal impulsive control of systems arising from linear compartment models for drug distribution in the human body is considered. A system of linear, time-invariant, homogeneous differential equations is given along with a set of continuous constraints on state and control. The object is to develop a constructive algorithm for the computation of the optimal control relative to a convex cost functional. It is first shown that under suitable hypotheses, satisfying the continuous constraints is equivalent to satisfying the constraints at a finite set of abstractly definedcritical points. Once these critical points have been determined, the solution of the optimal control problem is found as the solution of a finite-dimensional convex programming problem. The set of critical points can often be determineda priori solely from the qualitative behavior of the solutions of the system. A class of such problems, generalizing the so-calledplateau effect, is considered in detail. It is shown that the solution achieving the plateau effect is indeed optimal in certain cases. In a subsequent paper, an iterative algorithm will be given for the solution of these problems when the critical points cannot all be determineda priori.This work was supported in part by the National Science Foundation under Grant No. GP-20130.