Elastoplastic deformation during projectile–wall collision

The Smoothed Particle Hydrodynamics method for elastic solid deformation is modified to include von Mises plasticity with linear isotropic hardening and is then used to investigate high speed collisions of elastic and elastoplastic bodies. The Lagrangian mesh-free nature of SPH makes is very well suited to these extreme deformation problems eliminating issues relating to poor element quality at high strains that limits finite element usage for these types of problems. It demonstrates excellent numerical stability at very high strains (of more than 200%). SPH can naturally track history dependent material properties such as the cumulative plastic strain and the degree of work hardening produced by its strain history. The high speed collisions modelled here demonstrate that the method can cope easily with collisions of multiple bodies and can also naturally resolve self-collisions of bodies undergoing high levels of plastic strain. The nature and the extent of the elastic and plastic deformation of a rectangular body impacting on an elastic wall and of an elastic projectile impacting on a thin elastic wall are investigated. The final plastically deformed shapes of the projectile and wall are compared for a range of material properties and the evolution of the maximum plastic strain throughout each collision and the coefficient of restitution are used to make quantitative comparisons. Both the elastoplastic projectile–elastic wall and the elastic projectile–elastoplastic wall type collisions have two distinct plastic flow regimes that create complex relationships between the yield stress and the responses of the solid bodies.     

Time optimal factorizations on compact Lie groups

In this paper we study the relationship between factorization problems on SU(2ⁿ) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control over the heat field in thin bodies local constraints on control

Asymptotic solutions of problems of optimal locally constrained control over the heat field in thin bodies are constructed and justified. These problems relate to the critical case of singularly perturbed systems (the degenerate problem has a family of solutions). Dnepropetrovsk Technical University of Railway Transport, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1200–1208, September, 1996.     

Control of distributed time delay systems with Dirichlet boundary conditions

Various optimization problems associated with the optimal controlof distributed parameter systems with time lags appearing inthe boundary conditions have been studied recently by Wang (1975),Knowles (1978), Wong (1987), Kowalewski (1987a,b, 1988a,b,c,1990a,b,c,d, 1991, 1993a,b,c,d, 1995) and Kowalewski & Duda(1992). In this paper optimal boundary control problems fordistributed systems described by linear partial differentialequations of parabolic and hyperbolic type in which constanttime delays appear in the state equations are considered. Sufficientconditions for the existence of a unique solutions of such equationswith the Dirichlet boundary conditions are proved. The performancefunctional has the quadratic form. The time horizon T is fixed.Finally, we impose some constraints on the control. Making useof Lions' scheme (Lions 1971) necessary and sufficient conditionsof optimality for the Dirichlet problem with the quadratic performancefunctional and constrained control are derived. The flow chartof the algorithm, which can be used in the numerical solvingof certain optimization problems for distributed parameter systems,is also presented.     

Optimal processes in smooth-convex minimization problems

The paper elaborates a general method for studying smooth-convex conditional minimization problems that allows one to obtain necessary conditions for solutions of these problems in the case where the image of the mapping corresponding to the constraints of the problem considered can be of infinite codimension. On the basis of the elaborated method, the author proves necessary optimality conditions in the form of an analog of the Pontryagin maximum principle in various classes of quasilinear optimal control problems with mixed constraints; moreover, the author succeeds in preserving a unified approach to obtaining necessary optimality conditions for control systems without delays, as well as for systems with incommensurable delays in state coordinates and control parameters. The obtained necessary optimality conditions are of a constructive character, which allows one to construct optimal processes in practical problems (from biology, economics, social sciences, electric technology, metallurgy, etc.), in which it is necessary to take into account an interrelation between the control parameters and the state coordinates of the control object considered. The result referring to systems with aftereffect allows one to successfully study many-branch product processes, in particular, processes with constraints of the “bottle-neck” type, which were considered by R. Bellman, and also those modern problems of flight dynamics, space navigation, building, etc. in which, along with mixed constraints, it is necessary to take into account the delay effect. The author suggests a general scheme for studying optimal process with free right endpoint based on the application of the obtained necessary optimality conditions, which allows one to find optimal processes in those control systems in which no singular cases arise. The author gives an effective procedure for studying the singular case (the procedure for calculating a singular control in quasilinear systems with mixed constraints. Using the obtained necessary optimality conditions, the author constructs optimal processes in concrete control systems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 42, Optimal Control, 2006.     

A consistent direct transcription method for optimal control problems in multibody dynamics

The present work deals with optimal control problems governed by differential-algebraic equations (DAEs). In particular, the control effort, which is necessary for moving a multibody system from one configuration to another, will be minimized. The orientation of the rigid bodies will be described using directors, which facilitates the integration of the equations of motion with an energy-momentum consistent time-stepping scheme [1]. This type of structure-preserving integrators offer outstanding numerical stability and robustness properties in comparison to the often applied generalized coordinates formulation. In the context of optimal control, other kinds of consistent integrators have been applied previously in [2] and [3]. We will test the different formulations with two numerical examples, a 3-link manipulator and a satellite. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the approximate synthesis of the optimal control of stochastic quasilinear systems with aftereffect : PMM vol. 42, no. 6, 1978, pp. 978–988

Control problems for quasilinear deterministic systems without time lag were analyzed in [1, 2]. In the present paper the control of quasilinear stochastic systems, whose theory has been presented in [3–6], is studied. The approximate synthesis of the control of stochastic systems with aftereffect is of importance since the construction of their exact optimal control is successful only in exceptional cases [7, 8]. In the paper an approximate optimal control synthesis algorithm is proposed and a method for obtaining error bounds, different from ones previously obtained [9, 10], is developed.     

On the Optimal Stabilisation for a set of Programmed Motions of the Bilinear Control System

During recent years there has been considerable interest in using bilinear systems [1, 2] as mathematical models to represent the dynamic behavior of a wide class of engineering, biological and economic systems. Moreover, there are some methods [3] which may approximate nonlinear control systems by bilinear systems. For the first time Zubov has proposed a method of stabilization control synthesis for a set of programmed motions in linear systems [4]. In papers [5, 6] this method has been developed and used to solve the same problem for bilinear systems. In the present paper the following problems are considered. First, synthesis of nonlinear control as feedback under which the bilinear control system has a given set of programmed and asymptotic stable motions. Because this control is not unique, the second problem concerns optimal stabilization. In this paper a method for the design of nonlinear optimal control is suggested. This control is constructed in the form of a convergent series. The theorem on the sufficient conditions to solve this problem is represented.     

Rotation-Symmetric Referencebodys For Energy Efficient Flow Around Bodies

Topology optimization is used to optimize problems of flow around bodies and problems of guided flow. Within the context of research, optimization criteria are developed to increase the energy efficiency of these problems [1] [2] [3] [4] [5]. In order to evaluate the new criteria in respect to the increasing of energy efficiency, reference bodies for different Reynolds numbers in combination with given design space limitations are needed. Therefore, an optimal body at Reynolds number against 0 was analytically determined by Bourot [6]. At higher Reynolds numbers, in the range of laminar and turbulent flows, no analytical solution is known. Accordingly, reference bodies are calculated by CFD calculations at three technical relevant Reynolds numbers (1.000, 32.000, 100.000) in combination with parameter optimization. The cross section of the bodies is described by a parameterized model. To get the optimal body, a parameter optimization based on a “brute force”; algorithm is used to optimize with regard to the friction loss and pressure loss in order to minimize the total loss (c_d-value). The result is an optimal parameter constellation, depending on the Reynolds number. Within the results, it is possible to develop the optimal geometries. The identified characteristics of the flow field around these bodies are used as base for new optimization criteria. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of numerical optimal control to nonlinear hybrid systems

This paper develops a technique for numerically solving hybrid optimal control problems. The theoretical foundation of the approach is a recently developed methodology by S.C. Bengea and R.A. DeCarlo [Optimal control of switching systems, Automatica. A Journal of IFAC 41 (1) (2005) 11–27] for solving switched optimal control problems through embedding. The methodology is extended to incorporate hybrid behavior stemming from autonomous (uncontrolled) switches that results in plant equations with piecewise smooth vector fields. We demonstrate that when the system has no memory, the embedding technique can be used to reduce the hybrid optimal control problem for such systems to the traditional one. In particular, we show that the solution methodology does not require mixed integer programming (MIP) methods, but rather can utilize traditional nonlinear programming techniques such as sequential quadratic programming (SQP). By dramatically reducing the computational complexity over existing approaches, the proposed techniques make optimal control highly appealing for hybrid systems. This appeal is concretely demonstrated in an exhaustive application to a unicycle model that contains both autonomous and controlled switches; optimal and model predictive control solutions are given for two types of models using both a minimum energy and minimum time performance index. Controller performance is evaluated in the presence of a step frictional disturbance and parameter uncertainties which demonstrates the robustness of the controllers.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

A constructive method of establishing the validity of the theory of systems with non-retaining constraints

The formal axiomatic approach to establishing the validity of the theory of constrained systems has obvious disadvantages: the source of the initial axioms (such as the Befreiungsprinzip and the conditions for constraints to be ideal) remains unclear. A constructive method is proposed for establishing the validity of the main principles of the dynamics of unilaterally constrained systems (including systems with collisions). The idea of the method is related to the analysis of physical methods for realising constraints (stiff systems, anisotropic viscosity, and apparent additional masses). This approach yields simple equations of motion, suitable for the entire time interval and more accurately incorporating the actual dynamics. Several problems of the mechanics of oscillatory systems with collisions are solved by the method. In particular, conditions are determined for the stability of periodic oscillatory modes and a study is made of the evolution of motion with inelastic collisions when the coefficient of restitution is close to unity. Total integrability is established and a qualitative analysis is presented of the problem of parabolic billiards in a uniform force field.     

Optimal control of systems with discontinuous differential equations

In this paper we discuss the problem of verifying and computing optimal controls of systems whose dynamics is governed by differential systems with a discontinuous right-hand side. In our work, we are motivated by optimal control of mechanical systems with Coulomb friction, which exhibit such a right-hand side. Notwithstanding the impressive development of nonsmooth and set-valued analysis, these systems have not been closely studied either computationally or analytically. We show that even when the solution crosses and does not stay on the discontinuity, differentiating the results of a simulation gives gradients that have errors of a size independent of the stepsize. This means that the strategy of “optimize the discretization” will usually fail for problems of this kind. We approximate the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side. For these smoothed approximations, we show that the resulting gradients approach the true gradients provided that the start and end points of the trajectory do not lie on the discontinuity and that Euler’s method is used where the step size is “sufficiently small” in comparison with the smoothing parameter. Numerical results are presented for a crude model of car racing that involves Coulomb friction and slip showing that this approach is practical and can handle problems of moderate complexity.     

Pontryagin's Principle of Mixed Control-State Constrained Optimal Control Governed by Fluid Dynamic Systems

This article deals with Pontryagin's minimum principle for optimal control problems governed by an abstract fluid dynamical system with mixed control-state constraints. Two techniques are mainly used to obtain our results: ?-perturbation for admissible control set and diffuse perturbation for admissible control. Using these results, the necessary conditions for L ²-local optimal solution of our control problems can be studied under some assumptions. In the end of this article, these results are applied to some special fluid dynamical systems which contain the fluid systems driven by its boundary and magnetohydrodynamics(MHD).     

Esistenza e condizioni necessarie per controlli ottimi di sistemi definiti da equazioni integrali

Summary In the first two sections of this paper existence theorems are proved for optimal control problems described either by Fredholm systems of integral equations (with control appearing nonlinearly) on a given (possibly infinite) measure space, eventually in presence of eigenvalues, or by Urysohn systems, both with pointwise and integral constraints on controls and states. Controllability is assumed throughout the paper. In the last section necessary conditions are proved for optimal controls of Fredholm systems (a minimum principle in pointwise, or ? differential ? form), and theorems (that seem to be new also for the corresponding ordinary differential control systems) are deduced about bang-bang and regularization properties of the optimal control (s) (which turns out to be a continuous function except for a finite number of points, if positivity and semicontinuity assumptions hold about the data): in some particular cases the optimal control will be piecewise constant. All these results are independent on the known ones.

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Lavoro eseguito nell'ambito del Centro di Matematica e Fisica Teorica del C.N.R. presso l'Università di Genova.

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Entrata in Redazione il 6 maggio 1971.     

Calculation of an optimal region of operation for dual response systems fitted from experimental data

While there is abundant literature on Response Surface Methodology (RSM) about how to seek optimal operating settings for Dual Response Systems (DRS) using various optimisation approaches, the inherent sampling variability of the fitted responses has typically been neglected. That is, the single global optimum settings for the fitted response represent the expected value of the fitted functions since the true response systems are, in general, noisy and unknown in many engineering and scientific experiments. This paper presents an approach for DRS based on Monte Carlo simulation of the system under study. For each simulated set of responses, a new global optimisation algorithm for DRS is utilised to compute the global optimal factor settings. Repetition of this process constructs an optimal region in the control factor space that provides more useful information to a process engineer than a single optimal—expected—solution. It is shown how the optimal region can be used as an indicator of how trustworthy this single solution is, and as a set of alternative solutions from where an engineer can select other process settings in case limitations not considered by the DRS model prevent the adoption of the single expected optimum. Application to Taguchi's Robust Parameter Design problems illustrates the proposed method.     

Weak maximum principle for optimal control problems of nonsmooth systems

In this paper, by considering vector-valued maximum type functions satisfying Lipschitz condition, and optimal control systems with continuous-time which is governed by systems of ordinary differential equation, we derive results similar to Pontryagin’s maximum principle and properties concerning the generalized Jacobian set for optimal control problems of these systems.     

The achievable region method in the optimal control of queueing systems; formulations,bounds and policies

We survey a new approach that the author and his co-workers have developed to formulate stochastic control problems (predominantly queueing systems) asmathematical programming problems. The central idea is to characterize the region of achievable performance in a stochastic control problem, i.e., find linear or nonlinear constraints on the performance vectors that all policies satisfy. We present linear and nonlinear relaxations of the performance space for the following problems: Indexable systems (multiclass single station queues and multiarmed bandit problems), restless bandit problems, polling systems, multiclass queueing and loss networks. These relaxations lead to bounds on the performance of an optimal policy. Using information from the relaxations we construct heuristic nearly optimal policies. The theme in the paper is the thesis that better formulations lead to deeper understanding and better solution methods. Overall the proposed approach for stochastic control problems parallels efforts of the mathematical programming community in the last twenty years to develop sharper formulations (polyhedral combinatorics and more recently nonlinear relaxations) and leads to new insights ranging from a complete characterization and new algorithms for indexable systems to tight lower bounds and nearly optimal algorithms for restless bandit problems, polling systems, multiclass queueing and loss networks.     

Virtual and effective control for distributed systems and decomposition of everything

For the numerical approximation of the solution of boundary value problems (BVP), decomposition techniques are very important, in particular in view of parallel computations. The same is true, in principle, for optimal control of distributed systems, i.e., systems governed (modelled) by partial differential equations (PDE). Very many techniques have been studied for the approximation of BVP, such as DDM (domain decomposition method), decomposition of operators (splitting up, for instance). In contrast, not so many techniques of decomposition have been used in control problems for distributed systems, as pointed out in the contributions of Benamou [1], Benamou and Després [2], and Lagnese and Leugering [11]. However, it has been observed by Pironneau and Lions [24], [26] that by using so-calledvirtual controls, systematic DDM can be obtained, and that problems of optimal control and analysis of BVP can be considered in the same framework. We then deal with virtual control problems for BVP, virtual and effective control problems for the control of PDE (cf. Pironneau and Lions [25]). Using the idea of virtual control in other guises, Glowinski, Lions and Pironneau [9] have shown how to obtain new decomposition methods for the energy spaces (cf. Section 3), and Pironneau and Lions [27] have shown how to obtain systematically operator decomposition in BVP. In the present paper, we show (without assuming prior knowledge) how to apply the virtual control ideas in several different guises to the “decomposition of everything” for PDE of evolution and for their control. In this way, one can decompose the geometrical domain, the energy space and the operator. This is briefly presented in Sections 2, 3 and 4. We show in Section 5 how one can simultaneously apply two of the decomposition techniques and also indicate briefly how virtual control ideas can be used in case of bilinear control. The content of this paper is presented here for the first time. It is part of a systematic program which is in progress, developed with several colleagues. I wish to thank particularly F. Hecht, R. Glowinski, J. Periaux, O. Pironneau, H. Q. Chen and T. W. Pan. Of course we do not claim by any means that the methods based on “virtual control” are “better” than the many decomposition techniques already available (no attempt has been made to compile a Bibliography on these topics). Numerical works in progress show that the methods are “not bad”, but no serious benchmarking has been made yet. Possible interest lies in the fact thatone technique, with some variants, seems to lead to all possible Decompositions of Everything.