Shapes and measures

It is a well-known fact that a measurable set a shape can beconsidered as a measure; the aim of this work is to solve anoptimal-shape problem in such a way that it also answers thequestion of whether measures can be considered as shapes. Thispaper introduces a new method for solving problems of optimalshape design; by a process of embedding, the problem is replacedby another in which we seek to minimize a linear form over asubset of the product of two measure spaces defined by linearequalities. This minimization is global, and the theory allowsus to develop a computational method which enables us to findthe solution by finite-dimensional linear programming. The nearlyoptimal pair (C, dC) is obtained via the optimal pair of measuresby an approximation procedure. It is sometimes necessary toapply a standard minimization algorithm, because of some limitationsin the accuracy. Some examples are presented.     

Optimal control problems with unbounded constraint sets

We consider in this paper optimal control problems in which some of the constraint sets are unbounded. Firstly we deal with problems in which the control set is unbounded, so that ‘impulses’ are allowed as admissible controls, discontinuous functions as admissible trajectories. The second type of problem treated is that of infinite horizons, the time set being unbounded. Both class of problems are treated in a similar way. Firstly, a problem is transformed into a semi-infinite linear programming problem by embedding the spacesof admissible trajectory-control pairs into spaces of measures. Then this is mapped into an appropriate nonstandard structure, where a near-minimizer is found for the non-standard optimization; this entity is mapped back as a minimizer for the original problem. An appendix is including introducing the basic concepts of nonstandard analysis

Numerical methods are presented for the estimation of the minimizing measure, and the construction of nearly optimal trajectory-control pairs. Examples are given involving multiplicative controls     

An approach for solving nonlinear programming problems

In this paper we use measure theory to solve a wide range of the nonlinear programming problems. First, we transform a nonlinear programming problem to a classical optimal control problem with no restriction on states and controls. The new problem is modified into one consisting of the minimization of a special linear functional over a set of Radon measures; then we obtain an optimal measure corresponding to functional problem which is then approximated by a finite combination of atomic measures and the problem converted approximately to a finite-dimensional linear programming. Then by the solution of the linear programming problem we obtain the approximate optimal control and then, by the solution of the latter problem we obtain an approximate solution for the original problem. Furthermore, we obtain the path from the initial point to the admissible solution.     

Coefficient estimation in a first-order nonlinear hyperbolic Cauchy problem

This paper deals with the estimation and approximation of coefficient function in a first-order, nonlinear, hyperbolic Cauchy problem. The estimation is accomplished by minimizing a functional which measures the error between a finite set of given observations and the corresponding values of the solution generated by the coefficient function. A class of admissible coefficient functions is defined, and it is proved that minimizing coefficient function always exists within this class. We also develop an approximation by a sequence of solutions of associated finite-dimensional minimization problems.     

Optimal control of the multidimensional diffusion equation

The existence and numerical estimation of a boundary control for then-dimensional linear diffusion equation is considered. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures. The existence of an optimal measure corresponding to the above problem is shown, and the optimal measure is approximated by a finite convex combination of atomic measures. This construction gives rise to a finite-dimensional linear programming problem, whose solution can be used to construct the combination of atomic measures, and thus a piecewise-constant control function which approximates the action of the optimal measure, so that the final state corresponding to the above control function is close to the desired final state, and the value it assigns to the performance criterion is close to the corresponding infimum. A numerical procedure is developed for the estimation of these controls, entailing the solution of large, finite-dimensional linear programming problems. This procedure is illustrated by several examples.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

Computing an interior point for inequalities using linear optimization

The problem of finding the middle of a feasible region defined by solutions to a set of linear inequalities is considered. The solution of this problem is formulated as a primal-dual pair of linear optimization problems whose solutions can be obtained using linear programming computations.     

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.     

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.     

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.     

The Global Optimization of Variational Problems with Discontinuous Solutions

We consider variational problems in which the slope of the admissible curves is not necessarily bounded, so that they admit discontinuous solutions. A problem is first reformulated as one consisting of the minimization of an integral in a space of functions satisfying a set of integral equalities; this is then transfered to a nonstandard framework, in which Loeb measures take the place of the functions and a near-minimizer can always be found. This is mapped back to the standard world by means of the standard part map; its image is a minimizer, so that the optimization is global. The minimizer is shown to be the solution of an infinite dimensional linear program and by well-proven approximation procedures a finite dimensional linear program is found by means of which nearly-optimal curves can be constructed for the original problem. A numerical example is given.     

Weak global controllability of nonlinear systems

The problem of the controllability of nonlinear systems in finite-dimensional state spaces is considered in a measure-theoretical framework, in which we deal with a set of measured defined by the boundary conditions and the differential equations of the problem. The property of weak controllability is then equivalent to the existence of a positive measure satisfying a set of linear equalities and inequalities. This problem is solved by considering the extension of an associated linear functional, defined in a suitable subspace of the space of functions being used. Necessary and sufficient conditions for weak controllability are obtained.     

Control the Fibre Orientation Distribution at the Outlet of Contraction

A two dimensional model of the orientation distribution of fibres in a paper machine headbox is studied. The goal is to control the fibre orientation distribution at the outlet of contraction by changing its shape. The mathematical formulation leads to an optimization problem with control in coefficients of a linear convection-diffusion equation as the state problem. Then, the problem is expressed as an optimal control problem governed by variational forms. By using an embedding method, the class of admissible shapes is replaced by a class of positive Radon measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this linear programming problem. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to shape variation design to a one dimensional headbox. The usefulness of this idea is that the method is not iterative and it does not need any initial guess of the solution.      

An L \infty/ L 1-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L_\infty norm and in the L₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

An <Emphasis Type="Italic">L</Emphasis><Subscript>\infty</Subscript>/<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-Constrained Quadratic Optimization Problem with Applications to Neural Networks

Pattern formation in associative neural networks is related to a quadratic optimization problem. Biological considerations imply that the functional is constrained in the L _\infty norm and in the L ₁ norm. We consider such optimization problems. We derive the Euler–Lagrange equations, and construct basic properties of the maximizers. We study in some detail the case where the kernel of the quadratic functional is finite-dimensional. In this case the optimization problem can be fully characterized by the geometry of a certain convex and compact finite-dimensional set.     

A new approach for asymptotic stability system of the nonlinear ordinary differential equations

In this paper, we use measure theory for considering asymptotically stable of an autonomous system [1] of first order nonlinear ordinary differential equations(ODE’s). First, we define a nonlinear infinite-horizon optimal control problem related to the ODE. Then, by a suitable change of variable, we transform the problem to a finite-horizon nonlinear optimal control problem. Then, the problem is modified into one consisting of the minimization of a linear functional over a set of positive Radon measures. The optimal measure is approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this linear programming problem is used to find a piecewise-constant control, and by using the approximated control signals, we obtain the approximate trajectories and the error functional related to it. Finally the approximated trajectories and error functional is used to for considering asymptotically stable of the original problem.     

On an optimal shape design problem to control a thermoelastic deformation under a prescribed thermal treatment

A shape optimization problem concerned with thermal deformation of elastic bodies is considered. In this article, measure theory approach in function space is derived, resulting in an effective algorithm for the discretized optimization problem. First the problem is expressed as an optimal control problem governed by variational forms on a fixed domain. Then by using an embedding method, the class of admissible shapes is replaced by a class of positive Borel measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this finite-dimensional linear programming problem. Numerical examples are also given.     

On the optimal value function for certain linear programs with unbounded optimal solution sets

Often, the coefficients of a linear programming problem represent estimates of true values of data or are subject to systematic variations. In such cases, it is useful to perturb the original data and to either compute, estimate, or otherwise describe the values of the functionf which gives the optimal value of the linear program for each perturbation. If the right-hand derivative off at a chosen point exists and is calculated, then the values off in a neighborhood of that point can be estimated. However, if the optimal solution set of either the primal problem or the dual problem is unbounded, then this derivative may not exist. In this note, we show that, frequently, even if the primal problem or the dual problem has an unbounded optimal solution set, the nature of the values off at points near a given point can be investigated. To illustrate the potential utility of our results, their application to two types of problems is also explained.This research was supported, in part, by the Center for Econometrics and Decision Sciences, University of Florida, Gainesville, Florida.The author would like to thank two anonymous reviewers for their most useful comments on earlier versions of this paper.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

Inverse Problems of Submodular Functions on Digraphs

In this paper, we study the inverse problem of submodular functions on digraphs. Given a feasible solution x* for a linear program generated by a submodular function defined on digraphs, we try to modify the coefficient vector c of the objective function, optimally and within bounds, such that x* becomes an optimal solution of the linear program. It is shown that the problem can be formulated as a combinatorial linear program and can be transformed further into a minimum cost circulation problem. Hence, it can be solved in strongly polynomial time. We also give a necessary and sufficient condition for the feasibility of the problem. Finally, we extend the discussion to the version of the inverse problem with multiple feasible solutions.