Solving some optimal path planning problems using an approach based on measure theory

In this paper we solve a collection of optimal path planning problems using a method based on measure theory. First we consider the problem as an optimization problem and then we convert it to an optimal control problem by defining some artificial control functions. Then we perform a metamorphosis in the space of problem. In fact we define an injection between the set of admissible pairs, containing the control vector function and a collision-free path defined on free space and the space of positive Radon measures. By properties of this kind of measures we obtain a linear programming problem that its solution gives rise to constructing approximate optimal trajectory of the original problem. Some numerical examples are proposed.     

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.     

A new computational approach for solving optimal control of linear PDEs problem

In this paper, we present a new computational approach for solving an internal optimal control problem, which is governed by a linear parabolic partial differential equation. Our approach is to approximate the PDE problem by a nonhomogeneous ordinary differential equation system in higher dimension. Then, the homogeneous part of ODES is solved using semigroup theory. In the next step, the convergence of this approach is verified by means of Toeplitz matrix. In the rest of the paper, the optimal control problem is solved by utilizing the solution of homogeneous part. Finally, a numerical example is given.     

Optimal control of systems of parabolic PDEs in exploitation of oil

Optimal control problem for the exploitation of oil is investigated. The optimal control problem under consideration in this paper is governed by weak coupled parabolic PDEs and involves with pointwise state and control constraints. The properties of solution of the state equations and the continuous dependence of state functions on control functions are investigated in a suitable function space; existence of optimal solution of the optimal control problem is also proved.     

On the general inverse problem of optimal control theory

The general inverse problem of optimal control is considered from a dynamic programming point of view. Necessary and sufficient conditions are developed which two integral criteria must satisfy if they are to yield the same optimal feedback law, the dynamics being fixed. Specializing to the linear-quadratic case, it is shown how the general results given here recapture previously obtained results for quadratic criteria with linear dynamics.Dedicated to R. Bellman     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Numerical solution of a time-optimal parabolic boundary-value control problem

A special time-optimal parabolic boundary-value control problem describing a one-dimensional heat-diffusion process is solved numerically. Using a bang-bang principle recently proved by Lempio, this problem can be transformed in such a way that the variables are jumps of bang-bang controls. A discretization is performed in two steps, and the convergence of the approximate solutions is proved. Finally, an algorithm to solve the discrete problem is developed and some numerical results are discussed.The author would like to thank Prof. F. Lempio, who pointed out this problem to him, and Prof. K. Glashoff for many helpful comments and suggestions.     

Numerical treatment of a parabolic boundary-value control problem

The purpose of this paper is to present a method to compute optimal controls for a class of one-dimensional heat-diffusion processes. The approach used is in the spirit of the Ritz method and approximates the given problem with simpler tasks which are solved by means of algorithms based on the principles of semi-infinite programming. General convergence properties of the procedures are shown. Some illustrative numerical examples are also given.This research was supported by NSF Grant No. GK-31833 and by The Swedish Institute of Applied Mathematics, Stockholm, Sweden.     

An applicable method for solving the shortest path problems

A theorem of Hardy, Littlewood, and Polya, first time is used to find the variational form of the well known shortest path problem, and as a consequence of that theorem, one can find the shortest path problem via quadratic programming. In this paper, we use measure theory to solve this problem. The shortest path problem can be written as an optimal control problem. Then the resulting distributed control problem is expressed in measure theoretical form, in fact an infinite dimensional linear programming problem. The optimal measure representing the shortest path problem is approximated by the solution of a finite dimensional linear programming problem.     

A dual dynamic programming for minimax optimal control problems governed by parabolic equation

In the article, minimax optimal control problems governed by parabolic equations are considered. We apply a new dual dynamic programming approach to derive sufficient optimality conditions for such problems. The idea is to move all the notions from a state space to a dual space and to obtain a new verification theorem providing the conditions, which should be satisfied by a solution of the dual partial differential equation of dynamic programming. We also give sufficient optimality conditions for the existence of an optimal dual feedback control and some approximation of the problem considered, which seems to be very useful from a practical point of view.     

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.      

Optimal control of swinging alliances in a parabolic competition model

A system of parabolic partial differential equations describes the interaction of three populations, modeling a dynamic competition/cooperation scenario. More precisely, two populations are always competing with each other, but the third population can switch the mode of alliance with the other two populations between cooperation and competition. The control is a function measuring the strength and nature of the alliance and the goal is to maximize the population with the swinging alliance while keeping the other two populations close to each other and minimizing the cost of the alliance action. Various scenarios are illustrated with numerical results.     

Numerical approximation of nonconvex optimal control problems defined by parabolic equations

In this paper, we consider an optimal control problem for distributed systems governed by parabolic equations. The state equations are nonlinear in the control variable; the constraints and the cost functional are generally nonconvex. Relaxed controls are used to prove existence and derive necessary conditions for optimality. To compute optimal controls, a descent method is applied to the resulting relaxed problem. A numerical method is also given for approximating a special class of relaxed controls, notably those obtained by the descent method. Convergence proofs are given for both methods, and a numerical example is provided.     

Singular limit of a transmission problem for the parabolic phase-field model

A transmission problem describing the thermal interchange between two regions occupied by possibly different fluids, which may present phase transitions, is studied in the framework of the Caginalp-Fix phase field model. Dirichlet (or Neumann) and Cauchy conditions are required. A regular solution is obtained by means of approximation techniques for parabolic systems. Then, an asymptotic study of the problem is carried out as the time relaxation parameter for the phase field tends to 0 in one of the domains. It is also proved that the limit formulation admits a unique solution in a suitable weak sense.     

An optimal control problem in cancer chemotherapy

We consider a mathematical model for the control of the growth of tumor cells which is formulated as a problem of optimal control theory. It is concerned with chemotherapeutic treatment of cancer and aims at the minimization of the size of the tumor at the end of a certain time interval of treatment with a limited amount of drugs. The treatment is controlled by the dosis of drugs that is administered per time unit for which also a limit is prescribed. It is shown that optimal controls are of bang-bang type and can be chosen at the upper limit, if the total amount of drugs is large enough.     

Eigen series solutions to terminal-state tracking optimal control problems and exact controllability problems constrained by linear parabolic PDEs

Terminal-state tracking optimal control problems for linear parabolic equations are studied in this paper. The control objectives are to track a desired terminal state and the control is of the distributed type. Explicit solution formulae for the optimal control problems are derived in the form of eigen series. Pointwise-in-time L² norm estimates for the optimal solutions are obtained and approximate controllability results are established. Exact controllability is shown when the target state vanishes on the boundary of the spatial domain. One-dimensional computational results are presented which illustrate the terminal-state tracking properties for the solutions expressed by the series formulae.     

Linear optimal control problem for discrete 2-D systems with constraints

Optimal control problem for linear two-dimensional (2-D) discrete systems with mixed constraints is investigated. The problem under consideration is reduced to a linear programming problem in appropriate Hubert space. The main duality relations for this problem is derived such that the optimality conditions for the control problem are specified by using methods of the linear operator theory. Optimality conditions are expressed in terms of solutions for conjugate system.     

Optimal information acquisition for a linear quadratic control problem

This paper develops a generalization of the linear quadratic control problem with partial information. As in the standard partial information setting, it is assumed that the state variable is only observed with noise. The idea in this paper is that the information level may be chosen optimally. In real life information is costly to acquire. It is therefore a trade off between the costs of getting detailed information and the increased value this information gives. We believe that the technique we present should have potential for application within both economics and engineering.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem