Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.     

A minimax optimal control problem

A control system x=f(t,x,u) is considered, and a cost functional ess supT ₀tT ₁ G(t, x(t),u(t)) is to be minimized. Necessary conditions for optimality (maximum principle and transversality conditions) are derived. It is also shown that an optimal control is optimal for the corresponding problem on a subinterval of [T ₀,T ₁], if a certain controllability condition is satisfied.     

On an optimal control problem

In this paper, we applied the finite differences method to the solution of variational problem of an inverse problem for the Scrödinger equation with a final functional. These types of problems arise in various fields in quantum-mechanical, nuclear physics and modern physics [2, 11]. Also, we prove two estimates for the differences scheme and convergence speed of difference approximations according to the functional. The inverse problems for the Schrödinger equation having different variational formulation were investigated in [7, 12, 13].     

Balance function for the optimal control problem

The balance-function concept for transforming constrained optimization problems into unconstrained optimization problems, for the purpose of finding numerical iterative solutions, is extended to the optimal control problem. This function is a combination orbalance between the penalty and Lagrange functions. It retains the advantages of the penalty function, while eliminating its numerical disadvantages. An algorithm is developed and applied to an orbit transfer problem, showing the feasibility and usefulness of this concept.These results are part of the author's doctoral thesis written under Professors H. Lo and D. Alspaugh of Purdue University.     

Bang-bang optimal control for the dam problem

The dam problem with general geometry is considered. Fluid is drawn from the bottomS ₁ at a ratek where 0 k N, _{S 1} k M; the objective is to minimize the total pressure of the fluid in the dam. A bang-bang principle is established for any optimal controlk ₀, that is,k ₀ = 0 on a setA andk ₀ =N on the complement setS ₁ A. In the case of a rectangular dam the structure ofA is determined and the uniqueness of the minimizerk ₀ is established.This work is partially supported by National Science Foundation Grants DMS-8501397 and DMS-8420896.     

On a student-related optimal control problem

In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.This paper is a modified and improved version of Ref. 4. It is based, in part, on research sponsored by NSF.     

Duality theorem for the stochastic optimal control problem

We prove a duality theorem for the stochastic optimal control problem with a convex cost function and show that the minimizer satisfies a class of forward–backward stochastic differential equations. As an application, we give an approach, from the duality theorem, to h$h$-path processes for diffusion processes.     

An optimal control problem for the Hoff equation

The sufficient and necessary conditions are given for existence of an optimal control in the bending problem for an I-beam.     

Remarks on a simple optimal control problem with monotone control functions

We consider the minimization of the mean-square deviation of a prescribed function from the class of monotone functions. Two problems are considered. The first problem places no restriction on the initial value of the controls, while the second problem assumes that all the control functions must start at a fixed initial value. Optimal controls are exhibited in both problems. Finally, we consider the situation with general payoff and dynamics and give the heuristic characterization of the value function for such problems.     

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     

Criteria of optimality in the infinite-time optimal control problem

The present paper gives a systematic presentation of different definitions of optimality in the infinite-time optimal control problem. Some of these definitions are new, while others have been used throughout the literature, sometimes with different names. The logical implications between them are clearly stated, corresponding comparison criteria for solutions are defined, and other relations as well as two types of equivalence relations are established.This work was supported in part by Simmons College Fund for Research, Grant No. 201.     

The problem of controlling resources on network graphs as an optimal control problem

The problem of optimal distribution of resources dedicated to a certain complex of inter-related tasks according to the criterion of minimum execution time of all tasks is described. A reenterable (reusable) resource is considered instead of a traditional separable-type resource supposing a fixed distribution among tasks. The problem is formalized in direct static and dynamic settings. The latter is a classical performance optimal control problem. The correctness of the formalization is substantiated.     

Design problem solved by optimal control theory

In this paper we present an application to airfoil design of an optimum design method based on optimal control theory. The method used here transforms the design problem by way of a change of variable into an optimal control problem for a distributed system with Neumann boundary control. This results in a set of variational inequalities which is solved by adding a penalty term to the differential equation. This is in turn solved by a finite element method.     

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.     

A non-well-posed problem in convex optimal control

This article is devoted to the study of a non-well-set problem of optimal control for a system governed by an elliptic partial differential equation. By generalizing some classical methods in convex optimization, we state and prove the system of necessary and sufficient conditions for such a problem.     

An optimal control problem in image processing

As a starting point, we present a control problem in mammographic image processing which leads to non-standard penalty terms and involves a degenerate parabolic PDE which has to be controlled in the coefficients. We then discuss the classical conditional gradient method from constrained optimization and propose a generalization for non-convex functionals which covers the conditional gradient method as well as the recently proposed iterative shrinkage method of Daubechies, Defrise and De Mol for the solution of linear inverse problems with sparsity promoting penalty terms. We prove that this new algorithm converges. This also gives a deeper understanding of the iterative shrinkage method. Further, we show an application to the above-mentioned control problem in image processing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)