Optimal control problems with no turning back

In this paper we consider the maximization of a payoff functional subject to a differential equality contraint over the class of monotonically increasing functions with values in [0, 1]. We will show that an optimal control exists, derive the system of inequalities (similar to a quasi-variational inequality) that the value function satisfies and derive various properties of the value function sufficient to characterize it. Furthermore, we derive a perturbation result using the theory of Lipschitz controls. Finally, we also consider the case when the control functions are of bounded total variation and relate the problems considered herein to the impulse control problem of Bensoussan-Lions.     

Stochastic initial boundary value problems subject to distributed and boundary noise and their optimal control

In this paper we consider a class of stochastic evolution equations arising from initial boundary value problems with both boundary and distributed noise. We prove existence and regularity of mild solutions. Then we consider a controlled version of the model and prove the existence of optimal controls and develop the necessary conditions of optimality for partially observed problems using relaxed controls.     

The closure of the sheaf of trajectories of a linear control system with integral constraints

We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.     

Approximate solutions of the bellman equation of deterministic control theory

We consider an infinite horizon discounted optimal control problem and its time discretized approximation, and study the rate of convergence of the approximate solutions to the value function of the original problem. In particular we prove the rate is of order 1 as the discretization step tends to zero, provided a semiconcavity assumption is satisfied. We also characterize the limit of the optimal controls for the approximate problems within the framework of the theory of relaxed controls.This work was done while the authors were visiting members of The Department of Mathematics of The University of Maryland at College Park.     

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     

A Technique for Stochastic Control Problems with Unbounded Control Set

We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.     

An optimal control problem for a parabolic equation in the class of smooth controls

We consider an optimal control problem for a parabolic equation with a differential constraint on the boundary. We study this problem in the class of smooth controls satisfying certain pointwise constraints. Such problems describe mass transfer processes in column-type apparatuses, taking into account the longitudinal mixing. Control functions in these problems represent flows of raw materials or finished products. For the problem under consideration we obtain a necessary optimality condition, propose a method for improving admissible controls, and carry out the numerical experiment.     

Optimal control of nonlinear hereditary systems

A function space Λ is introduced for the study of nonlinear hereditary differential equations. The properties of Λ include: it is a Banach space under the supremum norm, the continuous functions constitute a closed proper subspace, and the unit ball is sequentially compact in the weak-¹ topology. Existence, uniqueness, and continuous dependence results are obtained for solutions of a broad class of initial value problems. An optimization problem is formulated for systems which are affine in the control, and solutions are approximated by means of a sequence of problems which are finite-dimensional in the control.     

Optimal control of Markovian jump processes with partial information and applications to a parallel queueing model

We consider a stochastic control problem over an infinite horizon where the state process is influenced by an unobservable environment process. In particular, the Hidden-Markov-model and the Bayesian model are included. This model under partial information is transformed into an equivalent one with complete information by using the well-known filter technique. In particular, the optimal controls and the value functions of the original and the transformed problem are the same. An explicit representation of the filter process which is a piecewise-deterministic process, is also given. Then we propose two solution techniques for the transformed model. First, a generalized verification technique (with a generalized Hamilton–Jacobi–Bellman equation) is formulated where the strict differentiability of the value function is weaken to local Lipschitz continuity. Second, we present a discrete-time Markovian decision model by which we are able to compute an optimal control of our given problem. In this context we are also able to state a general existence result for optimal controls. The power of both solution techniques is finally demonstrated for a parallel queueing model with unknown service rates. In particular, the filter process is discussed in detail, the value function is explicitly computed and the optimal control is completely characterized in the symmetric case.     

Robust suboptimal control of nonlinear systems

In this paper, we consider a nonlinear dynamic system with uncertain parameters. Our goal is to choose a control function for this system that balances two competing objectives: (i) the system should operate efficiently; and (ii) the system’s performance should be robust with respect to changes in the uncertain parameters. With this in mind, we introduce an optimal control problem with a cost function penalizing both the system cost (a function of the final state reached by the system) and the system sensitivity (the derivative of the system cost with respect to the uncertain parameters). We then show that the system sensitivity can be computed by solving an auxiliary initial value problem. This result allows one to convert the optimal control problem into a standard Mayer problem, which can be solved directly using conventional techniques. We illustrate this approach by solving two example problems using the software MISER3.     

An adaptive parameter approach for an approximate solution of optimal control problems

In this paper, we consider a class of nonlinear optimal control problems (Bolza-problems) with constraints of the control vector, initial and boundary conditions of the state vectors. The time interval is fixed. Our approach to parametrize both the state functions and the control functions is described by general piecewise polynomials with unknown coefficients (parameters), where a fixed partition of the time interval is used. Here each of these functions in a suitable way individually will be approximated by such polynomials. The optimal control problem thus is reduced to a mathematical programming problem for these parameters. The existence of an optimal solution is assumed. Convergence properties of this method are not considered in this paper.     

Some linear nonautonomous control problems with quadratic cost

In this paper we consider two similar nonautonomous linear control problems which have quadratic cost functionals. We give necessary conditions for the problems to be optimized over an infinite interval and prove that the optimal controls are linear feedback controls. If the first problem is set in a real Hilbert space the feedback controls generate a uniformly asymptotically stable evolutionary process. In the second problem the controls generate an asymptotically stable system of neutral functional differential equations.     

Mayer and optimal stopping stochastic control problems with discontinuous cost

We study two classes of stochastic control problems with semicontinuous cost: the Mayer problem and optimal stopping for controlled diffusions. The value functions are introduced via linear optimization problems on appropriate sets of probability measures. These sets of constraints are described deterministically with respect to the coefficient functions. Both the lower and upper semicontinuous cases are considered. The value function is shown to be a generalized viscosity solution of the associated HJB system, respectively, of some variational inequality. Dual formulations are given, as well as the relations between the primal and dual value functions. Under classical convexity assumptions, we prove the equivalence between the linearized Mayer problem and the standard weak control formulation. Counter-examples are given for the general framework.     

One control problem of a pencil of trajectories

A linear control problemwith a nonsingleton initial set is dealt with. For this problem,we consider the problem of transforming this initial set along trajectories of a linear controlled system into some fixed terminal set in a finite time using a single control. Various aspects of this control problem are studied using the machinery of support functions.     

First-Order Optimality Conditions in Generalized Semi-Infinite Programming

In this paper, we consider a generalized semi-infinite optimization problem where the index set of the corresponding inequality constraints depends on the decision variables and the involved functions are assumed to be continuously differentiable. We derive first-order necessary optimality conditions for such problems by using bounds for the upper and lower directional derivatives of the corresponding optimal value function. In the case where the optimal value function is directly differentiable, we present first-order conditions based on the linearization of the given problem. Finally, we investigate necessary and sufficient first-order conditions by using the calculus of quasidifferentiable functions.     

A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains

We consider an exact boundary control problem for the wave equation with given initial and terminal data and Dirichlet boundary control. The aim is to steer the state of the system that is defined on a given domain to a position of rest in finite time. The optimal control that is obtained as the solution of the problem depends on the data that define the problem, in particular on the domain. Often for the numerical solution of the control problem, this given domain is replaced by a polygon. This is the motivation to study the convergence of the optimal controls for the polygon to the optimal controls for the given domain. To study the convergence, the values of the optimal controls that are defined on the boundaries of the approximating polygons are mapped in the normal directions of the polygon to control functions defined on the boundary of the original domain. This map has already been used by Bramble and King, Deckelnick, Guenther and Hinze and by Casas and Sokolowski. Using this map, we can show the strong convergence of the transformed controls as the polygons approach the given domain. An essential tool to obtain the convergence is a regularization term in the objective functions to increase the regularity of the state.     

Decomposition of linear-quadratic optimal control problems for two-steps systems

The paper deals with problems with state equations of two sequentially acting systems. Matching conditions for trajectories at the switch point are absent, however the minimized functional depends on values of a state trajectory in the left and right sides from the switch point. State trajectories have fixed left and right points, and, in general, they are discontinuous functions. The algorithm for solving this problem is given, which is based on sequential solving eight independent initial value problems. The formula for the minimal value of the performance index is also presented.     

Optimal discrete-valued control computation

In this paper, we consider an optimal control problem in which the control takes values from a discrete set and the state and control are subject to continuous inequality constraints. By introducing auxiliary controls and applying a time-scaling transformation, we transform this optimal control problem into an equivalent problem subject to additional linear and quadratic constraints. The feasible region defined by these additional constraints is disconnected, and thus standard optimization methods struggle to handle these constraints. We introduce a novel exact penalty function to penalize constraint violations, and then append this penalty function to the objective. This leads to an approximate optimal control problem that can be solved using standard software packages such as MISER. Convergence results show that when the penalty parameter is sufficiently large, any local solution of the approximate problem is also a local solution of the original problem. We conclude the paper with some numerical results for two difficult train control problems.     

离散优化问题最优值函数的连续性质

对优化问题的最优值研究是有意义的, 尽管有时并不知道怎样寻求最优值. 研究了几个重要的组合最优化问题的目标值随着输入值变化的连续化性质, 重点研究几个经典的、有代表性的离散优化问题：极小化最大完工时间的排序问题、背包问题、旅行商问题等, 以连续的数学分析思维模式审视离散问题. 最后, 研究了一些近似算法对应的目标函数的性质.