Optimal driving strategies for a train journey with non-zero track gradient and speed limits

In this paper we formulate and solve an important problem inapplied optimal control. A train is to be driven along a trackwith non-zero gradient, where speed limits are imposed. Thejourney is to be completed within a specified time using aslittle fuel as possible. We find key equations that determinestrategies of optimal type and present a general solution algorithm.Several specific examples will be given to illustrate the solutionprocedure.     

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.     

Direct methods with maximal lower bound for mixed-integer optimal control problems

Many practical optimal control problems include discrete decisions. These may be either time-independent parameters or time-dependent control functions as gears or valves that can only take discrete values at any given time. While great progress has been achieved in the solution of optimization problems involving integer variables, in particular mixed-integer linear programs, as well as in continuous optimal control problems, the combination of the two is yet an open field of research. We consider the question of lower bounds that can be obtained by a relaxation of the integer requirements. For general nonlinear mixed-integer programs such lower bounds typically suffer from a huge integer gap. We convexify (with respect to binary controls) and relax the original problem and prove that the optimal solution of this continuous control problem yields the best lower bound for the nonlinear integer problem. Building on this theoretical result we present a novel algorithm to solve mixed-integer optimal control problems, with a focus on discrete-valued control functions. Our algorithm is based on the direct multiple shooting method, an adaptive refinement of the underlying control discretization grid and tailored heuristic integer methods. Its applicability is shown by a challenging application, the energy optimal control of a subway train with discrete gears and velocity limits.      

Approximate Gradient Projection Method with Runge-Kutta Schemes for Optimal Control Problems

We consider an optimal control problem for systems governed by ordinary differential equations with control constraints. The state equation is discretized by the explicit fourth order Runge-Kutta scheme and the controls are approximated by discontinuous piecewise affine ones. We then propose an approximate gradient projection method that generates sequences of discrete controls and progressively refines the discretization during the iterations. Instead of using the exact discrete directional derivative, which is difficult to calculate, we use an approximate derivative of the cost functional defined by discretizing the continuous adjoint equation by the same Runge-Kutta scheme and the integral involved by Simpson's integration rule, both involving intermediate approximations. The main result is that accumulation points, if they exist, of sequences constructed by this method satisfy the weak necessary conditions for optimality for the continuous problem. Finally, numerical examples are given.     

Discrete approximation of relaxed optimal control problems

We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.     

Inventory control with fixed cost and price optimization in continuous time

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s, S, p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s, S, p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Se- lecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.     

Deterministic mean-variance-optimal consumption and investment

In dynamic optimal consumption–investment problems one typically aims to find an optimal control from the set of adapted processes. This is also the natural starting point in case of a mean-variance objective. In contrast, we solve the optimization problem with the special feature that the consumption rate and the investment proportion are constrained to be deterministic processes. As a result we get rid of a series of unwanted features of the stochastic solution including diffusive consumption, satisfaction points and consistency problems. Deterministic strategies typically appear in unit-linked life insurance contracts, where the life-cycle investment strategy is age dependent but wealth independent. We explain how optimal deterministic strategies can be found numerically and present an example from life insurance where we compare the optimal solution with suboptimal deterministic strategies derived from the stochastic solution.     

Solving Strategies and Well-Posedness in Linear Semi-Infinite Programming

In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.     

A Continuous Implementation of a Second-Variation Optimal Control Method for Space Trajectory Problems

The paper describes a continuous second-variation method to solve optimal control problems with terminal constraints where the control is defined on a closed set. The integration of matrix differential equations based on a second-order expansion of a Lagrangian provides linear updates of the control and a locally optimal feedback controller. The process involves a backward and a forward integration stage, which require storing trajectories. A method has been devised to store continuous solutions of ordinary differential equations and compute accurately the continuous expansion of the Lagrangian around a nominal trajectory. Thanks to the continuous approach, the method adapts implicitly the numerical time mesh and provides precise gradient iterates to find an optimal control. The method represents an evolution to the continuous case of discrete second-order techniques of optimal control. The novel method is demonstrated on bang–bang optimal control problems, showing its suitability to identify automatically optimal switching points in the control without insight into the switching structure or a choice of the time mesh. A complex space trajectory problem is tackled to demonstrate the numerical robustness of the method to problems with different time scales.     

Existence and explicit determination of optimal stopping times

We consider large classes of continuous time optimal stopping problems for which we establish the existence and form of the optimal stopping times. These optimal times are then used to find approximate optimal solutions for a class of discrete time problems.     

An inverse optimal problem in discrete-time stochastic control

In this paper, we study an inverse optimal problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain optimal control problem. Our results extend to the stochastic case the work of Dechert. In particular, we present a stochastic version of an important principle in welfare economics.     

Weakly invariant sets of hybrid systems

We consider a mathematical model of a hybrid system in which the continuous dynamics generated at any point in time by one of a given finite family of continuous systems alternates with discrete operations commanding either an instantaneous switching from one system to another, or an instantaneous passage from current coordinates to some other coordinates, or both operations simultaneously. As a special case, we consider a model of a linear switching system. For a hybrid system, we introduce the notion of a weakly invariant set and analyze its structure. We obtain a representation of a weakly invariant set as a union of sets of simpler structure. For the latter sets, we introduce special value functions, for which we obtain expressions by methods of convex analysis. For the same functions, we derive equations of the Hamilton-Jacobi-Bellman type, which permit one to pass from the problem of constructing weakly invariant sets to the control synthesis problem for a hybrid system.     

The problem of estimation in continuous/discrete dynamic systems

We consider the problem of optimal nonlinear estimation in a continuous/discrete dynamic system whose state vector is a piecewise-continuous function and the observations are represented by a collection of continuous and discrete processes. We obtain equations that determine the piecewise-continuous conditional probability density function of the process being estimated, on the basis of which we form optimal estimates, as well as an exact representation of the solution of these equations and corresponding estimation algorithms for the problem of linear optimal continuous/discrete estimation. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

On the relaxation gap for PDE mixed-integer optimal control problems

Mixed-integer optimal control problems require taking discrete and continuous control decisions for the optimization of a dynamical system. We consider dynamics governed by partial differential equations of evolution type and assess the problem by relaxation and rounding strategies. For this solution approach, we present a priori estimates for semilinear evolutions on Banach spaces concerning the optimality gap. The theoretical results show that the gap can be made arbitrary small. We demonstrate the numerical performance of the approach on benchmark problems of parabolic type motivated from thermal manufacturing and of hyperbolic type motivated from traffic flow control. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inventory control as a discrete system control for the fixed-order quantity system

This paper shows how to model a problem to find optimal number of replenishments in the fixed-order quantity system as a basic problem of optimal control of the discrete system. The decision environment is deterministic and the time horizon is finite. A discrete system consists of the law of dynamics, control domain and performance criterion. It is primarily a simulation model of the inventory dynamics, but the performance criterion enables various order strategies to be compared. The dynamics of state variables depends on the inflow and outflow rates. This paper explicitly defines flow regulators for the four patterns of the inventory: discrete inflow – continuous/discrete outflow and continuous inflow – continuous/discrete outflow. It has been discussed how to use suggested model for variants of the fixed-order quantity system as the scenarios of the model. To find the optimal process, the simulation-based optimization is used.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

Mean–variance target-based optimisation for defined contribution pension schemes in a stochastic framework

We solve a mean–variance optimisation problem in the accumulation phase of a defined contribution pension scheme. In a general multi-asset financial market with stochastic investment opportunities and stochastic contributions, we provide the general forms for the efficient frontier, the optimal investment strategy, and the ruin probability. We show that the mean–variance approach is equivalent to a “user-friendly” target-based optimisation problem which minimises a quadratic loss function, and provide implementation guidelines for the selection of the target. We show that the ruin probability can be kept under control through the choice of the target level. We find closed-form solutions for the special case of stochastic interest rate following the Vasiček (1977) dynamics, contributions following a geometric Brownian motion, and market consisting of cash, one bond and one stock. Numerical applications report the behaviour over time of optimal strategies and non-negative constrained strategies.     

An approach for an algorithmic solution of discrete optimal control problems and their game-theoretical extension

We consider time discrete systems which are described by a system of difference equations. The related discrete optimal control problems are introduced. Additionally, a gametheoretic extension is derived, which leads to general multicriteria decision problems. The characterization of their optimal behavior is studied. Given starting and final states define the decision process; applying dynamic programming techniques suitable optimal solutions can be gained. We generalize that approach to a special gametheoretic decision procedure on networks. We characterize Nash equilibria and present sufficient conditions for their existence. A constructive algorithm is derived. The sufficient conditions are exploited to get the algorithmic solution. Its complexity analysis is presented and at the end we conclude with an extension to the complementary case of Pareto optima.Dmitrii Lozovanu was Supported by BGP CRDF-MRDA MOM2-3049-CS-03.     

A stochastic target formulation for optimal switching problems in finite horizon

We consider a general optimal switching problem for a controlled diffusion and show that its value coincides with the value of a well-suited stochastic target problem associated to a diffusion with jumps. The proof consists in showing that the Hamilton–Jacobi–Bellman equations of both problems are the same and in proving a comparison principle for this equation. This provides a new family of lower bounds for the optimal switching problem, which can be computed by Monte-Carlo methods. This result has also a nice economical interpretation in terms of a firm's valuation.     

Numerical Computation of Optimal Feed Rates for a Fed-Batch Fermentation Model

In this paper, we consider a model for a fed-batch fermentation process which describes the biosynthesis of penicillin. First, we solve the problem numerically by using a direct shooting method. By discretization of the control variable, we transform the basic optimal control problem to a finite-dimensional nonlinear programming problem, which is solved numerically by a standard SQP method. Contrary to earlier investigations (Luus, 1993), we consider the problem as a free final time problem, thus obtaining an improved value of the penicillin output. The results indicate that the assumption of a continuous control which underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to the fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple shooting techniques. It turns out that this BVP is sensitive, which is due to the rigid behavior of the specific growth rate functions. By relaxation of the characteristic parameters, we obtain a simpler BVP, which can be solved by using the predicted control structure (Lim et al., 1986). Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution which satisfies all first-order and second-order necessary conditions of optimal control theory. The solution is similar to the one obtained by direct methods, but in addition it contains certain very small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.