We study the turnpike phenomenon for optimal control problems with mean-field dynamics that are obtained as the limit $N\rightarrow \infty$ of systems governed by a large number $N$ of ordinary differential equations. We show that the optimal control problems with large time horizons give rise to a turnpike structure of the optimal state and the optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary differential equations is preserved under the corresponding mean-field limit.

A stochastic model predictive control (SMPC) algorithm is developed to solve the problem of three-dimensional spacecraft rendezvous and docking with unbounded disturbance. In particular, we only assume that the mean and variance information of the disturbance is available. In other words, the probability density function of the disturbance distribution is not fully known. Obstacle avoidance is considered during the rendezvous phase. Line-of-sight cone, attitude control bandwidth, and thrust direction constraints are considered during the docking phase. A distributionally robust optimization based algorithm is then proposed by reformulating the SMPC problem into a convex optimization problem. Numerical examples show that the proposed method improves the existing model predictive control based strategy and the robust model predictive control based strategy in the presence of disturbance.

We study mean-variance hedging under portfolio constraints in a general semimartingale model. The constraints are formulated via predictable correspondences, meaning that the trading strategy is restricted to lie in a closed convex set which may depend on the state and time in a predictable way. To obtain the existence of a solution, we first establish the closedness in L2 of the space of all gains from trade (i.e. the terminal values of stochastic integrals with respect to the price process of the underlying assets). This is a first main contribution which enables us to tackle the problem in a systematic and unified way. In addition, using the closedness allows us to explain and generalise in a systematic way the convex duality results obtained previously by other authors via ad-hoc methods in specific frameworks.

In this paper, we consider a linear program with only equality constraints but containing interval and random coefficients. We first address the linear program with interval coefficients, and establish some structural properties of this linear program. On this basis, a solution method is proposed. We then move on to consider the linear program with random coefficients. Using the chance constraint approach and a new approach, the satisfaction degree approach, we obtain the two respective deterministic equivalent formulations. Then the results and the numerical solution methods obtained for these two linear models are applied to the original linear problem which contains both interval and random coefficients. By way of illustration, we consider a practical problem, where the optimal mixing proportions need to be determined for the mix slurry in the production process of aluminium with sintering. This gives rise to a linear program with interval and random coefficients. Its deterministic equivalent formulations are presented. Preliminary numerical examples show that the proposed models and the solution methods are promising.

The maximization of the long-term growth rate of expected utility is considered under drawdown constraints. In a general situation, the value and the optimal strategy of the problem are related to those of another ‘standard’ risk-sensitive-type portfolio optimization problem. Furthermore, an upside-chance maximization problem of a large deviation probability is stated as a ‘dual’ optimization problem. As an example, a ‘linear-quadratic’ model is studied in detail: the conditions to ensure the solvabilities of the problems are discussed and explicit expressions for the solutions are presented.

We introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems. The class of distributional resolvent linear systems includes all linear time-invariant systems that have been studied in the control literature.

We show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory.