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1. Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions

   https://doi.org/10.1051/cocv/2018013  

   Wei, Qingmeng; Yong, Jiongmin; Yu, Zhiyong (January 2019, ESAIM: Control, Optimisation and Calculus of Variations)

   An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved. 

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2. Chance Constraint based Design of Controllers for Linear Uncertain Systems

   Nandi, Souransu; Singh, Tarunraj (May 2017, 2017 American Control Conference)

   This paper considers the problem of state-tostate transition with state and control constraints, for a linear system with model parameter uncertainties. Polynomial chaos is used to transform the stochastic model to a deterministic surrogate model. This surrogate model is used to pose a chance constrained optimal control problem where the state constraints and the residual energy cost are represented in terms of the mean and variance of the stochastic states. The resulting convex optimization is illustrated on the problem of rest-to-rest maneuver of the benchmark floating oscillator. 

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3. A Risk Extended Version of Merton’s Optimal Consumption and Portfolio Selection

   https://doi.org/10.1287/opre.2021.2197  

   Bensoussan, Alain; Hoe, SingRu; Kim, Joohyun; Yan, Zhongfeng (March 2022, Operations Research)

   The objective of this paper is to study the optimal consumption and portfolio choice problem of risk-controlled investors who strive to maximize total expected discounted utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem. The control problem presented is no longer a standard stochastic control problem but rather, a mean field-type control problem. The optimal portfolio and consumption rules are obtained explicitly. Numerical results shed light on the importance of controlling variance risk. The optimal investment policy is nonmyopic, and consumption is not sacrificed. 

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4. Multi-agent infinite horizon persistent monitoring of targets with uncertain states in multi-dimensional environments

   Pinto, Samuel C; Andersson, Sean B; Hendrickx, Julien M; Cassandras, Christos G (January 2020, Proc. IFAC World Congress)

   This paper investigates the problem of persistent monitoring, where a finite set of mobile agents persistently visits a finite set of targets in a multi-dimensional environment. The agents must estimate the targets’ internal states and the goal is to minimize the mean squared estimation error over time. The internal states of the targets evolve with linear stochastic dynamics and thus the optimal estimator is a Kalman-Bucy Filter. We constrain the trajectories of the agents to be periodic and represented by a truncated Fourier series. Taking advantage of the periodic nature of this solution, we define the infinite horizon version of the problem and explore the property that the mean estimation squared error converges to a limit cycle. We present a technique to compute online the gradient of the steady state mean estimation error of the targets’ states with respect to the parameters defining the trajectories and use a gradient descent scheme to obtain locally optimal movement schedules. This scheme allows us to address the infinite horizon problem with only a small number of parameters to be optimized. 

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5. Finite-Horizon Separation-Based Covariance Control for Discrete-Time Stochastic Linear Systems

   https://doi.org/10.1109/CDC.2018.8619542  

   Bakolas, Efstathios (December 2018, 2018 IEEE Conference on Decision and Control (CDC))

   In this paper, we address a finite-horizon stochastic optimal control problem with covariance assignment and input energy constraints for discrete-time stochastic linear systems with partial state information. In our approach, we consider separation-based control policies that correspond to sequences of control laws that are affine functions of either the complete history of the output estimation errors, that is, the differences between the actual output measurements and their corresponding estimated outputs produced by a discrete-time Kalman filter, or a truncation of the same history. This particular feedback parametrization allows us to associate the stochastic optimal control problem with a tractable semidefinite (convex) program. We argue that the proposed procedure for the reduction of the stochastic optimal control problem to a convex program has significant advantages in terms of improved scalability and tractability over the approaches proposed in the relevant literature. 

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