This paper investigates a class of unified stochastic linear quadratic Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a {generalized} setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be \emph{indefinite} in cost functional. This setup is {innovative and has great theoretical and realistic significance} as its applications in mathematical finance {(e.g., portfolio selection in mean-variation model)}. Using some \emph{fully-coupled} variational analysis under person-by-person optimality principle, and mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with \emph{embedding representation}. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.
This paper studies asymptotic solvability of a linear quadratic (LQ) mean field social optimization problem with controlled diffusions and indefinite state and control weights. Starting with an $N$-agent model, we employ a rescaling approach to derive a low-dimensional Riccati ordinary differential equation (ODE) system, which characterizes a necessary and sufficient condition for asymptotic solvability. The decentralized control obtained from the mean field limit ensures a bounded optimality loss in minimizing the social cost having magnitude $O(N)$, which implies an optimality loss of $O(1/N…
This paper investigates a linear quadratic mean field leader-follower team problem, where the model involves one leader and a large number of weakly-coupled interactive followers. The leader and the followers cooperate to optimize the social cost. Specifically, for any strategy provided first by the leader, the followers would like to choose a strategy to minimize social cost functional. Using variational analysis and person-by-person optimality, we construct two auxiliary control problems. By solving sequentially the auxiliary control problems with consistent mean field approximations, we can…
A linear quadratic (LQ) stochastic optimization problem with delay involving weakly-coupled large population is investigated in this paper. Different to classic mean field (MF) game, here agents cooperate with each other to minimize the so-called \emph{social} objective. With the aid of \emph{delayed person-by-person optimality} principle, one arrives at an auxiliary LQ delayed control problem by decentralized information. A decentralized strategy is obtained by feat of an MF type anticipated forward-backward stochastic differential delay equation (AFBSDDE) consistency condition. The discounti…
In this paper, we study the social optimality for mean field linear quadratic control systems following the direct approach, where subsystems are coupled via individual dynamics and costs according to a network topology. A graph is introduced to represent the network topology of the large-population system, where nodes represent subpopulations called clusters and edges represent communication relationship. By the direct approach, we first seek the optimal controller under centralized information structure, which characterized by a set of forward-backward stochastic differential equations. Then…
This paper examines mean field linear-quadratic-Gaussian (LQG) social optimum control with volatility-uncertain common noise. The diffusion terms in the dynamics of agents contain an unknown volatility process driven by a common noise. We apply a robust optimization approach in which all agents view volatility uncertainty as an adversarial player. Based on the principle of person-by-person optimality and a two-step-duality technique for stochastic variational analysis, we construct an auxiliary optimal control problem for a representative agent. Through solving this problem combined with a con…
This paper considers a linear-quadratic (LQ) mean field control problem involving a major player and a large number of minor players, where the dynamics and costs depend on random parameters. The objective is to optimize a social cost as a weighted sum of the individual costs under decentralized information. We apply the person-by-person optimality principle in team decision theory to the finite population model to construct two limiting variational problems whose solutions, subject to the requirement of consistent mean field approximations, yield a system of forward-backward stochastic differ…
This paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems, where subsystems are coupled via individual dynamics and costs, and the state weight is not assumed with the definiteness condition. For the finite-horizon problem, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback-type control by decoupling the FBSDEs. For the infinite-horizon problem, by using solutions to two Riccati equations, we design a set of decentralized control laws, whi…
This paper studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semi-definite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSD…
This paper investigates a class of mixed stochastic linear-quadratic-Gaussian (LQG) social optimization and Nash game in the context of a large scale system. Two types of interactive agents are involved: a major agent and a large number of weakly-coupled minor agents. All minor agents are cooperative to minimize the social cost as the sum of their individual costs, whereas such social cost are conflictive to that of major agent. Thus, major agent and all minor agents are further competitive to reach some non-zero Nash equilibrium. The control processes enter both diffusion and drift terms of a…
This paper introduces the quadratically-constrained quadratic programming (QCQP) framework recently added in HPIPM alongside the original quadratic-programming (QP) framework. The aim of the new framework is unchanged, namely providing the building blocks to efficiently and reliably solve (more general classes of) optimal control problems (OCP). The newly introduced QCQP framework provides full features parity with the original QP framework: three types of QCQPs (dense, optimal control and tree-structured optimal control QCQPs) and interior point method (IPM) solvers as well as (partial) conde…
This paper is concerned with a new class of mean-field games which involve a finite number of agents. Necessary and sufficient conditions are obtained for the existence of the decentralized open-loop Nash equilibrium in terms of non-standard forward-backward stochastic differential equations (FBSDEs). By solving the FBSDEs, we design a set of decentralized strategies by virtue of two differential Riccati equations. Instead of the $\varepsilon$-Nash equilibrium in classical mean-field games, the set of decentralized strategies is shown to be a Nash equilibrium. For the infinite-horizon problem,…
This paper is concerned with a linear quadratic (LQ, for short) optimal control problem with fixed terminal states and integral quadratic constraints. A Riccati equation with infinite terminal value is introduced, which is uniquely solvable and whose solution can be approximated by the solution for a suitable unconstrained LQ problem with penalized terminal state. Using results from duality theory, the optimal control is explicitly derived by solving the Riccati equation together with an optimal parameter selection problem. It turns out that the optimal control is not only a feedback of the cu…
This paper studies social optima and Nash games for mean field linear quadratic control systems, where subsystems are coupled via dynamics and individual costs. For the social control problem, we first obtain a set of forward-backward stochastic differential equations (FBSDE) from variational analysis, and construct a feedback-type control by decoupling the FBSDE. By using solutions of two Riccati equations, we design a set of decentralized control laws, which is further proved to be asymptotically social optimal. Two equivalent conditions are given for uniform stabilization of the systems in …
In this paper, we consider linear quadratic optimal control with mean-field type for discrete-time stochastic systems with state and control dependent noise. An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic.
This paper considers the stochastic linear quadratic optimal control problem in which the control domain is nonconvex. By the functional analysis and convex perturbation methods, we establish a novel maximum principle. The application of the proposed maximum principle is illustrated through a work-out example.
Stochastic Optimal Linear Quadratic Regulation Control of Discrete-time Systems with Delay and Quadratic Constraints
School of Control Science and Engineering, Shandong University（山东大学控制科学与工程学院）；College of Electrical Engineering and Automation, Shandong University of Science and Technology（山东科技大学电气工程学院）
This paper is concerned with a linear quadratic optimal control problem of delayed backward stochastic differential equations. An explicit representation is derived for the optimal control, which is a linear feedback of the entire past history and the future state trajectory in a short period of time. This is one of the major distinctive features of the delayed backward stochastic linear quadratic optimal control problem. To obtain the optimal feedback, a new class of delayed Riccati equations is introduced and the unique solvability of their solutions are discussed in detail.
We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.
$\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Max Planck Institute for Dynamics of Complex Technical Systems（马克斯·普朗克复杂技术系统动力学研究所）；Virginia Tech（弗吉尼亚理工大学）
This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedbac…
This paper addresses an open problem in the area of linear quadratic optimal control. We consider the regular, infinite-horizon, stability-modulo-a-subspace, indefinite linear quadratic problem under the assumption that the dynamics are stabilizable. Our result generalizes previous works dealing with the same problem in the case of controllable dynamics. We explicitly characterize the unique solution of the algebraic Riccati equation that gives the optimal cost and optimal feedback control, as well as necessary and sufficient conditions for the existence of optimal controls.
A new method for solving optimal tracking control of linear quadratic time-varying systems with multiple time delays in state and input variables and with combined constraints is presented in this paper. By using the relations of Chebyshev wavelets, we simulate the optimal tracking problem to a static optimization one. This alternative method is applied on different optimal tracking systems and simulation results demonstrate the effectiveness of the proposed method.
This paper first presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Then, by introducing several sequences of bounded linear operators, the problem becomes an operator stochastic LQ problem, in which the optimal control is a linear state feedback. Furthermore, from the form of the optimal control, the problem changes to a matrix dynamic optimization problem. Solving this optimization problem, we obtain the optimal feedback gain and thus the optimal control. Finally, by completing the square, the o…
In this paper, we define and solve the Inverse Stochastic Optimal Control (ISOC) problem of the linear-quadratic Gaussian (LQG) and the linear-quadratic sensorimotor (LQS) control model. These Stochastic Optimal Control (SOC) models are state-of-the-art approaches describing human movements. The LQG ISOC problem consists of finding the unknown weighting matrices of the quadratic cost function and the covariance matrices of the additive Gaussian noise processes based on ground truth trajectories observed from the human in practice. The LQS ISOC problem aims at additionally finding the covarianc…
Long-Time Behaviors of Stochastic Linear-Quadratic Optimal Control Problems
University of Michigan（密歇根大学）；California State University San Marcos（圣马科斯加州州立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；University of Central Florida（中佛罗里达大学）
This paper is concerned with the stochastic linear-quadratic optimal control problem with Poisson jumps. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed indefinite. The notion of closed-loop strategies is introduced, and the optimal closed-loop strategy is characterized by a Riccati integral-differential equation and a backward stochastic differential equation with Poisson jumps.
This paper is concerned with linear quadratic optimal control problems for mean-field backward stochastic differential equations (MF-BSDEs, for short) with deterministic coefficients. The optimality system, which is a linear mean-field forward-backward stochastic differential equation with constraint, is obtained by a variational method. By decoupling the optimality system, two coupled Riccati equations and an MF-BSDE are derived. It turns out that the coupled two Riccati equations are uniquely solvable. Then a complete and explicit representation is obtained for the optimal control.
We consider a general linear control system and a general quadratic cost, where the state evolves continuously in time and the control is sampled, i.e., is piecewise constant over a subdivision of the time interval. This is the framework of a linear-quadratic optimal sampled-data control problem. As a first result, we prove that, as the sampling periods tend to zero, the optimal sampled-data controls converge pointwise to the optimal permanent control. Then, we extend the classical Riccati theory to the sampled-data control framework, by developing two different approaches: the first one uses …
This paper is concerned with mean-field stochastic linear-quadratic (MF-SLQ, for short) optimal control problems with deterministic coefficients. The notion of weak closed-loop optimal strategy is introduced. It is shown that the open-loop solvability is equivalent to the existence of a weak closed-loop optimal strategy. Moreover, when open-loop optimal controls exist, there is at least one of them admitting a state feedback representation, which is the outcome of a weak closed-loop optimal strategy. Finally, an example is presented to illustrate the procedure for finding weak closed-loop opti…
This paper is concerned with a discrete-time mean-field stochastic linear-quadratic optimal control problem arose from financial application. Through matrix dynamical optimization method, a group of linear feedback controls is investigated. The problem is then reformulated as an operator stochastic linear-quadratic optimal control problem by a sequence of bounded linear operators over Hilbert space, the optimal control with six algebraic Riccati difference equations is obtained by backward induction. The two above approaches are proved to be coincided by the classical method of completing the …
This paper studies a continuous-time stochastic linear-quadratic (SLQ) optimal control problem on infinite-horizon. A data-driven policy iteration algorithm is proposed to solve the SLQ problem. Without knowing three system coefficient matrices, this algorithm uses the collected data to iteratively approximate a solution of the corresponding stochastic algebraic Riccati equation (SARE). A simulation example is provided to illustrate the effectiveness and applicability of the algorithm.
In this paper, we consider the inverse optimal control problem for the discrete-time linear quadratic regulator, over finite-time horizons. Given observations of the optimal trajectories, and optimal control inputs, to a linear time-invariant system, the goal is to infer the parameters that define the quadratic cost function. The well-posedness of the inverse optimal control problem is first justified. In the noiseless case, when these observations are exact, we analyze the identifiability of the problem and provide sufficient conditions for uniqueness of the solution. In the noisy case, when …
This paper is concerned with an optimal control problem for a mean-field linear stochastic differential equation with a quadratic functional in the infinite time horizon. Under suitable conditions, including the stabilizability, the (strong) exponential, integral, and mean-square turnpike properties for the optimal pair are established. The keys are to correctly formulate the corresponding static optimization problem and find the equations determining the correction processes. These have revealed the main feature of the stochastic problems which are significantly different from the determinist…
In this paper, we study a linear-quadratic optimal control problem for mean-field stochastic differential equations driven by a Poisson random martingale measure and a multidimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying…
This paper studies social optimal control of mean field LQG (linear-quadratic-Gaussian) models with uncertainty. Specially, the uncertainty is represented by a uncertain drift which is common for all agents. A robust optimization approach is applied by assuming all agents treat the uncertain drift as an adversarial player. In our model, both dynamics and costs of agents are coupled by mean field terms, and both finite- and infinite-time horizon cases are considered. By examining social functional variation and exploiting person-by-person optimality principle, we construct an auxiliary control …
This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati…
This paper presents a novel value iteration (VI) algorithm for finding the optimal control for a kind of infinite-horizon stochastic linear quadratic (SLQ) problem with unknown systems. First, an off-line algorithm is estabilished to obtain the optimal feedback control of our problem. Then, based on the off-line algorithm, the VI-based model-free algorithm and its convergence proof is provided. The main feature of the model-free algorithm is that a stabilizing control is not needed to initiate the algorithm. Finally, we validate our results with a simulation example.
