Dmitry Khlopin
In an infinite-horizon optimal control problem the stability of the adjoint variable implying the vanishing of the adjoint variable at infinity along optimal solution.
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The transversality conditions for infinite-horizon optimal control problem with a free right endpoint and the stability of the adjoint variable (in Russian)
研究无限时域自由右端点最优控制问题，证明伴随变量的Lyapunov稳定性蕴含其沿最优解在无穷远处趋于零。
The transversality conditions in infinite horizon problems and the stability of adjoint variable
研究无限时域自由右端点一致超越最优控制的必要条件，证明了伴随变量总变差无界时庞特里亚金最大值原理的Clarke形式，并提出了伴随变量部分Lyapunov稳定下的横截条件及其修正。
In this paper we deal with infinite horizon optimal control problems. Basing on weak variations in an extremal problem in weighted function spaces we prove necessary conditions in form of the adjoint equation and a variational inequality. The obtained optimality conditions imply several transversality conditions and the validity of Mangasarian type sufficiency conditions. Arising difficulties in the presence of state constraints are discussed.
This paper is concerned with a discounted optimal control problem of partially observed forward-backward stochastic systems with jumps on infinite horizon. The control domain is convex and a kind of infinite horizon observation equation is introduced. The uniquely solvability of infinite horizon forward (backward) stochastic differential equation with jumps is obtained and more extended analysis, especially for the backward case, is made. Some new estimates are first given and proved for the critical variational inequality. Then an ergodic maximum principle is obtained by introducing some infi…
This paper analyses the interplay between dissipativity and stability properties in continuous-time infinite-horizon Optimal Control Problems (OCPs). We establish several relations between these properties, which culminate in a set of equivalence conditions. Moreover, we investigate convergence and stability of the infinite-horizon optimal adjoint trajectories. The workhorse for our investigations is a notion of strict dissipativity in OCPs, which has been coined in the context of economic model predictive control. With respect to the link between stability and dissipativity, the present paper…
In this paper we consider ergodic optimal control of a diffusion process $\{X^u_t\}_{t \geq 0}$, taking values in $\bR^n$, where both drift and volatility are controlled. We establish a novel strong duality between the existence of a unique solution to the infinite horizon adjoint BSDE and strong dissipativity of $X^u$. We then proceed to show that the latter implies irreducibility, the strong Feller property and exponential ergodicity. We conclude by discussing the connection with ergodic BSDEs.
We develop a necessary stochastic maximum principle for a finite-dimensional stochastic control problem in infinite horizon under a polynomial growth and joint monotonicity assumption on the coefficients. The second assumption generalizes the usual one in the sense that it is formulated as a joint condition for the drift and the diffusion term. The main difficulties concern the construction of the first and second order adjoint processes by solving backward equations on an unbounded time interval. The first adjoint process is characterized as a solution to a backward SDE, which is well-posed t…
We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad's idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev--Kryazhimskii-type formulae for co-state arcs. The conn…
In the paper we consider the infinite horizon control problems on the interval with free right-hand endpoint. We obtain the necessary conditions of strict optimality. The method of the proof actually follows the classic paper by Halkin, and the boundary condition for infinity that we construct in our paper is a stronger variety of the Seierstad condition. The complete system of relations of the maximum principle that was obtained in the paper allows us to write the expression for the adjoint variable in the form of improper integral that depends only on the developing trajectory. S.M. Aseev, A…
In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, …
Necessary conditions of optimality in the form of the Pontryagin Maximum Principle are derived for the Bolza-type discounted problem with free right end. The optimality is understood in the sense of the uniformly overtaking optimality. Such process is assumed to exist, and the corresponding payoff of the optimal process (expressed in the form of improper integral) is assumed to converge in the Riemann sense. No other assumptions on the asymptotic behaviour of trajectories or adjoint variables are required. In this paper, we prove that there exists a corresponding limiting solution of the Pontr…
We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini--Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler--Lagrange condition and the maximum principle.
We present a temporal decomposition scheme for solving long-horizon optimal control problems. In the proposed scheme, the time domain is decomposed into a set of subdomains with partially overlapping regions. Subproblems associated with the subdomains are solved in parallel to obtain local primal-dual trajectories that are assembled to obtain the global trajectories. We provide a sufficient condition that guarantees convergence of the proposed scheme. This condition states that the effect of perturbations on the boundary conditions (i.e., initial state and terminal dual/adjoint variable) shoul…
Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weighted $L^2$-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weighted $L^2$-space. As an application, we investiga…
Infinite horizon discounted LQ optimal control problems for mean-field switching diffusions
本文针对带机制切换和平均场相互作用的随机系统，提出无限时域折扣LQ最优控制框架，通过两个代数Riccati方程构造最优反馈律并验证其最优性，为相关控制问题提供理论与数值工具。
The exponential turnpike property for periodic linear quadratic optimal control problems in infinite dimension
本文在无限维周期线性二次最优控制中，证明最优轨迹、控制及伴随状态随时间区间趋于无穷而指数收敛到周期轨道，基于周期系数Riccati方程的大时间行为。
This chapter compares different formulations for Economic nonlinear Model Predictive Control (EMPC) which are all based on an established dissipativity assumption on the underlying Optimal Control Problem (OCP). This includes schemes with and without stabilizing terminal constraints, respectively, or with stabilizing terminal costs. We recall that a recently proposed approach based on gradient correcting terminal penalties implies a terminal constraint on the adjoints of the OCP. We analyze the feasibility implications of these dual/adjoint terminal constraints and we compare our findings to a…
In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional…
Age-Structured Harvesting Models: A Structural Comparison of Rate-Control and Effort-Control Optimality Systems
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Optimal Control of H-Mode Tokamak Plasma Temperature based on Pontryagin's Principle
本文基于庞特里亚金原理研究H模托卡马克等离子体温度的最优控制，提出伴随最优控制与反馈控制器，实现基于实验数据的实时剖面跟踪。
Mean-covariance turnpikes in Wasserstein distributionally robust linear-quadratic control
针对带经验扰动数据的离散时间随机线性系统的长时域Wasserstein分布鲁棒线性二次控制问题，通过构造混合策略证明了相关量的指数turnpike性质，可降低长时域鲁棒控制的计算成本。
In this paper, we discuss selected adjoint approaches for the turbulent flow control. In particular, we focus on the application of adjoint solvers for the scope of noise reduction, in which flow solutions are obtained by large eddy and direct numerical simulations. Optimization results obtained with round and plane jet configurations are presented. The results indicate that using large control horizons poses a serious problem for the control of turbulent flows due to existence of very large sensitivity values with respect to control parameters. Typically these sensitivities grow in time and l…
Turnpike and Sparse Optimal Control for Semiautonomous Neural ODEs
研究控制仿射半自主神经ODE的长时间最优控制，证明指数转向性质、单边时间稀疏性及积分转向估计，数值实验验证三阶段转向轮廓和稀疏结构。
Controlling systems of ordinary differential equations (ODEs) is ubiquitous in science and engineering. For finding an optimal feedback controller, the value function and associated fundamental equations such as the Bellman equation and the Hamilton-Jacobi-Bellman (HJB) equation are essential. The numerical treatment of these equations poses formidable challenges due to their non-linearity and their (possibly) high-dimensionality. In this paper we consider a finite horizon control system with associated Bellman equation. After a time-discretization, we obtain a sequence of short time horizon p…
Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite Péclet number $Pe=U h/κ$ (where $U$ and $h$ are characteristic velocity and length scales, and $κ$ is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when $Pe \gg 1$, identifying initial perturbations of given energy which optimally and thoroughly mix fluids of initally different properties can be computationally challenging. To address this challenge, we consider the identification of initial perturbations in an idealiz…
Mirror Descent for Deterministic Optimal Control
本文针对有限时域确定性最优控制问题，提出显式镜像下降法，基于庞特里亚金最大值原理推导收敛速率，数值示例验证其在多场景下的表现。
Turnpike property of linear quadratic control problems with unbounded control operators
本文在一般假设下，通过用有界算子序列逼近容许无界控制算子并证明其收敛性，建立了无界控制算子线性二次控制问题的回旋公路性质。
In this paper we consider the one-phase Stefan problem with surface tension, set in a two-dimensional strip-like geometry, with periodic boundary conditions respect to the horizontal direction $x_1\in\mathbb{T}$. We prove that the system is locally null-controllable in any positive time, by means of a control supported within an arbitrary open and non-empty subset. We proceed by a linear test and duality, but quickly find that the linearized system is not symmetric and the adjoint has a dynamic coupling between the two states through the (fixed) boundary. Hence, motivated by a Fourier decompos…
ExactVariance of Random Return in Distributional LQR and Its Application to Mean-Variance Optimal Control
针对经典LQR忽视性能变异性的缺陷，本文推导了分布LQR中随机回报方差的精确闭式解，并提出伴随梯度下降算法求解均值-方差最优控制问题，以显式权衡期望回报与风险。
Exponential Turnpike in Optimal Boundary Control of Multi-Dimensional First-Order Hyperbolic Systems
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An Efficient On-Policy Deep Learning Framework for Stochastic Optimal Control
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Physics-enhanced reinforcement learning for real-time optimal control of dynamical systems
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We systematically develop a learning-based treatment of stochastic optimal control (SOC), relying on direct optimization of parametric control policies. We propose a derivation of adjoint sensitivity results for stochastic differential equations through direct application of variational calculus. Then, given an objective function for a predetermined task specifying the desiderata for the controller, we optimize their parameters via iterative gradient descent methods. In doing so, we extend the range of applicability of classical SOC techniques, often requiring strict assumptions on the functio…
A Sensitivity Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws
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State-Dependent Delays in Optimal Control and Hamilton--Jacobi Equations
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Re-examining the Legendre-Gauss-Lobatto Pseudospectral Methods for Optimal Control
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Sequential Bayesian Optimal Experimental Design in Infinite Dimensions via Policy Gradient Reinforcement Learning
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