Sergey Nikitin
The paper presents necessary and sufficient conditions for a nonlinear system to be stabilized by a feedback. The conditions are based on the ideas related to the well-known Pontryagin's maximum principle. That allows us to formulate the results in terms that are valid for continuous, discontinuous, stationary and time-dependent feedbacks.
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The theorem like Pontryagin's maximum principle for multiple integrals is proved. Unlike the usual maximum principle, the maximum should be taken not over all matrices, but only on matrices of rank one. Examples are given.
We provide an improvement of the maximum principle of Pontryagin of the Optimal Control problems. We establish differentiability properties of the value function of problems of Optimal Control with assumptions as low as possible. Notably, we lighten the assumptions by using Gâteaux and Hadamard differentials.
In the infinite-horizon and discrete-time framework we establish maximum principles of Pontryagin under assumptions which weaker than these ones of existing results. We avoid several assumptions of continuity and of Fr{é}chet-differentiability and of linear independence. MSC 2010: 49J21, 65K05, 39A99.
This note is an addendum to [1,2], pointing out the differences between these papers and raising open questions.
Applying the Tubular Neighborhood Theorem, we give a short and new proof of the Pontryagin Maximum Principle on a smooth manifold. The idea is as follows. Given a control system on a manifold $M$, we embed it into an open subset of some $\mathbb R^n$, and extend the control system to the open set. Then, we apply the Pontryagin Maximum Principle on $\mathbb R^n$ to the extended system and project the consequence to $M$.
Based on Pontryagin Maximum Principle (PMP), this paper establishes a generalized PMP aiming at control system with with extra input/output terms. The paper details the adaptive target and gives a proof of the generalized theorem. Transformation and application of this specific method is showed and thereafter an example explained its feasibility.
Based on Pontryagin Maximum Principle (PMP), this paper established a generalized PMP aiming at non-feedback control system with stochastic initial conditions. We proved the conclusion and show its coming back to PMP when the randomness collapses. Through the generalized PMP, a general method is introduced to solve expectation maximum problem of these systems and thereafter an example showed its feasibility.
We establish Pontryagin principles for a Mayer's optimal control problem governed by a functional differential equation. The control functions are piecewise continuous and the state functions are piecewise continuously differentiable. To do that, we follow the method created by Philippe Michel for systems governed by ordinary differential equations, and we use properties of the resolvent of a linear functional differential equation.
A numerical solution approach for non-smooth optimal control problems based on the Pontryagin maximum principle
本文研究带有齐次Dirichlet边界条件的线性椭圆偏微分方程约束的非光滑最优控制问题，提出一种基于Pontryagin最大值原理的优化方法，证明了原理差异沿迭代序列消失，并通过数值实验验证了理论发现。
Pontryagin-type maximum principle for a controlled sweeping process with nonsmooth and unbounded sweeping set
本文将文献[30]中针对受控清扫过程的Pontryagin型最大值原理推广至清扫集非光滑且无界的情形，即该集合为有限个光滑函数零下水平集的交集。
This paper gives a brief contact-geometric account of the Pontryagin maximum principle. We show that key notions in the Pontryagin maximum principle---such as the separating hyperplanes, costate, necessary condition, and normal/abnormal minimizers---have natural contact-geometric interpretations. We then exploit the contact-geometric formulation to give a simple derivation of the transversality condition for optimal control with terminal cost.
In this paper we give a new approach to introduce switching strategies in a special class of optimal multiprocesses. Defined as the set of partitions of the time interval, switching strategies become the character of a classical control. With this approach we overcome the restrictive comparison of multiprocesses with the same switching strategy. The necessary conditions for strong local optimizer in form of the Pontryagin maximum principle are demonstrated in illustrative examples.
In this note our aim is to give a proof of the Pontryagin maximum principle for a general optimal control problem with running state constraints and smooth dynamics. Our proof is based on the classical Ekeland variational principle. The main result (and its proof) of this note are not new and are already well-known in the literature. The aim of the author is only to provide a complete and detailed proof of this classical theorem in the case of smooth dynamics. If you have any remarks or questions, do not hesitate to contact the author at loic.bourdin@unilim.fr.
In this paper, we prove a Pontryagin Maximum Principle for constrained optimal control problems in the Wasserstein space of probability measures. The dynamics, is described by a transport equation with non-local velocities and is subject to end-point and running state constraints. Building on our previous work, we combine the classical method of needle-variations from geometric control theory and the metric differential structure of the Wasserstein spaces to obtain a maximum principle stated in the so-called Gamkrelidze form.
We derive a Maximum Principle for optimal control problems with constraints given by the coupling of a system of ODEs and a PDE of Vlasov-type. Such problems arise naturally as $Γ$-limits of optimal control problems subject to ODE constraints, modeling, for instance, external interventions on crowd dynamics. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward-backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solu…
We discuss a mathematical framework for analysis of optimal control problems on infinite-dimensional manifolds. Such problems arise in study of optimization for partial differential equations with some symmetry. It is shown that some nonsmooth analysis methods and Lagrangian charts techniques can be used for study of global variations of optimal trajectories of such control systems and derivation of maximum principle for them.
We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using the formalism of subdifferential calculus in Wasserstein spaces. We show that the geometric approach based on needle variations and on the evolution of the covector (here replaced by the evolution of a mesure on the dual space) can be translated into this formalism.
In this work, we address some optimal control problems related to the evolution of two isothermal, incompressible, immisible fluids in a two dimensional bounded domain. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled nonlocal Cahn-Hilliard-Navier-Stokes equations. We describe the first order necessary conditions of optimality via Pontryagin minimum principle and prove second order necessary and sufficient conditions of optimality for the problem.
In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. This result was recently obtained in [L. Bourdin and E. Tr{é}lat , Optimal sampled-data control, and generalizations on time scales, arXiv:1501.07361, 2015]. Then we discuss the maximization condition for optimal sampled-data controls that can be seen as an average of the weak maximization condition stated in the classical Pontryagin maximum principle for optimal (permanent) controls. Finally, applying this theorem, we solve …
We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packages of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is "compressed" in one universal optimality condition by using the concept of centered family of compacta.
The aim of this paper is to establish Pontryagin's principles in a dicrete-time infinite-horizon setting when the state variables and the control variables belong to infinite dimensional Banach spaces. In comparison with previous results on this question, we delete conditions of finiteness of codi-mension of subspaces. To realize this aim, the main idea is the introduction of new recursive assumptions and useful consequences of the Baire category theorem and of the Banach isomorphism theorem.
Pontryagin maximum principle for fractional delay differential equations and controlled weakly singular Volterra delay integral equations
该文研究Pontryagin最大值原理在分数阶延迟微分方程、Volterra延迟积分方程两类系统中的应用，探讨对应控制问题的最优求解方法，并辅以示例验证研究结果。
This paper concerns a class of infinite horizon optimal control problems with state constraints. By extending the needle variation method to the infinite horizon case we obtain a complete set of necessary optimality conditions for a strong local minimizer in form of the Pontryagin maximum principle and the validity of several transversality conditions at infinity. To this purpose, we develop an approach in the space of continuous functions converging at infinity. The considerations including new results on optimality and transversality conditions, on Banach spaces of continuous functions, on t…
In this article we derive a strong version of the Pontryagin Maximum Principle for general nonlinear optimal control problems on time scales in finite dimension. The final time can be fixed or not, and in the case of general boundary conditions we derive the corresponding transversality conditions. Our proof is based on Ekeland's variational principle. Our statement and comments clearly show the distinction between right-dense points and right-scattered points. At right-dense points a maximization condition of the Hamiltonian is derived, similarly to the continuous-time case. At right-scattere…
The Pontryagin Maximum Principle for Nonlinear Infinite Horizon Optimal Control Problems with State Constraints
本文研究非线性无限时间最优控制问题，提出基于针形变分技术的方法，推导出强局部极小值的庞特里亚金最大值原理、Arrow型充分条件及多种横截条件。
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。
We provide an improvment of the maximum principle of Pon-tryagin of the optimal control problems, for a system governed by an ordinary differential equation, in presence of final constraints, in the setting of the piece-wise differentiable state functions (valued in a Banach space) and of piecewise continuous control functions (valued in a metric space). As Michel we use the needlelike variations, but we introduce tools of functional analysis and a recent multiplier rule of the static optimization to make our proofs. Mathematical Subject Classification 2010: 49K15, 47H10
Controllability maximization problem under sparsity constraints is a node selection problem that selects inputs that are effective for control in order to minimize the energy to control for desired state. In this paper we discuss the equivalence between the sparsity constrained controllability metrics maximization problems and their convex relaxation. The proof is based on the matrix-valued Pontryagin maximum principle applied to the controllability Lyapunov differential equation.
Optimal control of quantum system in fermion fields: Pontryagin-type maximum principle(I)
本文研究费米子场中量子随机系统的最优控制，证明Pontryagin型极大值原理，并利用非交换鞅不等式与鞅表示定理建立费米子Brown运动驱动倒向量子随机微分方程解的存在唯一性。
Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems
本文针对非相称分数阶最优控制问题，提出并证明了庞特里亚金最大值原理，并给出实例验证其适用性。
Infinite Dimensional Multipliers and Pontryagin Principles for Discrete-Time Problems
本文改进了无限时间框架下无限维状态和控制空间的庞特里亚金原理，通过有限时间缩减方法和泛函分析引理实现目标。
Memory-limited partially observable stochastic control (ML-POSC) is the stochastic optimal control problem under incomplete information and memory limitation. In order to obtain the optimal control function of ML-POSC, a system of the forward Fokker-Planck (FP) equation and the backward Hamilton-Jacobi-Bellman (HJB) equation needs to be solved. In this work, we firstly show that the system of HJB-FP equations can be interpreted via the Pontryagin's minimum principle on the probability density function space. Based on this interpretation, we then propose the forward-backward sweep method (FBSM)…
The Pontryagin maximum principle and $Q$-functions in rough environments
本文提出了一种新的微分方法，用于推导粗糙环境中松弛控制的庞特里亚金极大原理和$Q$-函数，并开发了策略改进算法以处理熵成本约束。
First-Order Pontryagin Maximum Principle for Risk-Averse Stochastic Optimal Control Problems
本文针对受终时不等式约束且成本为一般非光滑有限相干风险度量的风险厌恶随机最优控制问题，在标准布朗运动驱动的经典随机微分方程框架下推导出一阶庞特里亚金最优性条件，避免二阶伴随方程和弱版本庞特里亚金极大值原理。
In this study, we consider an optimal control problem driven by a stochastic differential equation with state constraints. Here, the state constraints mean the constraints about the path of state. In order to show the maximum principe for the optimal control problem under state constraints, we investigate a new near optimal control problem. In the following, we establish the stochastic maximum principle for the new optimal control problem under multi-time state constraints. In the end, we give a production planning example to verify the main results of this study.
Control in the coefficients of an elliptic differential operator: topological derivatives and Pontryagin maximum principle
本文研究控制位于椭圆算子主部的最优控制问题，利用拓扑导数方法在无需系数和解的梯度连续性假设下证明庞特里亚金极大值原理，并在二维情形通过优化椭圆形状得到含新型变分不等式的更强最优性条件。
Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreove…
We prove a duality relation and an integration by parts formula for fractional operators with a general analytical kernel. Based on these basic results, we are able to prove a new Gronwall's inequality and continuity and differentiability of solutions of control differential equations. This allow us to obtain a weak version of Pontryagin's maximum principle. Moreover, our approach also allow us to consider mixed problems with both integer and fractional order operators and derive necessary optimality conditions for isoperimetric variational problems and other problems of the calculus of variat…
In this paper, we discuss a new general formulation of fractional optimal control problems whose performance index is in the fractional integral form and the dynamics are given by a set of fractional differential equations in the Caputo sense. We use a new approach to prove necessary conditions of optimality in the form of Pontryagin maximum principle for fractional nonlinear optimal control problems. Moreover, a new method based on a generalization of the Mittag-Leffler function is used to solving this class of fractional optimal control problems. A simple example is provided to illustrate th…
Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach
本文用拉格朗日方法研究确定性平均场类型自由端点最优控制问题，推导庞特里亚金极大值原理，并证明Kantorovich与Eulerian形式下的局部极小值对应，最后以线性二次调节器为例说明最优策略为线性反馈。
We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontry…
This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for op…
A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds
本文在光滑流形上建立了离散时间最优控制问题的几何庞特里亚金极大原理，考虑了状态、控制动作以及最优控制轨迹频谱的约束，并基于[Cha11]中连续时间最优控制的思路进行了证明。
We present a new geometric unfolding of a prototype problem of optimal control theory, the Mayer problem. This approach is crucially based on the Stokes Theorem and yields to a necessary and sufficient condition that characterizes the optimal solutions, from which the classical Pontryagin Maximum Principle is derived in a new insightful way. It also suggests generalizations in diverse directions of such famous principle.
Pontryagin's Principle Based Algorithms for Optimal Control Problems of Parabolic Equation
本文提出基于Pontryagin原理的增广逐次逼近法，用于求解半线性抛物方程的状态约束最优控制问题，并证明收敛性及数值有效性。
This article considers a discrete-time robust optimal control problem on matrix Lie groups. The underlying system is assumed to be perturbed by exogenous unmeasured bounded disturbances, and the control problem is posed as a min-max optimal control wherein the disturbance is the adversary and tries to maximise a cost that the control tries to minimise. Assuming the existence of a saddle point in the problem, we present a version of the Pontryagin maximum principle (PMP) that encapsulates first-order necessary conditions that the optimal control and disturbance trajectories must satisfy. This P…
On a mean-field Pontryagin minimum principle for stochastic optimal control
本文提出了一种新的均场庞特里亚金最优化原理，用于解决随机最优控制问题，通过引入辅助函数简化了求解过程，并在多个系统中进行了数值测试。
The Pontryagin Maximum Principle for Training Convolutional Neural Networks
针对带$L^0$正则化的卷积神经网络训练问题，提出基于离散时间庞特里亚金最大值原理的批量序贯二次哈密顿算法，通过自适应增强参数与分层近似最大化，理论分析其收敛性并经数值实验验证全批量和小批量模式下的有效性。
A stochastic Pontryagin maximum principle on the Sierpinski gasket
本文研究Sierpinski垫片上的随机控制问题，通过热核估计推导阶比较引理，结合BSDEs技术建立含两个必要性方程的随机庞特里亚金极大值原理，揭示分形空间测度奇异性的影响，并以线性调节器问题为例验证。
The fundamental theorem of the theory of optimal control, the Pontryagin maximum principle (PMP), is extended to the setting of almost Lie (AL) algebroids, geometrical objects generalizing Lie algebroids. This formulation of the PMP yields, in particular, a scheme comprising reductions of optimal control problems similar to the reduction for the rigid body in analytical mechanics. On the other hand, in the presented approach the reduced and unreduced PMPs are parts of the same universal formalism. The framework is based on a very general concept of homotopy of measurable paths and the geometry…
Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise with Heading and Throttle Controls using Pontryagin's Maximum Principle
本研究基于庞特里亚金最大原理解决任意风场下商用飞机巡航的时间-燃料最优轨迹问题，通过 Zermelo 导航恒等式和反馈函数分别确定最优航向角与油门设定，并利用切换点算法验证了最优性条件。
Discrete time Pontryagin maximum principle for optimal control problems under state-action-frequency constraints
本文研究了在状态、动作和最优控制轨迹频率谱约束下离散时间最优控制问题，提出了庞特里亚金最大值原理，并给出了两种不同的证明方法，特别针对控制仿射非线性及线性系统模型提供了特例。
The main purpose of this paper is to give a solution to a long-standing unsolved problem in stochastic control theory, i.e., to establish the Pontryagin-type maximum principle for optimal controls of general infinite dimensional nonlinear stochastic evolution equations. Both drift and diffusion terms can contain the control variables, and the control domains are allowed to be nonconvex. The key to reach it is to provide a suitable formulation of operator-valued backward stochastic evolution equations (BSEEs for short), as well as a way to define their solutions. Besides, both vector-valued and…
Quantum Pontryagin Principle under Continuous Measurements and Feedback
本文研究了在连续测量与反馈条件下量子庞特里亚金原理的理论发展，提出了在统计矩坐标中基于薛定谔图和海森堡图的庞特里亚金最大原理，并开发了更适合控制应用的LQG方案。
Optimal control of quantum stochastic systems in fermion fields: The Pontryagin-type maximum principle (II)
本文利用松弛转置方法求解费米子场量子随机最优控制中的二阶伴随方程，克服其在有界算子代数中无定义的困难，为Pontryagin型最大值原理奠定基础。
A frequency-constrained geometric Pontryagin maximum principle on matrix Lie groups
本文提出了一种在矩阵李群上结合频率约束的几何离散时间庞特里亚金极大值原理，该原理在问题 formulation 阶段直接对状态和控制动作施加点约束和频率约束，从而得到最优性的一阶必要条件，并通过射击技术解决两终点边界值问题，最后通过航天器姿态控制的数值实验验证了理论结果。
An optimal control problem associated with the dynamics of the orientation of a bipolar molecule in the plane can be understood by means of tools in differential geometry. For first time in the literature $k$-symplectic formalism is used to provide the optimal control problems associated to some families of partial differential equations with a geometric formulation. A parallel between the classic formalism of optimal control theory with ordinary differential equations and the one with particular families of partial differential equations is established. This description allows us to state and…
Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
本文探讨了标量凸守恒律与庞特里亚金最优化原理之间的联系，提出了一种求解守恒律的最小值解方法，并证明该解等价于熵解，扩展了Lax-Oleinik公式至非均匀凸通量函数。
We study an optimal control problem arising from a resource allocation problem in cellular metabolism. A minimalistic model that describes the production of enzymatic vs. non-enzymatic biomass components from a single nutrient source is introduced. The basic growth modes with this model are linear growth, where only the non-enzymatic component is produced, and exponential growth with only enzymatic components being produced. Using Pontryagin's maximum principle, we derive the optimal growth trajectory depending on the model's parameters. It turns out that depending on the parameters, either a …
Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method
该研究推导了高斯粗糙路径驱动RDE系统的随机庞特里亚金最大值原理，提出首个非线性随机最优控制间接打靶法，收敛速度较直接法快10倍。
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