Johan Jönsson, Magnus Perninge
In this work we show that one can solve a finite horizon non-Markovian impulse control problem with control dependant dynamics. This dynamic satisfies certain functional Lipschitz conditions and is path dependent in such a way that the resulting trajectory becomes a flow.
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We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional $L^p$-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.
Minimax Impulse Control Problems in Finite Horizon
考虑有限时间内脉冲控制极小极大问题，当费用函数$(C(t,x,ξ)>0)$ .证明了问题的值函数的存在性。此外，还刻画了该值函数为Isaacs拟变分不等式的唯一黏性解.这个问题与金融数学中的一个应用有关。
We consider a two-player zero-sum deterministic differential game where each player uses both continuous and impulse controls in infinite-time horizon. We assume that the impulses supposed to be of general term and the costs depend on the state of the system. We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equations (PDEs) of the game. We prove under Isaacs condition that the upper and lower value functions coincide.
We consider a robust impulse control problem in finite horizon where the underlying uncertainty stems from an impulsively and continuously controlled functional stochastic differential equation (FSDE) driven by Brownian motion. We assume that the controller acts upon the system by impulses while the adversary player (nature) acts through continuous controls. We look for a weak solution which leads us to consider a system of sequentially interconnected, obliquely reflected backward stochastic differential equations (RBSDEs) with stochastic Lipschitz coefficients. We show existence of solutions …
Deterministic Minimax Impulse Control in Finite Horizon: the Viscosity Solution Approach
我们在这里研究脉冲控制极大极小问题。我们允许成本泛函和动力学是无界的，因此价值函数可能是无界的。我们证明了问题的值函数是连续的。此外，还刻画了该值函数为Isaacs拟变分不等式的唯一黏性解.这个问题与金融数学中的一个应用有关。
In the present paper, we study a two-player zero-sum deterministic differential game with both players adopting impulse controls, in infinite time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle (DPP) that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI QVI for which, under a proportional property assumption on the maximizer cost, the value functions are the unique viscosity solution. …
Impulse Control of Multi-dimensional Jump Diffusions in Finite Time Horizon
本文研究了有限时间范围内多维跳跃扩散的一类脉冲控制问题。遵循Stroock和Varadhan \cite{StroockVaradhan06}的基本数学框架，本文首先严格地建立了动态规划原理（DPP）的一种适当形式。证明了该值函数是含积分-微分算子的Hamilton-Jacobi-Belleman（HJB）方程的黏性解.最后，在跳跃无限活动但有限变差和扩散是一致椭圆的假设下，证明了该值函数是唯一黏性解，且对$1< p< \infty$具有$W_{loc}^{(2,1),p}$正则性
Numerical approximations of the value of zero-sum stochastic differential impulse controls game in finite horizon
本文研究有限时域零和随机微分脉冲控制博弈，基于经典Policy Iteration算法求Nash均衡数值解及最优策略对，并严格证明逼近方案收敛至粘性解。
We consider stochastic impulse control problems when the impulses cost functions are arbitrary. We use the dynamic programming principle and viscosity solutions approach to show that the value function is a unique viscosity solution for the associated Hamilton-Jacobi-Bellman equation (HJB) partial differential equation (PDE) of stochastic impulse control problems
In this article, we are discussing a more vital concept of controllability, termed total controllability. We have considered a nonlocal semilinear functional evolution equations with non-instantaneous impulses and finite delay in Hilbert spaces. A set of sufficient conditions of total controllability is obtained for the evolution system under consideration, by imposing the theory of C_0-semigroup and Banach fixed point theorem. We also established the total controllability results for a functional integro-differential equation. Finally, an example is given to demonstrate the feasibility of der…
This paper studies an optimal stochastic impulse control problem in a finite horizon with a decision lag, by which we mean that after an impulse is made, a fixed number units of time has to be elapsed before the next impulse is allowed to be made. The continuity of the value function is proved. A suitable version of dynamic programming principle is established, which takes into account the dependence of state process on the elapsed time. The corresponding Hamilton-Jacobi-Bellman (HJB) equation is derived, which exhibit some special feature of the problem. The value function of this optimal imp…
In this paper we consider impulse control of continuous time Markov processes with average cost per unit time functional. This problem is approximated using impulse control problems stopped at the first exit time from increasing sequence of open sets. We find solution to Bellman equation corresponding to the original problem and show that stopped impulse control problems approximate optimal value of the cost functional.
This paper considers an optimal impulse control problem of dynamical systems generated by a flow. The performance criteria are total costs over the infinite time horizon. Apart from the main performance to be minimized, there are multiple constraints on performance functionals of a similar type. Under a natural set of compactness-continuity conditions on the system primitives, we establish a linear programming approach, and prove the existence of a stationary optimal control strategy out of a more general class of randomized strategies. This is done by making use of the tools from Markov decis…
This paper considers the problem of two-player zero-sum stochastic differential game with both players adopting impulse controls in finite horizon under rather weak assumptions on the cost functions ($c$ and $χ$ not decreasing in time). We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equation (PDE) of the game. We prove that the upper and lower value functions coincide.
Constrained optimal impulse control and inventory model
本文提出了一种受约束的最优冲击控制与库存模型，通过将问题转化为马尔可夫决策过程，解决无限时间 horizon 的优化问题并展示其在库存管理中的应用。
Innovation, commutation et contrôle impulsionnel en horizon infini
考虑了无限时域上的脉冲控制问题.为了解决这个问题，我们将双障碍反射向后随机微分方程的结果推广到无限水平情形。Snell包络的性质可以减少我们的问题，以显示存在一对可测的连续过程。
Zero-sum Stochastic Differential Games of Impulse Control with Random Intervention Costs
针对双方均实施脉冲控制的有限时段零和随机微分博弈，本文提出控制随机化表述以证明上下值函数相等，并允许干预成本依赖状态与时间且无需系数单调性假设。
This paper considers a new class of deterministic finite-time horizon, two-player, zero-sum differential games (DGs) in which the maximizing player is allowed to take continuous and impulse controls whereas the minimizing player is allowed to take impulse control only. We seek to approximate the value function, and to provide a verification theorem for this class of DGs. We first, by means of dynamic programming principle (DPP) in viscosity solution (VS) framework, characterize the value function as the unique VS to the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) double-obstacle equation. Ne…
The impulses, cutting entropy functional (EF) measure on trajectories Markov diffusion process, integrate information path functional (IPF) composing discrete information Bits extracted from observing random process. Each cut brings memory of the cutting entropy, which provides both reduction of the process entropy and discrete unit of the cutting entropy a Bit. Consequently, information is memorized entropy cutting in random observations which process interactions. The origin of information associates with anatomy creation of impulse enables both cut entropy and stipulate random process gener…
This paper focuses on zero-sum stochastic differential games in the framework of forward-backward stochastic differential equations on a finite time horizon with both players adopting impulse controls. By means of BSDE methods, in particular that of the notion from Peng's stochastic \textit{% backward semigroups}, we prove a dynamic programming principle for both the upper and the lower value functions of the game. The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle. As a consequence,…
Optimal Control of Volterra Equations with Impulses
考虑一类由具有脉冲项的Volterra积分方程控制的系统的最优控制问题.脉冲既作用于状态又作用于控制;控制由离散时间的切换组成。成本函数包括综合成本率（连续部分）和离散脉冲时间处的切换成本（离散部分）。我们证明了类似于离散最大值原理的形式的必要最优性条件。对于脉冲常微分方程所支配的系统的特殊情况，我们得到了脉冲Volterra方程最优性必要条件最优性条件的一个特殊情况，即脉冲最大值原理.
Impulse Control of Multidimensional Jump Diffusions
本文研究了一类无限时域折扣成本脉冲控制问题的值函数的正则性，其中被控过程是一个可能具有“无限活动”跳跃的多维跳跃扩散过程.令人惊讶的是，尽管有这些跳跃，我们获得了相同程度的规律性扩散的情况下，至少当跳跃满足一定的可积性条件。
We consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multi-dimensional heat equation with a periodic terminal state constraint. This problem has a unique optimal solution, which can be characterized by an optimality system derived from the Pontryagin maximum principle. We define two approximations of this optimal control problem. The first one is an impulse approximation, and consists of considering a system of linear heat equations with impulse control. The second one is obtained by the sample-and-hold procedur…
This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder's fixed point theorem. Moreover, in this paper, we define proper identification of resolven…
Discrete time stochastic impulse control with delay
该研究针对离散时间下带执行延迟的无限时域脉冲控制问题，运用基于Snell包络的概率方法，针对两类效用函数构建最优策略并证明有界ε-最优策略存在，为处理相关系统延迟提供鲁棒框架。
Impulse control maximising average cost per unit time: a non-uniformly ergodic case
本文针对非一致遍历的Feller-Markov过程，研究单位时间平均成本最大化的脉冲控制问题，允许无界脉冲成本，证明最优值与初始状态无关并给出最优或近似最优策略。
Long-run impulse control with generalised discounting
本文研究广义贴现对Feller-Markov过程长期脉冲控制的影响，证明贴现与未贴现问题最优值相同，且未贴现最优策略对贴现准则最优或近似最优，从而提供时间一致解并简化Bellman方程。
We consider an optimal stochastic impulse control problem over an infinite time horizon motivated by a model of irreversible investment choices with fixed adjustment costs. By employing techniques of viscosity solutions and relying on semiconvexity arguments, we prove that the value function is a classical solution to the associated quasi-variational inequality. This enables us to characterize the structure of the continuation and action regions and construct an optimal control. Finally, we focus on the linear case, discussing, by a numerical analysis, the sensitivity of the solution with resp…
We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling.
The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of the turnpike is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present condition…
Asymptotics of impulse control problem with multiplicative reward
本文研究一般马尔可夫过程带乘性奖励的长期脉冲控制问题，构造贝尔曼方程解并给出验证，结合概率性质与Krein-Rutman定理，利用有界域近似和二进时间网格。
Threshold-Safe Shock Absorption in a Compartmental Voter-Flow Model:\ A Conservative Impulse-Control Benchmark
针对简化分区选民流模型，提出一个确定性阈值安全问题，通过脉冲控制层获得显式安全基准，并证明标量储层模型是非线性选民流动力学的保守包络。
Long-Term Average Impulse and Singular Control of a Growth Model with Two Revenue Sources
本文研究了长期平均冲击控制及相关奇异控制问题，针对一维扩散过程，探讨了可再生能源管理与金融投资优化中的长期平均收益最大化方法，分析了冲击与奇异控制间的联系。
The Hamilton-Jacobi-Bellman equation (HJB) associated with the time inhomogeneous singular control problem is a parabolic partial differential equation, and the existence of a classical solution is usually difficult to prove. In this paper, a class of finite horizon stochastic singular control problems of one dimensional diffusion is solved via a time inhomogeneous zero-sum game (Dynkin game). The regularity of the value function of the Dynkin game is investigated, and its integrated form coincides with the value function of the singular control problem. We provide conditions under which a cla…
This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the schemes are half-order accurate for HJBQVIs with Lipschitz coefficients, and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse co…
We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the value function of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth fun…
Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates
本文研究了零售商在批发市场购买能源并转售给最终消费者时，如何通过干预来调整价格以最大化收入，提出了一个无限时间 horizon 的随机冲击控制问题，并给出了最优定价策略的分析结果及对干预成本的依赖性研究。
A Finite Horizon Optimal Multiple Switching Problem
考虑了有限时间内系统的最优多次切换问题，当系统的状态（包括切换成本）是一般适应随机过程时。该问题被制定为一个扩展的脉冲控制问题，并完全解决使用概率工具，如Snell包络的过程和反射向后随机微分方程。最后，当系统的状态是一个马尔可夫扩散过程，我们证明了最优问题的值函数的向量是一个粘性解的变分不等式系统与互联的障碍。
This paper studies input-to-state stability (ISS) of general nonlinear time-delay systems subject to delay-dependent impulse effects. Sufficient conditions for ISS are constructed by using the method of Lyapunov functionals. It is shown that, when the continuous dynamics are ISS but the discrete dynamics governing the delay-dependent impulses are not, the impulsive system as a whole is ISS provided the destabilizing impulses do not occur too frequently. On the contrary, when the discrete dynamics are ISS but the continuous dynamics are not, the delayed impulses must occur frequently enough to …
Feedback Nash Equilibria in Differential Games with Impulse Control
本文研究一方连续控制、另一方脉冲控制的有限时域非零和微分博弈，基于HJB方程与QVIs给出反馈纳什均衡的验证定理及干预次数上界，并首次解析刻画线性二次型情形。
System level synthesis is a promising approach that formulates structured optimal controller synthesis problems as convex problems. This work solves the distributed linear-quadratic regulator problem under communication constraints directly in infinite-dimensional space, without the finite-impulse response relaxation common in related work. Our method can also be used to construct optimal distributed Kalman filters with limited information exchange. We combine the distributed Kalman filter with state-feedback control to perform localized LQG control with communication constraints. We provide a…
An optimal control problem with an infinite horizon quadratic cost functional for a linear system with a known additive disturbance is considered. The feature of this problem is that a weight matrix of the control cost in the cost functional is singular. Due to this singularity, the problem can be solved neither by application of the Pontriagin's Maximum Principle, nor using the Hamilton-Jacobi-Bellman equation approach, i.e. this problem is singular. Since the weight matrix of the control cost, being singular, is not in general zero, only a part of the control coordinates is singular, while t…
Turnpike Property of a Linear-Quadratic Optimal Control Problem in Large Horizons with Regime Switching II: Non-Homogeneous Cases
本文研究大时域非齐次线性随机系统带机制切换的二次最优控制问题，将强收费公路性质从齐次情形推广至非齐次情形，并发现部分结果在无切换时亦为新结论。
This work introduces a controller synthesis method via system level synthesis for nonlinear systems characterized by polynomial dynamics. The resulting framework yields finite impulse response, time-invariant, closed-loop transfer functions with guaranteed disturbance cancellation. Our method generalizes feedback linearization to enable partial feedback linearization, where the cancellation of the nonlinearity is spread across a finite-time horizon. This provides flexibility to use the system dynamics to attenuate disturbances before cancellation via control, reducing the cost of control compa…
Necessary Conditions In Infinite-Horizon Control Problem That Need No Asymptotic Assumptions
研究带渐近终端约束的无限时域最优控制问题，推导无需渐近行为假设的协态弧必要边界条件，结合庞特里亚金最大值原理可提升协态弧，单点集时与已有表示一致，并用示例验证结果。
We investigate variational problems with recursive integral functionals governed by infinite-dimensional differential inclusions with an infinite horizon and present an existence result in the setting of nonreflexive Banach spaces. We find an optimal solution in a Sobolev space taking values in a Banach space under the Cesari type condition. We also investigate sufficient conditions for the existence of solutions to the initial value problem for the differential inclusion.
In this note, we present some complementary results on the infinite horizon optimal control for linear time-delay systems. We formally establish some properties of the matrices arising in the Bellman functional, and we prove that no concentrated delay term is present in the optimal control law.
New form of sufficient optimality condition is obtained in comparison with the Mangasarian sufficiency theorem. Both finite and infinite values of objective functional are allowed since concepts of overtaking and weakly overtaking optimality are implied. Examples, where new conditions can be applied are presented. The conditions are shown to be both necessary and sufficient, when Hamiltonian is linear with respect to state and control.
A class of infinite horizon optimal control problems involving $L^p$-type cost functionals with $0<p\leq 1$ is discussed. The existence of optimal controls is studied for both the convex case with $p=1$ and the nonconvex case with $0<p<1$, and the sparsity structure of the optimal controls promoted by the $L^p$-type penalties is analyzed. A dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers.
Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated. The controls are subject to a cost-functional which promotes sparsity in time. The focus is put on deriving first order optimality conditions. This is achieved without relying on a well-defined control-to-state mapping in a neighborhood of minimizers. The technique of proof is based on the approximation of the original problem by a family of finite horizon problems. The optimality conditions allow to deduce sparsity properties of the optimal controls in time.
The minimization of energy-like cost functionals is addressed in the context of optimal control problems. For a general class of dynamical systems, with possibly unstable and nonlinear free dynamics, it is shown that a sequence of solutions of finite time-horizon optimal control problems approximates a solution of the analog infinite time-horizon problem. The latter solution and corresponding optimal cost value function are not assumed to be known a-priori. Numerical simulations are presented validating the theoretical findings for several examples, including systems governed by both ordinary …
The topics treated in this thesis are inherently two-fold. The first part considers the problem of a market maker optimally setting bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. This stochastic control problem under partial information is solved by means of stochastic filtering, control and PDMPs theory. The value function is characterized as the unique continuous viscosity solution of its dynamic programming eq…
Second-Order Conditions for Infinite-Horizon Semilinear Parabolic Control Problems without Tikhonov Regularization
本文研究了带有Neumann边界条件、控制约束和无限时间 horizon 的半线性抛物型最优控制问题，提出二阶最优性条件并证明有限时间问题的局部最优状态在时间趋于无穷时逼近无限时间问题的局部最优状态。
Infinite Horizon Optimal Control Problems with Discount Factors
本文研究了受半线性抛物方程约束的无限 horizon 最优控制问题，探讨了不同折扣因子对二阶优化条件的影响，并证明了有限 horizon 问题对无限 horizon 问题的收敛性。
Necessary optimality conditions in the form of the maximum principle for control problems with infinite time horizon are considered. Both finite and infinite values of objective functional are allowed since the concept of overtaking or weakly overtaking optimality is used. New form of optimality condition is obtained and compared with the transversality conditions usually used in the literature. The examples, where these transversality conditions may fail while the new condition holds are presented. For Ramsey problem of capital accumulation a simple form of necessary optimality conditions is …
Optimal Stopping of BSDEs with Constrained Jumps and Related Zero-Sum Games
本文提出非线性Snell包络刻画带约束跳跃BSDE的最优停止值，并证明其与零和随机微分博弈的值等价，通过惩罚技术与控制随机化建立存在性与正则性。
A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some…
We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a n…
Path-dependent Hamilton--Jacobi equations: Uniqueness results for viscosity solutions defined via families of compact sets
本文研究了路径依赖的哈密顿-雅可比方程，证明了在特定条件下粘性解的唯一性，通过引入新的惩罚函数构造了协变光滑测试函数，并放宽了哈密顿量的假设条件。
A Note on Reflected BSDEs in Infinite Horizon with Stochastic Lipschitz Coefficients
本文将无限时域反射BSDE的结果推广到生成元具有随机Lipschitz系数的情形，并将其应用于生成元线性增长的FSDE稳健最优停止问题。
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