Catherine Donnelly
We prove a sufficient stochastic maximum principle for the optimal control of a regime-switching diffusion model. We show the connection to dynamic programming and we apply the result to a quadratic loss minimization problem, which can be used to solve a mean-variance portfolio selection problem.
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This paper presents three versions of maximum principle for a stochastic optimal control problem of Markov regime-switching forward-backward stochastic differential equations with jumps (FBSDEJs). A general sufficient maximum principle for optimal control for a system driven by a Markov regime-switching forward and backward jump-diffusion model is developed. After, an equivalent maximum principle is proved. Malliavin calculus is also employed to derive a general stochastic maximum principle. The latter does not require concavity of Hamiltonian. Applications of the stochastic maximum principle …
In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum prin…
In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward-backward stochastic differential equations with jumps and partial information. First, we prove a sufficient maximum principle for non zero-sum stochastic differential game problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for non zero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a proble…
A stochastic maximum principle for singular mean-field regime-switching optimal control
本文提出了一种适用于均场切换奇异随机最优控制问题的随机最大原理，并应用于带有交易成本的银行间借贷模型分析。
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.
An infinite horizon sufficient stochastic maximum principle for regime switching diffusions and applications
本文提出了一种无限时间 horizon 的随机最大原理，用于 regime switching 扩散的最优控制问题，并通过线性二次生产计划问题展示了其应用。
In this paper, we establish a general stochastic maximum principle for optimal control for systems described by a continuous-time Markov regime-switching stochastic recursive utilities model. The control domain is postulated not to be convex, and the diffusion terms depend on control variables. To this end, we first study a kind of classical forward stochastic optimal control problems. Afterwards, based on previous results, we introduce two groups of new first and second-order adjoint equations. The corresponding variational equations for forward-backward stochastic differential equations are …
This paper is devoted to a global stochastic maximum principle for conditional mean-field forward-backward stochastic differential equations (FBSDEs, for short) with regime switching. The control domain is unnecessarily convex and the driver of backward stochastic differential equations (BSDEs, for short) could depend on $Z$. Different from the case of non-recursive utility, the first-order and second-order adjoint equations are both high-dimensional linear BSDEs. Based on the adjoint equations, we reveal the relations among the terms of the first- and second-order Taylor's expansions. A gener…
This report provides a comprehensive complexity study of line switching in the Linear DC model for the feasibility problem and the optimization problems of maximizing the load that can be served (maximum switching flow, MSF) and minimizing generation cost (optimal transmission switching, OTS). Our results show that these problems are NP-complete and that there is no fully polynomial-time approximation scheme for planar networks with a maximum-node degree of 3. Additionally, we demonstrate that the problems are still NP-hard if we restrict the network structure to cacti with a maximum degree of…
Sufficient Conditions on the Existence of Switching Observers for Nonlinear Time-Varying Systems
针对包括三角结构在内的一类非线性时变系统，在比现有文献更弱的假设下，通过构造时变非因果动态的切换序列实现状态确定。
Bang--bang trajectories with a double switching time: sufficient strong local optimality conditions
本文针对具有多个切换点的bang-bang极值轨迹，给出了强拓扑下局部最优的充分条件，该条件是先前特定情形条件的自然推广。
We study local structure of time-optimal controls and trajectories for a 3-dimensional control-affine system with a 2-dimensional control parameter with values in the disk. In particular, we give sufficient conditions, in terms of Lie bracket relations, for optimal controls to be smooth or to have only isolated jump discontinuities.
This paper deals with the stability analysis problem of discrete-time switched linear systems with ranged dwell time. A novel concept called L-switching-cycle is proposed, which contains sequences of multiple activation cycles satisfying the prescribed ranged dwell time constraint. Based on L-switching-cycle, two sufficient conditions are proposed to ensure the global uniform asymptotic stability of discrete-time switched linear systems. It is noted that two conditions are equivalent in stability analysis with the same $L$-switching-cycle. These two sufficient conditions can be viewed as gener…
Maximum Principle of Stochastic Optimal Control Problems with Model Uncertainty
本文在模型不确定性和马尔可夫机制切换框架下，通过变分不等式、线性化与弱收敛技术建立了随机最优控制的必要最大值原理及充分条件，并应用于风险最小化投资组合选择问题。
Sufficient conditions for the genericity of feedback stabilisability of switching systems via Lie-algebraic solvability
本文基于公共二次Lyapunov函数，利用李代数可解性推导了离散时间切换线性系统在任意切换下反馈镇定的一般性充分条件，并给出了高效的数值实现方法。
This paper studies switching stabilization problems for continuous-time switched linear systems. We consider four types of switching stabilizability defined under different assumptions on the switching control input. The most general switching stabilizability is defined as the existence of a measurable switching signal under which the resulting time-varying system is asymptotically stable. Discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length. In addition, we define feedback stabilizability in Filippov …
In this note, we discuss the problem of consensus finding in communication networks of agents with dynamically switching topologies. In particular, we consider the case of directed networks with unbalanced matrices of communication rates. We formulate sufficient conditions for consensus finding in terms of strong connectivity of the underlying directed graphs and prove that, given these conditions, consensus is found asymptotically. Moreover, we show that this consensus is an emergent property of the system, being encoded in its dynamics and not just an invariant of its initial configuration.
In this paper we analyse local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. In the case of k equal to n-1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities.
This paper studies a class of zero-sum stopping game in a regime switching model. A verification theorem as a sufficient criterion for Nash equilibriums is established based on a set of variational inequalities (VIs). Under an appropriate regularity condition for solutions to the VIs, a suitable system of algebraic equations is derived via the so-called smooth-fit principle. Explicit Nash equilibrium stopping rules of threshold-type for the two players and the corresponding value function of the game in closed form are obtained. Numerical experiments are reported to demonstrate the dependence …
In this paper, we consider a linear quadratic (LQ) leader-follower stochastic differential game for regime switching diffusions with mean-field interactions. One of the salient features of this paper is that conditional mean-field terms are included in the state equation and cost functionals. Based on stochastic maximum principles (SMPs), the follower's problem and the leader's problem are solved sequentially and an open-loop Stackelberg equilibrium is obtained. Further, with the help of the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and…
This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing result…
In this paper we consider an infinite horizon zero-sum differential game where the dynamics of each player and the running cost are also depending on the evolution of some discrete (switching) variables. In particular, such switching variables evolve according to the switching law of a so-called thermostatic delayed relay, applied to the players' states. We first address the problem of the continuity of both lower and upper value function. Then, by a suitable representation of the problem as a coupling of several exit-time differential games, we characterize those value functions as, respectiv…
We consider an optimal control problem for a system governed by a Volterra integral equation with impulsive terms. The impulses act on both the state and the control; the control consists of switchings at discrete times. The cost functional includes both, an integrated cost rate (continuous part) and switching costs at the discrete impulse times (discrete part). We prove necessary optimality conditions of a form analogous to a discrete maximum principle. For the particular case of a system governed by impulsive ordinary differential equations, we obtain an impulsive maximum principle as a spec…
When is a set of LMIs a sufficient condition for stability?
本文研究了离散时间切换系统的稳定性，刻画了作为稳定性充分条件的LMI集合的结构，并证明了识别特定LMI集合是否蕴含系统稳定性的问题是PSPACE完全的。
Infinite Horizon Linear Quadratic Mean Field Problems with Common Noise and Regime Switching via Conditional McKean-Vlasov FBSDEs
本文通过条件 McKean-Vlasov FBSDEs 方法，研究了包含共同噪声和制度切换的无限时间线性二次均场问题，推导了控制与博弈问题的最优解条件。
We consider a linear consensus system with n agents that can switch between r different connectivity patterns. A natural question is which switching law yields the best (or worst) possible speed of convergence to consensus? We formulate this question in a rigorous manner by relaxing the switched system into a bilinear consensus control system, with the control playing the role of the switching law. A best (or worst) possible switching law then corresponds to an optimal control. We derive a necessary condition for optimality, stated in the form of a maximum principle (MP). Our approach, combine…
Turnpike properties for zero-sum stochastic linear quadratic differential games of Markovian regime switching system
本文研究马尔可夫机制切换系统中零和随机线性二次微分博弈的长期行为，建立最优三元组的转辙性质，证明有限时域均衡策略可逼近无限时域策略，并扩展了现有结果。
The finite state semi-Markov process is a generalization over the Markov chain in which the sojourn time distribution is any general distribution. In this article we provide a sufficient stochastic maximum principle for the optimal control of a semi-Markov modulated jump-diffusion process in which the drift, diffusion and the jump kernel of the jump-diffusion process is modulated by a semi-Markov process. We also connect the sufficient stochastic maximum principle with the dynamic programming equation. We apply our results to finite horizon risk-sensitive control portfolio optimization problem…
This paper is concerned with optimal switching over multiple modes in continuous time and on a finite horizon. The performance index includes a running reward, terminal reward and switching costs that can belong to a large class of stochastic processes. Particularly, the switching costs are modelled by right-continuous with left-limits processes that are quasi-left-continuous and can take both positive and negative values. We provide sufficient conditions leading to a well known probabilistic representation of the value function for the switching problem in terms of interconnected Snell envelo…
In leader-follower multi-agent networks with switching topologies, choosing a subset of agents as leaders is a critical step to achieve desired performances. In this paper, we concentrate on the problem of selecting a minimum-size set of leaders that ensure the tracking of a reference signal in a highorder linear multi-agent network with a set of given topology dependent dwell time (TDDT). First, we derive a sufficient condition that guarantees the states of all agents converging to an expected state trajectory. Second, by exploiting submodular optimization method, we formulate the problem of …
Pinning dynamic systems of networks with Markovian switching couplings and controller-node set
本文研究了具有随机切换耦合和随机选择控制器节点集的耦合动态系统pinsing控制问题，通过构造Lyapunov函数建立指数稳定性条件，并通过两个场景验证了理论结果。
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent results on norm estimates for doubly reflected backward stochastic differential equations. This theory is then demonstrated numerically for the special cases of cancellable call and put options in a Black-Scholes market.
Multiparametric continuous-time optimal control via Pontryagin's Maximum Principle: explicit solutions and comparisons with discrete-time formulations
本文提出一种连续时间多参数框架，用于线性二次最优控制，通过直接求解Pontryagin最优性条件避免离散化问题，展示连续时间方法在减少关键区域复杂度和揭示系统动态方面优于离散时间方法。
This paper is concerned with cost optimization of an insurance company. The surplus of the insurance company is modeled by a controlled regime switching diffusion, where the regime switching mechanism provides the fluctuations of the random environment. The goal is to find an optimal control that minimizes the total cost up to a stochastic exit time. A weaker sufficient condition than that of (Fleming and Soner 2006, Section V.2) for the continuity of the value function is obtained. Further, the value function is shown to be a viscosity solution of a Hamilton-Jacobian-Bellman equation.
Stability of stochastic dynamic systems of a random structure with Markov switchings in the presence of concentration points
本文研究了存在浓度点时具有Markov切换的随机结构随机微分方程的稳定性，利用Lyapunov函数及对跳跃幅度和时间的约束证明了渐近稳定性，并通过实例验证。
We present a data-driven model predictive control scheme for chance-constrained Markovian switching systems with unknown switching probabilities. Using samples of the underlying Markov chain, ambiguity sets of transition probabilities are estimated which include the true conditional probability distributions with high probability. These sets are updated online and used to formulate a time-varying, risk-averse optimal control problem. We prove recursive feasibility of the resulting MPC scheme and show that the original chance constraints remain satisfied at every time step. Furthermore, we show…
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