Hybrid Stochastic Systems: Numerical Methods, Limit Results, And Controls

This dissertation is concerned with the so-called stochastic hybrid systems, which are featured by the coexistence of continuous dynamics and discrete events and their interactions. Such systems have drawn much needed attentions in recent years. One of the main reasons is that such systems can be used to better reflect the reality for a wide range of applications in networked systems, communication systems, economic systems, cyber-physical systems, and biological and ecological systems, among others. Our main interest is centered around one class of such hybrid systems known as switching diffusions. In such a system, in addition to the driving force of a Brownian motion as in a stochastic system represented by a stochastic differential equation (SDE), there is an additional continuous-time switching process that models the environmental changes due to random events. In the first part, we develops numerical schemes for stochastic differential equations with Markovian switching (Markovian switching SDEs). By utilizing a special form of It^o\u27s formula for switching SDEs and special structural of the jumps of the switching component we derived a new scheme to simulate switching SDEs in the spirit of Milstein\u27s scheme for purely SDEs. We also develop a new approach to establish the convergence of the proposed algorithm that incorporates martingale methods, quadratic variations, and Markovian stopping times. Detailed and delicate analysis is carried out. Under suitable conditions which are natural extensions of the classical ones, the convergence of the algorithms is established. The rate of convergence is also ascertained. The second part is concerned with a limit theorem for general stochastic differential equations with Markovian regime switching. Given a sequence of stochastic regime switching systems where the discrete switching processes are independent of the state of the systems. The continuous-state component of these systems are governed by stochastic differential equations with driving processes that are continuous increasing processes and square integrable martingales. We establish the convergence of the sequence of systems to the one described by a state independent regime-switching diffusion process when the two driving processes converge to the usual time process and the Brownian motion in suitable sense. The third part is concerned with controlled hybrid systems that are good approximations to controlled switching diffusion processes. In lieu of a Brownian motion noise, we use a wide-band noise formulation, which facilitates the treatment of non-Markovian models. The wide-band noise is one whose spectrum has band width wide enough. We work with a basic stationary mixing type process. On top of this wide-band noise process, we allow the system to be subject to random discrete event influence. The discrete event process is a continuous time Markov chain with a finite state space. Although the state space is finite, we assume that the state space is rather large and the Markov chain is irreducible. Using a two-time-scale formulation and assuming the Markov chain also subjects to fast variations, using weak convergence and singular perturbation test function method we first proved that the when controlled by nearly optimal and equilibrium controls, the state and the corresponding costs of the original systems would converge to those of controlled diffusions systems. Using the limit controlled dynamic system as a guidance, we construct controls for the original problem and show that the controls so constructed are near optimal and nearly equilibrium

Closed-loop Equilibria for Mean-Field Games in Randomly Switching Environments with General Discounting Costs

This work is devoted to finding the closed-loop equilibria for a class of mean-field games (MFGs) with infinitely many symmetric players in a common switching environment when the cost functional is under general discount in time. There are two key challenges in the application of the well-known Hamilton-Jacobi-Bellman and Fokker-Planck (HJB-FP) approach to our problems: the path-dependence due to the conditional mean-field interaction and the time-inconsistency due to the general discounting cost. To overcome the difficulties, a theory for a class of systems of path-dependent equilibrium Hamilton-Jacobi-Bellman equations (HJBs) is developed. Then closed-loop equilibrium strategies can be identified through a two-step verification procedure. It should be noted that the closed-loop equilibrium strategies obtained satisfy a new form of local optimality in the Nash sense. The theory obtained extends the HJB-FP approach for classical MFGs to more general conditional MFGs with general discounting costs

An optimal control problem for the continuity equation arising in smart charging

This paper is focused on the mathematical modeling and solution of the optimal charging of a large population of identical plug-in electric vehicles (PEVs) with mixed state variables (continuous and discrete). A mean field assumption is formulated to describe the evolution interaction of the PEVs population. The optimal control of the resulting continuity equation of the mixed system under state constraints is investigated. We prove the existence of a minimizer. We then characterize the solution as the weak solution of a system of two coupled PDEs: a continuity equation and of a Hamilton-Jacobi equation. We provide regularity results of the optimal feedback control

Nash and Wardrop equilibria in aggregative games with coupling constraints

We consider the framework of aggregative games, in which the cost function of each agent depends on his own strategy and on the average population strategy. As first contribution, we investigate the relations between the concepts of Nash and Wardrop equilibrium. By exploiting a characterization of the two equilibria as solutions of variational inequalities, we bound their distance with a decreasing function of the population size. As second contribution, we propose two decentralized algorithms that converge to such equilibria and are capable of coping with constraints coupling the strategies of different agents. Finally, we study the applications of charging of electric vehicles and of route choice on a road network.Comment: IEEE Trans. on Automatic Control (Accepted without changes). The first three authors contributed equall

Markov decision processes and discrete-time mean-field games constrained with costly observations: Stochastic optimal control constrained with costly observations

In this thesis, we consider Markov decision processes with actively controlled observations. Optimal strategies involve the optimisation of observation times as well as the subsequent action values. We first consider an observation cost model, where the underlying state is observed only at chosen observation times at a cost. By including the time elapsed from observations as part of the augmented Markov system, the value function satisfies a system of quasi-variational inequalities (QVIs). Such a class of QVIs can be seen as an extension to the interconnected obstacle problem. We prove a comparison principle for this class of QVIs, which implies uniqueness of solutions to our proposed problem. Penalty methods are then utilised to obtain arbitrarily accurate solutions. Finally, we perform numerical experiments on three applications which illustrate this model. We then consider a model where agents can exercise control actions that affect their speed of access to information. The agents can dynamically decide to receive observations with less delay by paying higher observation costs. Agents seek to exploit their active information gathering by making further decisions to influence their state dynamics to maximise rewards. We also extend this notion to a corresponding mean-field game (MFG). In the mean field equilibrium, each generic agent solves individually a partially observed Markov decision problem in which the way partial observations are obtained is itself also subject to dynamic control actions by the agent. Based on a finite characterisation of the agents’ belief states, we show how the mean field game with controlled costly information access can be formulated as an equivalent standard mean field game on a suitably augmented but finite state space. We prove that with sufficient entropy regularisation, a fixed point iteration converges to the unique MFG equilibrium and yields an approximate ε-Nash equilibrium for a large but finite population size. We illustrate our MFG by an example from epidemiology, where medical testing results at different speeds and costs can be chosen by the agents

NASA Workshop on Distributed Parameter Modeling and Control of Flexible Aerospace Systems

Although significant advances have been made in modeling and controlling flexible systems, there remains a need for improvements in model accuracy and in control performance. The finite element models of flexible systems are unduly complex and are almost intractable to optimum parameter estimation for refinement using experimental data. Distributed parameter or continuum modeling offers some advantages and some challenges in both modeling and control. Continuum models often result in a significantly reduced number of model parameters, thereby enabling optimum parameter estimation. The dynamic equations of motion of continuum models provide the advantage of allowing the embedding of the control system dynamics, thus forming a complete set of system dynamics. There is also increased insight provided by the continuum model approach