# Research Interests

## Dynamic Data Driven Framework

Accurate analysis and prediction of the behavior of a complicated system is difficult. Today, applications and systems simulations are fairly complex, but still are lacking the ability to accurately describe such systems. Even elaborate complex models of systems produce simulations that diverge from or fail to predict the real behaviors of those systems.

Dynamic Data Driven Application Systems is a paradigm in which computation and instrumentation aspects of an application system are dynamically integrated in a feedback control loop, in the sense that instrumentation data can be dynamically incorporated into the executing model of the application, and in reverse the executing model can control the instrumentation. New capabilities include modeling approaches, algorithm developments, systems software, and instrumentation methods, and as well as the need for synergistic multidisciplinary research among these areas. Dynamic data driven systems brings together practitioners of application domains, researchers in mathematics, statistics, electrical engineering, and computer sciences, as well as designers involved in the development of instrumentation systems and methods.

The framework can be widely applicable to the dynamical control systems, where there is high uncertainty and a lack of system knowledge. The framework can also be enriched by mathematical theories such as chaos theory. In future research, I plan to develop dynamic data driven framework for multiagent network systems to overcome the challenges of stochastic optimal control of a large scale systems.

## Swarm control

Swarm concepts, multiple agents operating in formation near one another, are of growing interest in the multiagent community as a solution to this challenge of simultaneous measurement. The capabilities needed to support swarm missions go beyond operator-specified geometry, alignment, or separation, but also cross-link communication with maintaining position in the formation.

Large swarms of agents will introduce new space mission capabilities and complexities. However, the conventional approach of planning and commanding individual satellites does not scale for swarms of tens or hundreds of satellites. Instead, a swarm must operate as a unit, responding to high-level commands and constraints. The complexity of this large-scale network dynamical systems is due to the natural scale of these systems and often necessitates hierarchical decentralized and distributed architectures for analyzing and controlling these systems due to their high system dimensionality and communication connection constraints.

## Stochastic semistability with application to network consensus and random communication noise

In large-scale network systems, the communication graph topology of the network can involve probabilistic variations in the information transfer between agents. This stochastic modeling can be used to capture communication uncertainty and attenuation errors between the agents in the network

As part of my research, I extended my deterministic semistability work on discrete-time systems to the case of stochastic systems. In particular, I developed necessary and sufficient Lyapunov conditions for stochastic semistability by using a new difference operator for the Lyapunov function, which I show is an analog of the infinitesimal generator used in the study of continuous-time stochastic dynamical system. These results are then used to develop semistable consensus protocols for discrete-time networks with communication uncertainty capturing measurement noise and attenuation errors in the information transfer between agents. The proposed distributed control architecture involves the exchange of information between agents guaranteeing that the closed-loop dynamical network is stochastically semistable to an equipartitioned equilibrium state representing a state of almost sure consensus consistent with basic discrete-time thermodynamic principles.

In future research, I plan to focus on robustness properties of the proposed protocols, as well as asynchronism, system time delays, and dynamic network topologies for addressing possible information asynchrony between agents, message transmission and processing delays, and communication link failures and communication dropouts.

## Finite and fixed time stability and stabilization

The notions of asymptotic and exponential stability in dynamical systems theory imply convergence of the system trajectories to an equilibrium state over the infinite horizon. In many applications, however, it is desirable that a dynamical system possesses the property that trajectories that converge to a Lyapunov stable equilibrium state must do so in finite time rather than merely asymptotically.

As part of my research effort, I have addressed finite time stability for discrete autonomous systems by developing Lyapunov and converse Lyapunov theorems for finite-time stability and showed that the settling-time function capturing the finite settling time behavior of the dynamical system and the regularity properties of the Lyapunov function are related. Moreover, I have proposed Lyapunov theorems for different types of fixed time stability for discrete autonomous systems, where the minimum bound on the settling-time function is characterized independently of the system initial conditions. Then, building on the analysis framework, I developed a framework for optimal nonlinear feedback control for finite time and fixed time stabilization for nonlinear discrete-time controlled dynamical systems by establishing connections between the Lyapunov functions guaranteeing finite time and fixed time stability, and the solution to the Bellman equation.

In future research, I aim to extend the framework of finite time stability for both continuous-time and discrete-time systems to address various control problems such as finite time consensus of heterogeneous network systems, stochastic optimal control of hybrid dynamical systems, and system robotics. Moreover, I plan to work on \emph{fixed time} stability and stabilization, which guarantees finite time stability in prescribed time, of dynamical systems to deal with control problems that guarantee predefined convergence times for network consensus, formation control of unmanned vehicles, and manufacturing process optimization. Finally, I plan to merge my developed frameworks with machine learning algorithms to guarantee online learning and optimality of my developed controller architectures.

## Hybrid dynamical systems

Modern complex aerospace systems can give rise to hybrid and impulsive systems that possess hierarchical hybrid control architectures characterized by continuous-time dynamics at the lower levels of the hierarchy and discrete time dynamics at the higher levels of the hierarchy. More specifically, these systems consist of three elements, a continuous-time differential equation, which governs the motion of the dynamical system between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs; and a criterion for determining when the states of the system are to be reset.

In this research, I developed a finite-time stability and stabilization framework for state dependent nonlinear impulsive systems by deriving new sufficient Lyapunov conditions for finite time stability and stabilization of impulsive dynamical systems. Specifically, I developed sufficient Lyapunov conditions for finite time stability of impulsive dynamical systems using both a scalar differential Lyapunov inequality on the continuous-time dynamics as well as a scalar difference Lyapunov inequality on the discrete-time resetting dynamics. Furthermore, using the proposed finite time stability results, I designed universal hybrid finite time stabilizing control laws for impulsive dynamical systems.

In future research, I plan to explore finite time and fixed time Lyapunov stability architectures for hybrid systems with system time delays, which are critical in industrial applications. Moreover, I will extend my stability analysis and synthesis results on hybrid and impulsive systems to model various real-world problems including automation, heterogeneous network systems, and hybrid control for swarm dynamics.

## Asymptotic and finite time semistability for network systems

Asymptotic and finite time stability are not the appropriate notions of stability for systems having a continuum of equilibria. Two notions that are of particular relevance to such systems are convergence and semistability. For this class of systems, I developed a framework for semistability and finite-time semistability of discrete-time systems. Specifically, I developed the first Lyapunov and converse Lyapunov theorems for semistability and finite time semistability. Moreover, using this framework, I designed semistable and finite time semistable consensus protocols for discrete dynamical networks based on thermodynamic principles.

In future research, I plan to extend this work to hybrid network and multiagent systems that involve intermittent communication between agents. Furthermore, I plan to develop new analysis structures using geometric concepts in order to circumvent theories relying on Lyapunov tests which can be difficult to apply for complex large-scale multilayered networks.

## Optimal finite time stochastic control

As compared to continuous-time stochastic dynamical systems, stability analysis of the discrete-time stochastic dynamical systems has received far less attention. In my recent research, I addressed finite time stability in probability of discrete-time stochastic dynamical systems. Specifically, I provide Lyapunov theorems for finite time stability in probability for Ito-type stationary nonlinear stochastic difference equations involving Lyapunov difference conditions on the minimum of the Lyapunov function itself along with a fractional power of the Lyapunov function. In addition, I established sufficient conditions for almost sure lower semicontinuity of the stochastic settling-time capturing the average settling time behavior of the discrete-time nonlinear stochastic dynamical system.

Furthermore, I developed a stochastic finite-time optimal control framework by exploiting connections between stochastic Lyapunov theory for finite time stability in probability and stochastic Bellman theory. In particular, I showed that finite time stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic finite time stability and optimality. This control architecture was applied to the optimal stochastic finite time stabilization of spacecraft dynamics.

In future research, I will focus on bridging the gap between stochastic optimal control theory and practical applications. Specifically, I plan to merge my stochastic optimal control framework with data-driven reinforcement learning control for learning the optimal control policy online and in fixed time.

## Thermodynamics, large-scale systems, and multi-agent systems

My current research has concentrated on coordinated control of large-scale stochastic multi-agent networked systems. In many applications involving multiagent systems, a group of agents are required to agree on certain quantities of interest. In such applications, it is important to develop information consensus protocols for a network of dynamic agents.

As part of this ongoing research effort, I am addressing semistable (i.e., Lyapunov stable plus convergent) consensus problems for nonlinear stochastic network systems with switching communication topologies to address communication dropouts. In particular, I am developing a thermodynamic-based stochastic control framework in order to consider random communication disturbances between agents in the network, wherein the evolution of each link of the random network follows a Markov process. This will necessitate the development of almost sure consensus of multiagent systems with nonlinear stochastic dynamics under distributed nonlinear consensus protocols.