This study introduces an approach to obtain a neighboring extremal optimal control (NEOC) solution for a closed-loop optimal control problem, applicable to a wide array of nonlinear systems and not necessarily quadratic performance indices. The approach involves investigating the variation incurred in the functional form of a known closed-loop optimal control law due to small, known parameter variations in the system equations or the performance index. The NEOC solution can formally be obtained by solving a linear partial differential equation, akin to those encountered in the iterative solution of a nonlinear Hamilton-Jacobi equation. Motivated by numerical procedures for solving these latter equations, we also propose a numerical algorithm based on the Galerkin algorithm, leveraging the use of basis functions to solve the underlying Hamilton-Jacobi equation of the original optimal control problem. The proposed approach simplifies the NEOC problem by reducing it to the solution of a simple set of linear equations, thereby eliminating the need for a full re-solution of the adjusted optimal control problem. Furthermore, the variation to the optimal performance index can be obtained as a function of both the system state and small changes in parameters, allowing the determination of the adjustment to an optimal control law given a small adjustment of parameters in the system or the performance index. Moreover, in order to handle large known parameter perturbations, we propose a homotopic approach that breaks down the single calculation of NEOC into a finite set of multiple steps. Finally, the validity of the claims and theory is supported by theoretical analysis and numerical simulations.

-- Multi-armed bandit algorithms provide solutions for sequential decision-making where learning takes place by interacting with the environment. In this work, we model a distributed optimization problem as a multi-agent kernelized multi-armed bandit problem with a heterogeneous reward setting. In this setup, the agents collabo-ratively aim to maximize a global objective function which is an average of local objective functions. The agents can access only bandit feedback (noisy reward) obtained from the associated unknown local function with a small norm in reproducing kernel Hilbert space (RKHS). We present a fully decentralized algorithm, Multi-agent IGP-UCB (MA-IGP-UCB), which achieves a sub-linear regret bound for popular classes for kernels while preserving privacy. It does not necessitate the agents to share their actions, rewards, or estimates of their local function. In the proposed approach, the agents sample their individual local functions in a way that benefits the whole network by utilizing a running consensus to estimate the upper confidence bound on the global function. Furthermore, we propose an extension, Multi-agent Delayed IGP-UCB (MAD-IGP-UCB) algorithm, which reduces the dependence of the regret bound on the number of agents in the network. It provides improved performance by utilizing a delay in the estimation update step at the cost of more communication. HE problem of distributed optimization deals with the optimization of a function over a network of agents in which the whole function is not completely known to any single agent [1], [2]. In fact, the "global" function can be expressed as an average of "local" functions associated with each agent which are independent of one another. In particular, our interest lies in the case when these local functions are non-convex, unknown, and expensive to compute or record. To form a feasible problem, we assume that these local functions belong to a reproducing kernel Hilbert space (RKHS), which is a very common assumption in the literature [3]- [5]. When dealing with unknown functions, the problem for each agent can be broken down into two segments: sampling and optimization.

This paper provides a solution to the problem of safe region formation control with reference velocity tracking for a second-order multi-agent system without velocity measurements. Safe region formation control is a control problem where the agents are expected to attain the desired formation while reaching the target region and simultaneously ensuring collision and obstacle avoidance. To tackle this control problem, we break it down into two distinct objectives: safety and region formation control, to provide a completely distributed algorithm. Region formation control is modeled as a high-level abstract objective, whereas safety and actuator saturation are modeled as a low-level objective designed independently, without any knowledge of the former, and being minimally invasive. Our approach incorporates connectivity preservation, actuator saturation, safety considerations, and lack of velocity measurement from other agents with second-order system dynamics which are important constraints in practical applications. Both internal safety for collision avoidance among agents and external safety for avoiding unsafe regions are ensured using exponential control barrier functions. We provide theoretical results for asymptotic convergence and numerical simulation to show the approach's effectiveness.