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The paper studies the convergence properties of (continuous-time) best-response dynamics from game theory. Despite their fundamental role in game theory, best-response dynamics are poorly understood in many games of interest due to the discontinuous, set-valued nature of the best-response map. The paper focuses on elucidating several important properties of best-response dynamics in the class of multiagent games known as potential games--a class of games with fundamental importance in multiagent systems and distributed control. It is shown that in almost every potential game and for almost every initial condition, the best-response dynamics (i) have a unique solution, (ii) converge to pure-strategy Nash equilibria, and (iii) converge at an exponential rate. [ABSTRACT FROM AUTHOR]

This paper studies the impulse control of a general Markov process under the average (or ergodic) cost when the impulse instants are restricted to be the arrival times of an exogenous process, and this restriction is referred to as a constraint. A detailed setting is described, a characterization of the optimal cost is obtained as a solution of an HJB equation, and an optimal impulse control is identified. [ABSTRACT FROM AUTHOR]

Markov chain Monte Carlo (MCMC) has been widely used to approximate the expectation of the statistic of a given probability measure π on a finite set, and the asymptotic variance is a typical approach to evaluating the performance of MCMC methods. In this paper, we provide a lower bound of the worst-case analysis of the asymptotic variance over general Markov chains with invariant probability π, reversible as well as nonreversible ones, and construct an optimal transition matrix that achieves this lower bound. In fact, for any statistic f to be evaluated, an MCMC with the constructed optimal transition matrix produces a smaller asymptotic variance than independent sampling. [ABSTRACT FROM AUTHOR]

In this paper we treat null-controllability properties for the linear Kuramoto- Sivashinsky equation on a network with two types of boundary conditions. More precisely, the equation is considered on a star-shaped tree with Dirichlet and Neumann boundary conditions. By using the moment theory we can derive null-controllability properties with boundary controls acting on the external vertices of the tree. In particular, the controllability holds if the anti-diffusion parameter of the equation does not belong to a critical countable set of real numbers. We point out that the critical set for which the null-controllability fails differs from the first model to the second one. [ABSTRACT FROM AUTHOR]

This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The actual optimized cost functional measures the volume and perimeter of the structure. In the implementation, shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations. [ABSTRACT FROM AUTHOR]

This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming, for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn-Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behavior of some common opinion dynamic models in a continuum of agents. We show that as long as the interactions among the agents are symmetric, the distribution of the agents' opinions converges. We also investigate whether convergence occurs in a stronger sense than merely in distribution, namely, whether the opinion of almost every agent converges. We show that while this is not the case in general, it becomes true under plausible assumptions on interagent interactions, namely that agents with similar opinions exert a nonnegligible pull on each other, or that the interactions are entirely determined by their opinions via a smooth function. [ABSTRACT FROM AUTHOR]

In this paper, the robust controllability for linear systems with constrained controls x = Ax+Bu, u ∊ Ω is studied under the assumption that Ω is an arbitrary subset satisfying the only condition 0 ∊ clcoΩ and the system matrices are subjected to structured perturbations. The notion of the structured local and global controllability radii are defined. Based on properties of convex processes, some general formulas for the complex and the real controllability radii, for both local and global controllability concepts, are established with respect to structured affine perturbations and multiperturbations of matrices A,B. As particular cases, the general results are applied to linear systems with bounded controls and single-input linear systems, yielding some explicit and computable formulas of controllability radii. Examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The finite-time stabilization problem is considered for a class of multi-input multioutput nonlinear systems composed of several different subsystems that are coupled with unknown nonlinearities, possibly superlinear and nonvanishing at the origin. A novel decentralized, continuous finite-time output-feedback control algorithm is presented by dynamically compensating the unknown nonlinear couplings and applying a saturation technique. A crucial strategy is to estimate the uncertain nonlinearities and the unmeasured state by means of a finite-time extended state observer. The effectiveness of the proposed decentralized finite-time output-feedback design is validated by rigorous mathematical analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

A multilevel self-similar domination-monotonicity structure is proposed, and a kind of coupled forward-backward stochastic differential equations (FBSDEs) with such structure is proved to be uniquely solvable. Then, this kind of FBSDEs is used to characterize the unique equilibrium of a linear-quadratic generalized Stackelberg game with multilevel hierarchy in a closed form. [ABSTRACT FROM AUTHOR]

We study the minimization of the cost functional F(μ)=∥u−ud∥Lp(Ω)+α∥μ∥M(Ω), where the controls µ are taken in the space of finite Borel measures and u ∈ W1,1(Ω) satisfies the equation −∆u + g(u) = µ in the sense of distributions in Ω for a given nondecreasing continuous function g : R → R such that g(0) = 0. We prove that F has a minimizer for every desired state ud ∈ L1(Ω) and every control parameter α > 0. We then show that when ud is nonnegative or bounded, every minimizer of F has the same property. [ABSTRACT FROM AUTHOR]

We consider a diffusion control problem where the controller totally determines the state's diffusion coefficient but has no influence on the state's drift rate. By using the Pontryagin maximum principle we characterize an optimal control in terms of the adjoint forward-backward stochastic differential equation (FBSDE), turning out to be fully coupled. We use the method of decoupling fields for proving that the adjoint FBSDE possesses a solution. [ABSTRACT FROM AUTHOR]

We consider a class of linear-quadratic-Gaussian mean-field games having a major agent and numerous heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets Γk of Rm. The decentralized strategies of individual agents and the consistency condition system are represented in a unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on Γk . The well-posedness of the consistency system is established in both the local and global cases through the contraction mapping and discounting methods, respectively. A related ε -Nash-equilibrium property is also verified. [ABSTRACT FROM AUTHOR]

We study a distributed framework for stochastic optimization which is inspired by models of collective motion found in nature (e.g., swarming) with mild communication requirements. Specifically, we analyze a scheme in which each one of N > 1 independent threads implements, in a distributed and unsynchronized fashion, a stochastic gradient-descent algorithm which is perturbed by a swarming potential. Assuming the overhead caused by synchronization is not negligible, we show the swarming-based approach exhibits better performance than a centralized algorithm (based upon the average of N observations) in terms of (real-time) convergence speed. We also derive an error bound that is monotone decreasing in network size and connectivity. We characterize the scheme's finite-time performances for both convex and nonconvex objective functions. [ABSTRACT FROM AUTHOR]

We study an optimal investment/consumption problem in a model economy capturing market and credit risk dependencies. Stochastic factors drive both the default intensity and the volatility of the stocks in the portfolio. Using the martingale approach, we analyze the recursive system of nonlinear Hamilton-Jacobi-Bellman equations associated with the dual problem. We transform such a system into an equivalent system of semilinear PDEs, for which we establish existence and uniqueness of a bounded global classical solution. We obtain explicit representations for the optimal strategy, consumption path, and wealth process, in terms of the solution to the recursive system of semilinear PDEs. We numerically analyze the sensitivity of the optimal investment strategies to risk aversion, default risk, and volatility. [ABSTRACT FROM AUTHOR]

Optimal control problems for semilinear parabolic equations with control costs involving the total bounded variation seminorm are analyzed. This choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps. It is an appropriate choice if a simple structure of the optimal controls is desired, which, however, is still sufficiently flexible so that good tracking properties can be maintained. Existence of optimal controls, necessary and sufficient optimality conditions, and sparsity properties of the derivatives are obtained. Convergence of a finite element approximation is analyzed and numerical examples illustrating structural properties of the optimal controls are provided. [ABSTRACT FROM AUTHOR]