Showing papers in "Siam Journal on Control and Optimization in 2016"
TL;DR: This paper is devoted to studying the trajectory and state controllability of Boolean control networks (BCNs) with time delay, and uses the semi-tensor product of matrices to convert the delayed BCNs into an equivalent algebraic description.
Abstract: This paper is devoted to studying the trajectory and state controllability of Boolean control networks (BCNs) with time delay. In contrast to BCNs without time delay, the dynamics of delayed BCNs are determined by a sequence of initial states, named here trajectories. Trajectory controllability means that there exists a control signal steering a system from an initial trajectory to a desired trajectory, while state controllability means that there exists a control signal steering an initial state to a given state. Here, both trajectory controllability and state controllability will be studied. It should be noted that in this paper, trajectory controllability does not mean tracking or following a given trajectory. In fact it means to control BCNs to a destination trajectory of length $\mu$ at the $k$-th step. Using the semi-tensor product of matrices, the delayed BCNs are first converted into an equivalent algebraic description, and then some necessary and sufficient conditions are derived for the trajecto...
168 citations
TL;DR: In this paper, a stochastic linear quadratic (LQ) optimal control problem is considered and the notions of open-loop and closed-loop solvabilities are introduced.
Abstract: This paper is concerned with a stochastic linear quadratic (LQ) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue of how negative the control weighting matrix-valued function $R(\cdot)$ can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, some illustrative examples are presented.
129 citations
TL;DR: This paper studies the stability of remotely controlled and observed single-input controllable linear systems under power-constrained pulse-width modulated denial-of-service (DoS) signals and proposes a first type of resilient control and triggering strategies which is provably capable of beating partially known jamming signals.
Abstract: This paper studies the stability of remotely controlled and observed single-input controllable linear systems under power-constrained pulse-width modulated denial-of-service (DoS) signals. The effect of a DoS jamming signal is to corrupt the communication channels, thus preventing the data from being received at its destination. In this work, we first assume that the DoS signal is partially known, i.e., a uniform lower bound for the off time intervals and the on-to-off transiting time instants are known. We then propose a first type of resilient control and triggering strategies which is provably capable of beating partially known jamming signals. Building on this, we then present our joint control and identification algorithms, ${\sc JAMCOID for Periodic Signals}$ and ${\sc JAMCOID}$, which are provably able to guarantee the system stability under unknown jamming signals. More precisely, the ${\sc JAMCOID for Periodic Signals}$ algorithm is able to partly identify a periodic DoS signal with known uniform...
115 citations
TL;DR: In this paper, the authors consider second order dynamical systems with the problem of finding zeros of the sum of a maximally monotone operator and a coco-ercive one.
Abstract: We begin by considering second order dynamical systems of the from $\ddot x(t) + \gamma(t)\dot x(t) + \lambda(t)B(x(t))=0$, where $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space ${\cal H}$, $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function, and $\gamma:[0,+\infty)\rightarrow [0,+\infty)$ is a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator $B$. The framework allows us to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as a particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when $B$ is the gradient of a smooth convex function the value of the latter converges along the ergo...
83 citations
TL;DR: In this article, the authors consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior.
Abstract: In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.
79 citations
TL;DR: In this paper, the authors study mean field game systems under density constraints as optimality conditions of two optimization problems in duality, and define a weak notion of Nash equilibrium for their model.
Abstract: In this paper we study mean field game systems under density constraints as optimality conditions of two optimization problems in duality. A weak solution of the system contains an extra term, an additional price imposed on the saturated zones. We show that this price corresponds to the pressure field from the models of incompressible Euler equations a la Brenier. By this observation we manage to obtain a minimal regularity, which allows us to write optimality conditions at the level of single-agent trajectories and to define a weak notion of Nash equilibrium for our model.
78 citations
Journal Article•
TL;DR: In this paper, the authors show that if there exist indistinguishability obfuscators for a certain class of circuits then there do not exist EOWFs where extraction works for any adversarial program with auxiliary-input of unbounded polynomial length.
Abstract: A function f is extractable if it is possible to algorithmically \"extract,\" from any adversarial program that outputs a value y in the image of f; a preimage of y. When combined with hardness properties such as one-wayness or collision-resistance, extractability has proven to be a powerful tool. However, so far, extractability has not been explicitly shown. Instead, it has only been considered as a non-standard knowledge assumption on certain functions. We make two headways in the study of the existence of extractable one-way functions (EOWFs). On the negative side, we show that if there exist indistinguishability obfuscators for a certain class of circuits then there do not exist EOWFs where extraction works for any adversarial program with auxiliary-input of unbounded polynomial length. On the positive side, for adversarial programs with bounded auxiliary input (and unbounded polynomial running time), we give the first construction of EOWFs with an explicit extraction procedure, based on relatively standard assumptions (e.g., sub-exponential hardness of Learning with Errors). We then use these functions to construct the first 2-message zero-knowledge arguments and 3-message zero-knowledge arguments of knowledge, against the same class of adversarial verifiers, from essentially the same assumptions.
75 citations
TL;DR: A diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier--Stokes system with a convective nonlocal Cahn--Hilliard equation in two dimensions of space is studied.
Abstract: We study a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids coupling the Navier--Stokes system with a convective nonlocal Cahn--Hilliard equation in two dimensions of space. We apply recently proved well-posedness and regularity results in order to establish existence of optimal controls as well as first-order necessary optimality conditions for an associated optimal control problem in which a distributed control is applied to the fluid flow.
75 citations
TL;DR: A large population of homogeneous social networks is considered, characterized by a vector state, a vector-valued controlled input, and a vectors-valued exogenous disturbance, to establish a robust mean-field equilibrium.
Abstract: Emulation, mimicry, and herding behaviors are phenomena that are observed when multiple social groups interact. To study such phenomena, we consider in this paper a large population of homogeneous social networks. Each such network is characterized by a vector state, a vector-valued controlled input and a vector-valued exogenous disturbance. The controlled input of each network is to align its state to the mean distribution of other networks’ states in spite of the actions of the disturbance. One of the contributions of this paper is a detailed analysis of the resulting mean field game for the cases of both polytopic and L2 bounds on controls and disturbances. A second contribution is the establishment of a robust mean-field equilibrium, that is, a solution including the worst-case value function, the state feedback best-responses for the controlled inputs and worst-case disturbances, and a density evolution. This solution is characterized by the property that no player can benefit from a unilateral deviation even in the presence of the disturbance. As a third contribution, microscopic and macroscopic analyses are carried out to show convergence properties of the population distribution using stochastic stability theory.
73 citations
TL;DR: The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation and a fully discrete scheme is considered: piecewise constant functions for the control and, for the state, first-degree tensor product finite elements in space and a finite difference discretization in time.
Abstract: We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders $s \in (0,1)$ and $\gamma \in (0,1]$, respectively The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem with a dynamic boundary condition as state equation The rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation We consider a fully discrete scheme: piecewise constant functions for the control and, for the state, first-degree tensor product finite elements in space and a finite difference discretization in time We show convergence of this scheme and, under additional data regularity, derive a priori error estimates for the case $s \in (0,1)$ and $\gamma = 1$
69 citations
Journal Article•
TL;DR: In this article, the authors proposed an approach to solve the problem of energy-efficient computing for the U.S. Dept. of Energy's Advanced Scientific Computing Research (Award DE-SC0007099).
Abstract: United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Award DE-SC0007099)
TL;DR: In this paper, the problem of portfolio selection with uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation, and solved under the Black-Scholes model under the theory of $G$-Brownian motions.
Abstract: In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black--Scholes model for risky assets with an ambiguous correlation using the theory of $G$-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including constant relative risk aversion and constant absolute risk aversion utilities, when stochastic volatilities are fast mean reverting. We propose a practical trading strategy that combines information from the option implied volatilit...
TL;DR: In this paper, the authors consider mean field games in the presence of common noise relaxing the usual independence assumption of individual random noise and show the existence and uniqueness of a mean field game solution using the stochastic maximum principle.
Abstract: In this paper, we consider mean field games in the presence of common noise relaxing the usual independence assumption of individual random noise. We assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property. Our main result is showing existence and uniqueness of a mean field game solution using the stochastic maximum principle. The uniqueness is a result of a monotonicity property similar to that of Lasry and Lions. We use the Banach fixed point theorem to establish an existence over a small time duration and show that it can be extended over an arbitrary finite time duration.
TL;DR: An implicit lower bound for $\tau^*$ is provided which can be computed numerically and enables the design of stochastic delay feedback controls in order to stabilize unstable differential equations.
Abstract: This paper is concerned with the almost sure exponential stability of the multi-dimensional nonlinear stochastic differential delay equation (SDDE) with variable delays of the form $dx(t) = f(x(t-\delta_1(t)),t)dt + g(x(t-\delta_2(t)),t) dB(t)$, where $\delta_1, \ \delta_2: \mathbb{R}_+\to [0,\tau]$ stand for variable delays. We show that if the corresponding (nondelay) stochastic differential equation (SDE) $dy(t) = f(y(t),t)dt + g(y(t),t) dB(t)$ admits a Lyapunov function (which in particular implies the almost sure exponential stability of the SDE) then there exists a positive number $\tau^*$ such that the SDDE is also almost sure exponentially stable as long as the delay is bounded by $\tau^*$. We provide an implicit lower bound for $\tau^*$ which can be computed numerically. Moreover, our new theory enables us to design stochastic delay feedback controls in order to stabilize unstable differential equations.
TL;DR: A class of distributed stochastic gradient algorithms are proposed that solve the problem using only local computation and communication, and the implementation of the algorithms removes the need for performing the intermediate projections.
Abstract: In this paper, we consider an optimization problem, where multiple agents cooperate to minimize the sum of their local individual objective functions subject to a global inequality constraint. We propose a class of distributed stochastic gradient algorithms that solve the problem using only local computation and communication. The implementation of the algorithms removes the need for performing the intermediate projections. For strongly convex optimization, we employ a smoothed constraint incorporation technique to show that the algorithm converges at an expected rate of $\mathcal{O}(\ln T / T)$ (where $T$ is the number of iterations) with bounded gradients. For non-strongly convex optimization, we use a reduction technique to establish an $\mathcal{O}(1/\sqrt{T})$ convergence rate in expectation. Finally, a numerical example is provided to show the convergence of the proposed algorithms.
TL;DR: A novel structure of networked control systems (NCSs) with communication logic, which incorporates model-based networking systems, predictive control, and an event-triggered communication scheme into a unified framework to consider the bandwidth reduction of the network communications is proposed.
Abstract: This paper proposes a novel structure of networked control systems (NCSs) with communication logic, which incorporates model-based networked control systems (MB-NCSs), predictive control, and an event-triggered communication scheme into a unified framework to consider the bandwidth reduction of the network communications. Within this framework, first an event-triggered communication scheme at the sensor side is introduced to determine whether or not the sensor measurement should be transmitted to the controller through the imperfect forward paths. Second, at the controller, a model of the plant is used to predict future state behavior of the plant, and based on the predicted state between successful transmission instants, a novel predictive event-triggering scheme is proposed to compress the size of the packetized control signals transmitted from the controller side to the actuator side through feedback paths. Finally, a unified model of NCSs is established. Based on this model, a codesign condition of th...
TL;DR: It is proved that the semigroup corresponding to the system is polynomially or exponentially stable and the decay rate depends on the parameter $\alpha\in(0,1]$.
Abstract: In this paper, we study the stability of an elastic string system with local Kelvin--Voigt damping. Few results for this system are known in the literature. Under the assumption that the damping coefficient has a singularity at the interface of the damped and undamped regions and behaves like $x^\alpha$ near the interface, we prove that the semigroup corresponding to the system is polynomially or exponentially stable and the decay rate depends on the parameter $\alpha\in(0,1]$.
TL;DR: In this work, the situation where the agents are provided only with partial information on the major agent's state is considered for systems with nonlinear dynamics and cost functions, and an $\epsilon$-Nash MFG theory is developed for this MM-MFG setup.
Abstract: Mean field game (MFG) theory where there is a major agent and many minor agents (MM-MFG) has been formulated for both the linear quadratic Gaussian (LQG) case and for the case of nonlinear state dynamics and nonlinear cost functions. In this framework, even asymptotically (as the population size $N$ approaches infinity), and in contrast to the situation without major agents, the mean field term becomes stochastic due to the stochastic evolution of the state of the major agent; furthermore, the best response control actions of the minor agents depend on the state of the major agent as well as the stochastic mean field. In a decentralized environment, one is led to consider the situation where the agents are provided only with partial information on the major agent's state; in this work such a scenario is considered for systems with nonlinear dynamics and cost functions, and an $\epsilon$-Nash MFG theory is developed for this MM-MFG setup. The approach to the problem of partially observed MM-MFG systems ado...
TL;DR: The purpose of this paper is to derive filters for an arbitrary open quantum system driven by a light wave packet prepared in a continuous-mode multiphoton state.
Abstract: The purpose of this paper is to derive filters for an arbitrary open quantum system driven by a light wave packet prepared in a continuous-mode multiphoton state. A continuous-mode multiphoton stat...
TL;DR: This paper presents a computational framework for the numerical solution of the uncertain optimal control problem, wherein an independently drawn random sample is taken from the space of uncertain parameters, and the expectation in the objective functional is approximated by a sample average.
Abstract: In this paper, we introduce the uncertain optimal control problem of determining a control that minimizes the expectation of an objective functional for a system with parameter uncertainty in both dynamics and objective. We present a computational framework for the numerical solution of this problem, wherein an independently drawn random sample is taken from the space of uncertain parameters, and the expectation in the objective functional is approximated by a sample average. The result is a sequence of approximating standard optimal control problems that can be solved using existing techniques. To analyze the performance of this computational framework, we develop necessary conditions for both the original and approximate problems and show that the approximation based on sample averages is consistent in the sense of Polak [Optimization: Algorithms and Consistent Approximations, Springer, New York, 1997]. This property guarantees that accumulation points of a sequence of global minimizers (stationary poin...
TL;DR: In this paper, a new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility notion given in Definition 4 in [E. Fornasini and M. Valcher, IEEE Trans. Automat. Control, 58 (2013), pp. 1390--1401].
Abstract: A new concept of weighted pair graphs (WPGs) is proposed to represent a new reconstructibility definition for Boolean control networks (BCNs), which is a generalization of the reconstructibility definition given in Definition 4 in [E. Fornasini and M. Valcher, IEEE Trans. Automat. Control, 58 (2013), pp. 1390--1401]. Based on the WPG representation, an effective algorithm for determining the new reconstructibility notion for BCNs is designed with the help of the theories of finite automata and formal languages. We prove that a BCN is not reconstructible iff its WPG has a complete subgraph. In addition, we prove that a BCN is reconstructible in the sense of Definition 4 in [E. Fornasini and M. Valcher, IEEE Trans. Automat. Control, 58 (2013), pp. 1390--1401] iff its WPG has no cycles, which is simpler to check than the condition in Theorem 4 in [E. Fornasini and M. Valcher, IEEE Trans. Automat. Control, 58 (2013), pp. 1390--1401].
TL;DR: In this paper, the authors studied controllability of linear systems on Lie groups by taking into account the eigenvalues of an associated derivation, i.e., the reachable set of the neutral element is open and the derivation has only pure imaginary eigen values.
Abstract: Linear systems on Lie groups are a natural generalization of linear systems on Euclidian spaces. For such systems, this paper studies controllability by taking into consideration the eigenvalues of an associated derivation ${\mathcal{D}}$. When the state space is a solvable connected Lie group, controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation ${\mathcal{D}}$ has only pure imaginary eigenvalues. For bounded systems on nilpotent Lie groups such conditions are also necessary.
TL;DR: By using Lyapunov's direct method, it is proved that the controller proposed ensures exponential stability of the equilibrium and is applied to stabilize the Brockett integrator.
Abstract: This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.
TL;DR: The uniqueness in law for b depending on the mean-field, and the proof of the existence of a weak solution, relatively short in comparison with other work on SDEs, are proved.
Abstract: This work is devoted to the study of stochastic differential equations (SDEs) whose diffusion coefficient $\sigma(s,X_{\cdot\wedge s})$ is Lipschitz continuous with respect to the path of the solution process $X$, while its drift coefficient $b(s,X_{\cdot\wedge s},Q_{X_s})$ is only measurable with respect to $X$ and depends continuously (in the sense of the 1-Wasserstein metric) on the law of the solution process. Embedded in a mean-field game, the weak existence for such SDEs with mean-field term was recently studied by Lacker [Stochastic Process. Appl., 125 (2015), pp. 2856--2894] and Carmona and Lacker [Ann. Appl. Probab., 25 (2015), pp. 1189--1231] under only sequential continuity of $b(s,X_{\cdot\wedge s},Q_{X_s})$ in $Q_{X}$ with respect to a weak topology, but for uniqueness, Carmona and Lacker supposed that $b$ is independent of the mean-field term. We prove the uniqueness in law for $b$ depending on the mean-field, and the proof of the existence of a weak solution, relatively short in comparison ...
Journal Article•
TL;DR: In this paper, Babuska and Lipton introduced local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov.
Abstract: In this paper we introduce local approximation spaces for component-based static condensation (sc) procedures that are optimal in the sense of Kolmogorov. To facilitate simulations for large structures such as aircraft or ships, it is crucial to decrease the number of degrees of freedom on the interfaces, or “ports,” in order to reduce the size of the statically condensed system. To derive optimal port spaces we consider a (compact) transfer operator that acts on the space of harmonic extensions on a two-component system and maps the traces on the ports that lie on the boundary of these components to the trace of the shared port. Solving the eigenproblem for the composition of the transfer operator and its adjoint yields the optimal space. For a related work in the context of the generalized finite element method, we refer the reader to [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406]. We further introduce a spectral greedy algorithm to generalize the procedure to the parameter...
TL;DR: Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state and the optimal control problem is used to solve an inverse source problem.
Abstract: Measure valued optimal control problems governed by the linear wave equation are analyzed. The space of vector measures $\mathcal{M}(\Omega_c,L^2(I))$ is chosen as control space and the corresponding total variation norm as the control cost functional. The support of the controls (sparsity pattern) is time-independent, which is desired in many applications, e.g., inverse problems or optimal actuator placement. New regularity results for the linear wave equation are proven and used to show the well-posedness of the control problem in all three space dimensions. Furthermore first order optimality conditions are derived and structural properties of the optimal control are investigated. Higher regularity of optimal controls in time is shown on the basis of the regularity results for the state. Finally the optimal control problem is used to solve an inverse source problem.
TL;DR: The authors showed that the linear system formed from the pair of Wigner random matrices (W,b) is controllable for a large class of deterministic vectors (W and b) under the Littlewood-Offord theory.
Abstract: It is conjectured by Godsil [Anna. Comb., 16 (2012), pp. 733--744] that the relative number of controllable graphs compared to the total number of simple graphs on $n$ vertices approaches one as $n$ tends to infinity. We prove that this conjecture is true. More generally, our methods show that the linear system formed from the pair $(W,b)$ is controllable for a large class of Wigner random matrices $W$ and deterministic vectors $b$. The proof relies on recent advances in Littlewood--Offord theory developed by Rudelson and Vershynin [Geom. Funct. Anal., to appear; at Adv. Math., 218 (2008), 600--633] and [R. Vershynin, Random Structures Algorithms, 44 (2014), 135--182].
TL;DR: The standard hypothesis that there exists a stabilizing continuous-time state feedback for delay-free nonlinear systems is introduced and the well-known approximation scheme based on first order splines is used.
Abstract: It is standard worldwide, in both industrial and academic activities, to design in continuous time a stabilizing state feedback for a nonlinear system and then implement it by digital devices using sampling and zero-order holding. It is well known, from practice, that stability is rather preserved if sampling is performed at suitable high frequency. This fact has been proved in the literature, also from a theoretical point of view, for delay-free nonlinear systems. An analogous result has not been proved, from the theoretical point of view, for nonlinear retarded systems. Moreover, in the case of retarded systems, implementation by means of digital devices often requires some further approximation due to nonavailability in the buffer of the value of the system variables at some past times. In order to cope with this problem, we make use here of the well-known approximation scheme based on first order splines. We introduce the standard hypothesis that there exists a stabilizing continuous-time state feedba...
TL;DR: Under some analyticity assumptions on the corresponding kernel, it is shown that the equations are controllable under compactness-uniqueness arguments in a suitable functional setting.
Abstract: In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.
TL;DR: A theoretical framework for analyzing the evolutionary dynamics on complex networks and derive some fundamental principles of consensus state selection is developed.
Abstract: Evolutionary dynamics has been widely used to characterize the evolution and formation of behavioral consensus. Governed by evolutionary dynamics, a network of agents reaches consensus at a selected state of all mutants or all residents. Of special interest is the question of how agents select the global consensus state through local state updating. This paper aims at establishing a link between local state updating and global consensus state selection. We develop a theoretical framework for analyzing the evolutionary dynamics on complex networks and derive some fundamental principles of consensus state selection. More specifically, if the probability that an agent adopts a mutant in one-step updating is monotonically increasing with the fitness of the mutant, monotonically increasing with the mutant set, and submodular or supermodular with the mutant set, then the probability that the network of agents converges to the all-mutant state is monotonically increasing with the fitness of the mutant, monotonic...