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Showing papers in "Siam Journal on Control and Optimization in 2013"


Journal ArticleDOI
TL;DR: This paper studies synchronization via pinning control on general complex dynamical networks, such as strongly connected networks, networks with a directed spanning tree, weaklyconnected networks, and directed forests, finding that the strongly connected components with very few connections from other components should be controlled and the components with many connections fromother components can achieve synchronization even without controls.
Abstract: This paper studies synchronization via pinning control on general complex dynamical networks, such as strongly connected networks, networks with a directed spanning tree, weakly connected networks, and directed forests. A criterion for ensuring network synchronization on strongly connected networks is given. It is found that the vertices with very small in-degrees should be pinned first. In addition, it is shown that the original condition with controllers can be reformulated such that it does not depend on the form of the chosen controllers, which implies that the vertices with very large out-degrees may be pinned. Then, a criterion for achieving synchronization on networks with a directed spanning tree, which can be composed of many strongly connected components, is derived. It is found that the strongly connected components with very few connections from other components should be controlled and the components with many connections from other components can achieve synchronization even without controls...

310 citations


Journal ArticleDOI
TL;DR: Global exponential synchronization stability in an array of linearly diffusively coupled reaction-diffusion neural networks with time-varying delays is investigated by adding impulsive controller to a small fraction of nodes (pinning-impulsive controller).
Abstract: In this paper, global exponential synchronization stability in an array of linearly diffusively coupled reaction-diffusion neural networks with time-varying delays is investigated by adding impulsive controller to a small fraction of nodes (pinning-impulsive controller). In order to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes, a new analysis method is developed. By using the developed method, two known lemmas on stability of delayed functional differential equation with and without impulses, and Lyapunov stability theory, several novel and easily verified synchronization criteria guaranteeing the whole network will be pinned to a homogenous solution are derived. Moreover, the effects of the pinning-impulsive controller and the dynamics of the uncontrolled nodes and the diffusive couplings on the synchronization process are explicitly expressed in the obtained criteria. Our results also show that we can always de...

303 citations


Journal ArticleDOI
TL;DR: It is proved that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide a solution and existence and regularity of the corresponding value function are proved.
Abstract: The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions. We implement the Mean-Field Game strategy developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in the states and the controls, and the costs are convex, our assumptions on the nature of the dependence of all the coefficients upon the statistical distribution of the states of the individual players remains of a rather general nature. Our probabilistic approach calls for the solution of systems of forward-backward stochastic differential equations of a McKean--Vlasov type for which no existence result is known, and for which we prove existence and regularity of the corresponding value function. Finally, we prove that a solution of the Mean-Field Game problem as formulated by Lasry and Lions, does indeed provide appr...

295 citations


Journal ArticleDOI
TL;DR: Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients using a variational method and two Riccati differential equations are obtained which are uniquely solvable under certain conditions.
Abstract: Linear-quadratic optimal control problems are considered for mean-field stochastic differential equations with deterministic coefficients. By a variational method, the optimality system is derived, which is a linear mean-field forward-backward stochastic differential equation. Using a decoupling technique, two Riccati differential equations are obtained which are uniquely solvable under certain conditions. Then a feedback representation is obtained for the optimal control.

289 citations


Journal ArticleDOI
TL;DR: In this paper, the global and semiglobal convergence of ADRC for a class of multi-input multi-output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance is proved.
Abstract: In this paper, the global and semiglobal convergence of the nonlinear active distur- bance rejection control (ADRC) for a class of multi-input multi-output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved. As a result, a class of linear systems with external disturbance that can be dealt with by the ADRC is classified. The ADRC is then compared both analytically and numerically to the well-known in- ternal model principle. A number of illustrative examples are presented to show the efficiency and advantage of the ADRC in dealing with unknown dynamics and in achieving fast tracking with lower overstriking.

237 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of boundary stabilization for a quasilinear first-order hyperbolic PDE with actuation on only one end of the domain.
Abstract: In this work, we consider the problem of boundary stabilization for a quasilinear $2\times2$ system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves $H^2$ exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type $4\times4$ system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.

231 citations


Journal ArticleDOI
TL;DR: It is proved that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition and two small-gain theorems are proved that provide a construction of an ISS Lyap unov function for an interconnection of impulsive Systems if the ISS LyAPunov functions for subsystems are known.
Abstract: We prove that impulsive systems, which possess an input-to-state stable (ISS) Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems if the ISS Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via the linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.

219 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs, which allows for state variables associated to the edges and formalizes the interconnection of networks.
Abstract: In this paper we present a unifying geometric and compositional framework for modeling complex physical network dynamics as port-Hamiltonian systems on open graphs. The basic idea is to associate with the incidence matrix of any directed graph a Dirac structure relating the flow and effort variables associated to the edges and vertices of the graph, and to formulate energy-storing or energy-dissipating relations between the flow and effort variables of the edges and the internal vertices. This allows for state variables associated to the edges and formalizes the interconnection of networks. Examples from different origins, such as consensus algorithms, that share the same structure are shown. It is shown how the identified Hamiltonian structure offers systematic tools for the analysis and control of the resulting dynamics.

185 citations


Journal ArticleDOI
TL;DR: This paper studies large population dynamic games involving nonlinear stochastic dynamical systems with agents of the following mixed types, finding that even asymptotically (as the population size $N$ approaches infinity) the noise process of the major agent causes random fluctuation of the mean field behavior of the minor agents.
Abstract: This paper studies large population dynamic games involving nonlinear stochastic dynamical systems with agents of the following mixed types: (i) a major agent and (ii) a population of $N$ minor agents where $N$ is very large. The major and minor agents are coupled via both (i) their individual nonlinear stochastic dynamics and (ii) their individual finite time horizon nonlinear cost functions. This problem is analyzed by the so-called $\epsilon$-Nash mean field game theory. A distinct feature of the mixed agent mean field game problem is that even asymptotically (as the population size $N$ approaches infinity) the noise process of the major agent causes random fluctuation of the mean field behavior of the minor agents. To deal with this, the overall asymptotic ($N \rightarrow \infty$) mean field game problem is decomposed into (i) two nonstandard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the ...

171 citations


Journal ArticleDOI
TL;DR: Two distributed learning algorithms are presented where each sensor only remembers its own utility values and actions played during the last plays and are proven to be convergent in probability to the set of Nash equilibria and global optima of a certain coverage performance metric.
Abstract: Inspired by current challenges in data-intensive and energy-limited sensor networks, we formulate a coverage optimization problem for mobile sensors as a (constrained) repeated multiplayer game. Each sensor tries to optimize its own coverage while minimizing the processing/energy cost. The sensors are subject to the informational restriction that the environmental distribution function is unknown a priori. We present two distributed learning algorithms where each sensor only remembers its own utility values and actions played during the last plays. These algorithms are proven to be convergent in probability to the set of (constrained) Nash equilibria and global optima of a certain coverage performance metric, respectively. Numerical examples are provided to verify the performance of our proposed algorithms.

156 citations


Journal ArticleDOI
TL;DR: This work develops a BCN model for the genetic switch controlling the lambda phage development upon infection of a bacteria and suggests that this biological switch is designed in a way that guarantees minimal time respon...
Abstract: Boolean networks (BNs) are discrete-time dynamical systems with Boolean state variables. BNs have recently been attracting considerable interest as computational models for biological systems and, in particular, gene regulating networks. Boolean control networks (BCNs) are Boolean networks with Boolean inputs. We consider the problem of steering a BCN from a given state to a desired state in minimal time. Using the algebraic state-space representation (ASSR) of BCNs we derive several necessary conditions, stated in the form of maximum principles (MPs), for a control to be time-optimal. In the ASSR every state and input vector is a canonical vector. Using this special structure yields an explicit state-feedback formula for all time-optimal controls. To demonstrate the theoretical results, we develop a BCN model for the genetic switch controlling the lambda phage development upon infection of a bacteria. Our results suggest that this biological switch is designed in a way that guarantees minimal time respon...

Journal ArticleDOI
TL;DR: The long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling is studied and an exponential convergence to the Solution of the associated stationary ergodic mean game is shown.
Abstract: We study the long time average, as the time horizon tends to infinity, of the solution of a mean field game system with a nonlocal coupling. We show an exponential convergence to the solution of the associated stationary ergodic mean field game. Proofs rely on semiconcavity estimates and smoothing properties of the linearized system.

Journal ArticleDOI
TL;DR: The results answer the question on how much interaction is required for a multiagent network to converge despite a certain amount of input disturbance.
Abstract: This paper investigates consensus problems for continuous-time multiagent systems with time-varying communication graphs subject to input noise. Based on input-to-state stability and integral input-to-state stability, robust consensus and integral robust consensus are defined with respect to L∞ and L1 norms of the noise function, respectively. Sufficient and/or necessary connectivity conditions are obtained for the system to reach robust consensus or integral robust consensus under mild assumptions. The results answer the question on how much interaction is required for a multiagent network to converge despite a certain amount of input disturbance. The � -convergence time is obtained for the consensus computation on directed and K-bidirectional graphs.

Journal ArticleDOI
TL;DR: This paper concentrates on the consensus problem of a class of DTSO MASs and develops a method to cope with the corresponding IPGSM.
Abstract: Traditionally, the consensus of a discrete-time multiagent system (MAS) with a switching topology is transformed into the convergence problem of the infinite products of stochastic matrices, which can be resolved by using the Wolfowitz theorem. However, such a transformation is very difficult or even impossible for certain MASs, such as discrete-time second-order MASs (DTSO MASs), whose consensus can only be transformed into the convergence problem of the infinite products of general stochastic matrices (IPGSM). These general stochastic matrices are matrices with row sum 1 but their elements are not necessarily nonnegative. Since there does not exist a general theory or an effective technique for dealing with the convergence of IPGSM, establishing the consensus criteria for a DTSO MAS with a switching topology is rather difficult. This paper concentrates on the consensus problem of a class of DTSO MASs and develops a method to cope with the corresponding IPGSM. Moreover, it is pointed out that the method ...

Journal ArticleDOI
TL;DR: This paper analyzes the convergence of optimal control problems for an evolution equation in a finite time-horizon toward the limit steady state ones as T tends to infinity and shows that the optimal controls and states exponentially converge in the transient time.
Abstract: This paper analyzes the convergence of optimal control problems for an evolution equation in a finite time-horizon $[0, T]$ toward the limit steady state ones as $T\to \infty$. We focus on linear problems. We first consider linear time-independent finite-dimensional systems and show that the optimal controls and states exponentially converge in the transient time (as $T$ tends to infinity) to the ones of the corresponding steady state model. For this to occur suitable observability assumptions need to be imposed. We then extend the results to infinite-dimensional systems including the linear heat and wave equations with distributed controls.

Journal ArticleDOI
TL;DR: This work develops, for the first time, sampled-data distributed distribution of a class of parabolic systems governed by one-dimensional semilinear transport reaction equations with additive disturbances, and suggests a sampled- data controller design, where the sampling intervals in time and in space are bounded.
Abstract: We develop, for the first time, sampled-data distributed $H_{\infty}$ control of a class of parabolic systems. These systems are governed by one-dimensional semilinear transport reaction equations with additive disturbances. A network of stationary sensing devices provides spatially averaged state measurements over the $N$ sampling spatial intervals. We suggest a sampled-data controller design, where the sampling intervals in time and in space are bounded. Our sampled-data static output feedback enters the equation through $N$ shape functions (which are localized in the space) multiplied by the corresponding state measurements. Sufficient conditions for the internal exponential stability and for $L_2$-gain analysis of the closed-loop system are derived via direct Lyapunov method in terms of linear matrix inequalities (LMIs). By solving these LMIs, upper bounds on the sampling intervals that preserve the internal stability and the resulting $L_2$-gain can be found. Numerical examples illustrate the efficie...

Journal ArticleDOI
TL;DR: In this paper, the authors studied the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph and showed that the average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the Markov process is strongly connected.
Abstract: This paper discusses the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph. The switching is determined by a finite state Markov process, each topology corresponding to a state of the process. We address the cases where the dynamics of the agents is expressed both in continuous time and in discrete time. We show that, if the consensus matrices are doubly stochastic, average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the Markov process is strongly connected. The aim of this paper is to show how techniques from the theory of Markovian jump linear systems, in conjunction with results inspired by matrix and graph theory, can be used to prove convergence results for stochastic consensus problems.

Journal ArticleDOI
TL;DR: The same funnel controller is applicable to relative degree one systems, and allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funn...
Abstract: Tracking of reference signals $y_{\mathrm{ref}}(\cdot)$ by the output $y(\cdot)$ of linear (as well as a considerably large class of nonlinear) single-input, single-output systems is considered. The system is assumed to have strict relative degree two with (weakly) stable zero dynamics. The control objective is tracking of the error $e=y-y_{\mathrm{ref}}$ and its derivative $\dot{e}$ within two prespecified performance funnels, respectively. This is achieved by the so-called funnel controller $u(t)=-k_0(t)^2e(t)-k_1(t)\dot{e}(t)$, where the simple proportional error feedback has gain functions $k_0$ and $k_1$ designed in such a way to preclude contact of $e$ and $\dot{e}$ with the funnel boundaries, respectively. The funnel controller also ensures boundedness of all signals. We also show that the same funnel controller (i) is applicable to relative degree one systems, (ii) allows for input constraints provided a feasibility condition (formulated in terms of the system data, the saturation bounds, the funn...

Journal ArticleDOI
TL;DR: For this approximation, the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator is considered and a priori error estimates of optimal order are derived.
Abstract: We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The analytical setting of this problem uses $L^2$ controls and a “very weak” formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation, we derive a priori error estimates of optimal order, which are confirmed by numerical experiments. The proofs employ duality arguments and known results from the $L^p$ error analysis for the finite element Dirichlet projection.

Journal ArticleDOI
TL;DR: A conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates, and although the state is discretized, the control problem can still be formulated and solved in the measure space.
Abstract: Optimal control problems in measure spaces lead to controls that have small support, which is desirable, e.g., in the context of optimal actuator placement. For problems governed by parabolic partial differential equations, well-posedness is guaranteed in the space of square-integrable measure-valued functions, which leads to controls with a spatial sparsity structure. A conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates. In particular, although the state is discretized, the control problem can still be formulated and solved in the measure space. Numerical examples illustrate the structural features of the optimal controls.

Journal ArticleDOI
TL;DR: Several key findings about the PBA are provided, which lead to the main conclusion that the expected absolute residuals of successive search results, i.e., E(|X ∗ − Xn|), converge to 0 at a geometric rate.
Abstract: Bisection search is the most efficient algorithm for locating a unique point X ∗ ∈ (0, 1) when we are able to query an oracle only about whether X ∗ lies to the left or right of a point x of our choosing. We study a noisy version of this classic problem, where the oracle's response is correct only with probability p. The probabilistic bisection algorithm (PBA) introduced by Horstein (IEEE Trans. Inform. Theory, 9 (1963), pp. 136-143) can be used to locate X ∗ in this setting. While the method works extremely well in practice, very little is known about its theoretical properties. In this paper, we provide several key findings about the PBA, which lead to the main conclusion that the expected absolute residuals of successive search results, i.e., E(|X ∗ − Xn|), converge to 0 at a geometric rate.

Journal ArticleDOI
TL;DR: This paper establishes three versions of maximum principle for optimal control derived by forward-backward stochastic systems with correlated noises between the system and the observation and works out two illustrative examples within the frameworks of linear-quadratic control and recursive utility.
Abstract: In this paper, we study a partial information optimal control problem derived by forward-backward stochastic systems with correlated noises between the system and the observation. Utilizing a direct method, an approximation method, and a Malliavin derivative method, we establish three versions of maximum principle (i.e., necessary condition) for optimal control. To show their applications, we work out two illustrative examples within the frameworks of linear-quadratic control and recursive utility and then solve them via the maximum principles and stochastic filtering.

Journal ArticleDOI
TL;DR: This work investigates the convergence of constant step-size gradient descent algorithms for solving the problem of finding the global Riemannian center of mass of a set of data points on a Riem Mannian manifold and states a conjecture which it is argued is the best (in a sense described) convergence condition one can hope for.
Abstract: We study the problem of finding the global Riemannian center of mass of a set of data points on a Riemannian manifold. Specifically, we investigate the convergence of constant step-size gradient de...

Journal ArticleDOI
TL;DR: A generic mixed time-scale stochastic procedure consisting of simultaneous distributed learning and estimation, in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly, and it is shown that the agent estimates are asymptotically efficient.
Abstract: This paper considers the problem of distributed adaptive linear parameter estimation in multiagent inference networks. Local sensing model information is only partially available at the agents, and interagent communication is assumed to be unpredictable. The paper develops a generic mixed time-scale stochastic procedure consisting of simultaneous distributed learning and estimation, in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly. Under rather weak assumptions on the statistical model and the interagent communication, it is shown that, by properly tuning the consensus potential with respect to the innovation potential, the asymptotic information rate loss incurred in the learning process may be made negligible. As such, it is shown that the agent estimates are asymptotically efficient, in that their asymptotic covariance coincides with that of a centralized estimator (the inverse of the centralized Fisher information rate for Gaussian...

Journal ArticleDOI
TL;DR: A duality theory for robust utility maximization is established in this framework and the existence of a probability which is least favorable is proved.
Abstract: In this paper, we provide a framework in which we can set the problem of maximization of utility function, taking into account the model uncertainty and encompassing the case of the uncertain volatility model. The uncertainty is specified by a family of semimartingales laws which is typically nondominated. We establish a duality theory for robust utility maximization in this framework and prove the existence of a probability which is least favorable.

Journal ArticleDOI
TL;DR: It is proved that the null controllability of the wave equation with structural damping on the one-dimensional torus holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.
Abstract: We investigate the internal controllability of the wave equation with structural damping on the one-dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.

Journal ArticleDOI
TL;DR: It is shown that the dynamics admit a splitting into fast and slow subspaces and it is proved an averaging principle for the slow dynamics.
Abstract: We study balanced model reduction for stable bilinear systems in the limit of partly vanishing Hankel singular values. We show that the dynamics admit a splitting into fast and slow subspaces and prove an averaging principle for the slow dynamics. We illustrate our method with an example from stochastic control (density evolution of a dragged Brownian particle) and discuss issues of structure preservation and positivity.

Journal ArticleDOI
TL;DR: An Euler--Newton continuation method for tracking a solution trajectory is introduced and demonstrated to have $l^\infty$ accuracy of order $O(h^4)$, thus generalizing a known error estimate for equations.
Abstract: A finite-dimensional variational inequality parameterized by $t\in [0,1]$ is studied under the assumption that each point of the graph of its generally set-valued solution mapping is a point of strongly regularity. It is shown that there are finitely many Lipschitz continuous functions on $[0,1]$ whose graphs do not intersect each other such that for each value of the parameter the set of values of the solution mapping is the union of the values of these functions. Moreover, the property of strong regularity is uniform with respect to the parameter along any such function graph. An Euler--Newton continuation method for tracking a solution trajectory is introduced and demonstrated to have $l^\infty$ accuracy of order $O(h^4)$, thus generalizing a known error estimate for equations. Two examples of tracking economic equilibrium parametrically illustrate the theoretical results. (A correction is attached.)

Journal ArticleDOI
TL;DR: Possamai et al. as mentioned in this paper considered the classical Merton problem of lifetime consumption-portfolio optimization with small proportional transaction costs, where the first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case.
Abstract: We consider the classical Merton problem of lifetime consumption-portfolio optimization with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from homogenization theory and use convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our companion paper [D. Possamai, H. M. Soner, and N. Touzi, Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case, arXiv:1212.6275v2 [math.AP], preprint, 2012] using the same approach.

Journal ArticleDOI
TL;DR: A new regularity result is proved for the optimal state and the optimal control in an optimal control problem, where the control variable lies in a measure space and the state variable fulfills an elliptic equation.
Abstract: In this paper an optimal control problem is considered, where the control variable lies in a measure space and the state variable fulfills an elliptic equation. This formulation leads to a sparse structure of the optimal control. In this setting we prove a new regularity result for the optimal state and the optimal control. Moreover, a finite element discretization based on [E. Casas, C. Clason, and K. Kunisch, SIAM J. Control Optim., 50 (2012), pp. 1735--1752] is discussed and a priori error estimates are derived, which significantly improve the estimates from that paper. Numerical examples for problems in two and three space dimensions illustrate our results.