Showing papers in "Siam Journal on Control and Optimization in 2004"
TL;DR: By inductive arguments employing the entropy power inequality of information theory, and a new quantizer error bound, an explicit expression for the infimum stabilizing data rate is derived, under very mild conditions on the initial state and noise probability distributions.
Abstract: Feedback control with limited data rates is an emerging area which incorporates ideas from both control and information theory. A fundamental question it poses is how low the closed-loop data rate can be made before a given dynamical system is impossible to stabilize by any coding and control law. Analogously to source coding, this defines the smallest error-free data rate sufficient to achieve "reliable" control, and explicit expressions for it have been derived for linear time-invariant systems without disturbances. In this paper, the more general case of finite-dimensional linear systems with process and observation noise is considered, the object being mean square state stability. By inductive arguments employing the entropy power inequality of information theory, and a new quantizer error bound, an explicit expression for the infimum stabilizing data rate is derived, under very mild conditions on the initial state and noise probability distributions.
740 citations
TL;DR: The controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts, and are illustrated via application to a friction servo-motor.
Abstract: Stability analysis is developed for uncertain nonlinear switched systems. While being asymptotically stable and homogeneous of degree q < 0, these systems are shown to approach the equilibrium point in finite time. Restricted to second order systems, this feature is additionally demonstrated to persist regardless of inhomogeneous perturbations. Based on this fundamental property, switched control algorithms are then developed to globally stabilize uncertain minimum phase systems of uniform m-vector relative degree (2,...,2)T. The controllers constructed do not rely on the generation of sliding motions while providing robustness features similar to those possessed by their sliding mode counterparts. The proposed synthesis procedure is illustrated via application to a friction servo-motor.
533 citations
TL;DR: Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state by proving that these approximations have convergence order h2.
Abstract: An optimal control problem for a two-dimensional (2-d) elliptic equation is investigated with pointwise control constraints. This paper is concerned with discretization of the control by piecewise constant functions. The state and the adjoint state are discretized by linear finite elements. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h2.
268 citations
TL;DR: It is shown how control approaches already presented in the literature can be unified, and a new control methodology is presented and discussed that relies on the generalization to infinite dimensions of the concept of structural invariant and on the extension to distributed systems of the so-called control by interconnection methodology.
Abstract: The purpose of this paper is to show how the Timoshenko beam can be fruitfully described within the framework of distributed port Hamiltonian (dpH) systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system. After the dpH model of the beam is introduced, the control problem is discussed. In particular, it is shown how control approaches already presented in the literature can be unified, and a new control methodology is presented and discussed. This control methodology relies on the generalization to infinite dimensions of the concept of structural invariant (Casimir function) and on the extension to distributed systems of the so-called control by interconnection methodology. In this way, finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy, i.e., by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced.
199 citations
TL;DR: It is proved that it is possible to move from any steady-state to any other by means of a boundary control, provided that both are in the same connected component of the set of steady-states.
Abstract: We investigate the problem of exact boundary controllability of semilinear one-dimensional heat equations. We prove that it is possible to move from any steady-state to any other by means of a boundary control, provided that both are in the same connected component of the set of steady-states. The proof is based on an effective feedback stabilization procedure, which is implemented.
168 citations
TL;DR: An approximation technique involving auxiliary finite-horizon optimal control problems is described and used to prove new versions of the Pontryagin maximum principle.
Abstract: This paper suggests some further developments in the theory of first-order necessary optimality conditions for problems of optimal control with infinite time horizons. We describe an approximation technique involving auxiliary finite-horizon optimal control problems and use it to prove new versions of the Pontryagin maximum principle. Special attention is paid to the behavior of the adjoint variables and the Hamiltonian. Typical cases, in which standard transversality conditions hold at infinity, are described. Several significant earlier results are generalized.
120 citations
TL;DR: This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics and relates the traditional market-selected volatility risk premium approach and the preference-based valuation techniques.
Abstract: This paper is a contribution to the valuation of derivative securities in a stochastic volatility framework, which is a central problem in financial mathematics. The derivatives to be priced are of European type with the payoff depending on both the stock and the volatility. The valuation approach uses utility-based criteria under the assumption of exponential risk preferences. This methodology yields the indifference prices as solutions to second order quasilinear PDEs. Two sets of price bounds are derived that highlight the important ingredients of the utility approach, namely, nonlinear pricing rules with dynamic certainty equivalent characteristics, and pricing measures depending on correlation and the Sharpe ratio of the traded asset. The problem is further analyzed by asymptotic methods in the limit of the volatility being a fast mean-reverting process. The analysis relates the traditional market-selected volatility risk premium approach and the preference-based valuation techniques.
93 citations
TL;DR: An asymptotic expansion of a shape function with respect to the insertion of a small hole or obstacle inside a domain is obtained for the Stokes equations with general shape functions and arbitrarily shaped holes.
Abstract: The topological sensitivity analysis provides an asymptotic expansion of a shape function with respect to the insertion of a small hole or obstacle inside a domain. This expansion can then be used for shape optimization. In this paper, such an expansion is obtained for the Stokes equations with general shape functions and arbitrarily shaped holes. A numerical example illustrates the use of the topological sensitivity in a shape optimization problem.
92 citations
TL;DR: Existence and characterization of Nash equilibrium payoffs are proved for stochastic nonzero-sum differential games.
Abstract: Existence and characterization of Nash equilibrium payoffs are proved for stochastic nonzero-sum differential games
87 citations
TL;DR: These results provide a set of separation principles for input-output-to-state stability---characterizations of the property as conjunctions of weaker stability notions yield analogous results.
Abstract: We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS (input-to-state stability) framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability---characterizations of the property as conjunctions of weaker stability notions. When applied to the notion of integral ISS, these characterizations yield analogous results.
86 citations
TL;DR: A portfolio optimization problem which is formulated as a stochastic control problem, the subsolution-supersolution method is used to obtain existence of solutions of the DPE, and the solutions are used to derive the optimal investment and consumption policies.
Abstract: We consider a portfolio optimization problem which is formulated as a stochastic control problem. Risky asset prices obey a logarithmic Brownian motion, and interest rates vary according to an ergodic Markov diffusion process. The goal is to choose optimal investment and consumption policies to maximize the infinite horizon expected discounted hyperbolic absolute risk aversion (HARA) utility of consumption. A dynamic programming principle is used to derive the dynamic programming equation (DPE). The subsolution-supersolution method is used to obtain existence of solutions of the DPE. The solutions are then used to derive the optimal investment and consumption policies.
TL;DR: Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space.
Abstract: Conditions are given for the viability and the weak convergence of an inexact, relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a Hilbert space. In turn, new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming.
TL;DR: The primal-dual active set method has proved to be an efficient numerical tool in the context of diverse applications as mentioned in this paper, and it has been used for nonlinear problems with bilateral constraints.
Abstract: The primal-dual active set method has proved to be an efficient numerical tool in the context of diverse applications. So far it has been investigated mainly for linear problems. This paper is devoted to the study of global convergence of the primal-dual active set method for nonlinear problems with bilateral constraints. Utilizing the close relationship between the primal-dual active set method and semismooth Newton methods, local superlinear convergence of the method is investigated as well.
TL;DR: In this article, the authors investigated the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions and derived sufficient conditions for dual convergence for a convex program with positivity and equality constraints.
Abstract: In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the introduction of Bregman-type distances. Then, the general evolution problem is introduced, and global convergence is established under quasi-convexity conditions, with interesting refinements in the case of convex minimization. Some explicit examples of these gradient flows are discussed. Dual trajectories are identified, and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints. Some convergence rate results are established. In the case of a linear objective function, several optimality characterizations of the orbits are given: optimal path of viscosi...
TL;DR: Two conjectures which were posed in 1991 and 1994 are shown not to hold and a generator of the form A_e on a Hilbert space such that $(sI -A_e)$ is uniformly left-invertible, but its semigroup does not have this property.
Abstract: This paper concerns systems of the form $\dot{x}(t) = Ax(t)$, $y(t) = Cx(t)$, where $A$ generates a $C_0$-semigroup. Two conjectures which were posed in 1991 and 1994 are shown not to hold. The first conjecture (by G. Weiss) states that if the range of $C$ is one-dimensional, then $C$ is admissible if and only if a certain resolvent estimate holds. The second conjecture (by D. Russell and G. Weiss) states that a system is exactly observable if and only if a test similar to the Hautus test for finite-dimensional systems holds. The $C_0$-semigroup in both counterexamples is analytic and possesses a basis of eigenfunctions. Using the $(A, C)$-pair from the second counterexample, we construct a generator $A_e$ on a Hilbert space such that $(sI -A_e)$ is uniformly left-invertible, but its semigroup does not have this property.
TL;DR: The stability problem of the Wonham filter with respect to initial conditions is addressed, and new bounds for the exponential stability rates, which do not depend on the observations are given.
Abstract: The stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure.
TL;DR: In this paper, it was shown that if the informed player also controls the transitions, the game has a value, whereas if the uninformed player controls the transition, the maxmin value as well as the min-max value exist, but they may differ.
Abstract: We study stochastic games with incomplete information on one side, in which the transition is controlled by one of the players. We prove that if the informed player also controls the transitions, the game has a value, whereas if the uninformed player controls the transitions, the max-min value as well as the min-max value exist, but they may differ. We discuss the structure of the optimal strategies, and provide extensions to the case of incom- plete information on both sides.
TL;DR: A regularization of the optimization problems that is proposed in order to handle generic data, and a barrier-like term is introduced to the optimization problem to achieve strictly minimum phase filters for estimated covariance and cepstral data.
Abstract: Methods for determining ARMA(n,m) filters from covariance and cepstral estimates are proposed. In [C. I. Byrnes, P. Enqvist, and A. Lindquist, SIAM J. Control Optim., 41 (2002), pp. 23--59], we have shown that an ARMA(n,m) model determines and is uniquely determined by a window r0,r1,...,rn of covariance lags and c1,c2,...,cn of cepstral lags. This unique model can be determined from a convex optimization problem which was shown to be the dual of a maximum entropy problem. In this paper, generalizations of this problem are analyzed. Problems with covariance lags r0,r1,...,rn and cepstral lags c1,c2,...,cm of different lengths are considered, and by considering different combinations of covariances, cepstral parameters, poles, and zeros, it is shown that only zeros and covariances give a parameterization that is consistent with generic data.
However, the main contribution of this paper is a regularization of the optimization problems that is proposed in order to handle generic data. For the covariance and cepstral problem, if the data does not correspond to a system of desired order, solutions with zeros on the boundary occur and the cepstral coefficients are not interpolated exactly. In order to achieve strictly minimum phase filters for estimated covariance and cepstral data, a barrier-like term is introduced to the optimization problem. This term is chosen so that convexity is maintained and so that the unique solution will still interpolate the covariances but only approximate the cepstral lags. Furthermore, the solution will depend analytically on the covariance and cepstral data, which provides robustness, and the barrier term increases the entropy of the solution.
TL;DR: A representation of the limit occupational measures set of a control system in terms of the vector function defining the system's dynamics is established and applications in averaging of singularly perturbed control systems are demonstrated.
Abstract: A representation of the limit occupational measures set of a control system in terms of the vector function defining the system's dynamics is established. Applications in averaging of singularly perturbed control systems are demonstrated.
TL;DR: The existence of a Nash equilibrium feedback is established for a two-player nonzero-sum stochastic differential game with discontinuous feedback by studying a parabolic system strongly coupled by discontinuous terms.
Abstract: The existence of a Nash equilibrium feedback is established for a two-player nonzero-sum stochastic differential game with discontinuous feedback. This is obtained by studying a parabolic system strongly coupled by discontinuous terms.
TL;DR: A numerical method for homogenization of Hamilton-Jacobi equations is presented and implemented as an L∞calculus of variations problem and it is shown to be convergent and error estimates are provided.
Abstract: A numerical method for homogenization of Hamilton--Jacobi equations is presented and implemented as an $L^\infty$ calculus of variations problem. Solutions are found by solving a nonlinear convex optimization problem. The numerical method is shown to be convergent, and error estimates are provided. One and two dimensional examples are worked in detail, comparing known results with the numerical ones and computing new examples. The cases of nonstrictly convex Hamiltonians and Hamiltonians for which the cell problem has no solution are treated.
TL;DR: The controllability problem for a system that models the vibrations of a controlled tree-shaped network of vibrating elastic strings is studied and a weighted observability inequality with weights that may be explicitly computed in terms of the eigenvalues of the associated elliptic problem is proved.
Abstract: In this paper we study the controllability problem for a system that models the vibrations of a controlled tree-shaped network of vibrating elastic strings. The control acts through one of the exterior nodes of the network. With the help of the dAlembert representation formula for the solutions of the one-dimensional wave equation, we find certain linear relations between the traces of the solutions at the nodes of the network. These relations allow us to prove a weighted observability inequality with weights that may be explicitly computed in terms of the eigenvalues of the associated elliptic problem. We characterize the class of trees for which all those weights are different from zero, which leads to the spectral controllability of the system. Additionally, we consider the same one-node control problem for several networks that are controlled simultaneously.
TL;DR: This problem is shown to be equivalent to an auxiliary control problem defined as a combination of an optimal stopping problem and a classical control problem, and it is proved that the value functions for these two problems are equal.
Abstract: In this paper, we study an optimal singular stochastic control problem. By using a time transformation, this problem is shown to be equivalent to an auxiliary control problem defined as a combination of an optimal stopping problem and a classical control problem. For this auxiliary control problem, the controller must choose a stopping time (optimal stopping), and the new control variables belong to a compact set. This equivalence is obtained by showing that the (discontinuous) state process governed by a singular control is given by a time transformation of an auxiliary state process governed by a classical bounded control. It is proved that the value functions for these two problems are equal. For a general form of the cost, the existence of an optimal singular control is established under certain technical hypotheses. Moreover, the problem of approximating singular optimal control by absolutely continuous controls is discussed in the same class of admissible controls.
TL;DR: This model admits a relatively simple solution, under which thevalue of the perpetual convertible bond, as a function of the value of the underlying firm, is determined by a nonlinear ordinary differential equation.
Abstract: A firm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The firm may at any time call the bond. Because calls and conversions often occur far from maturity, it is not unreasonable to model this situation with a perpetual convertible bond, i.e., a convertible coupon-paying bond without maturity. This model admits a relatively simple solution, under which the value of the perpetual convertible bond, as a function of the value of the underlying firm, is determined by a nonlinear ordinary differential equation.
TL;DR: The active control of sound is analyzed in the framework of the mathematical theory of optimal control and the existence of a unique solution and a finite element approximation when the source term is a Dirac delta measure.
Abstract: The active control of sound is analyzed in the framework of the mathematical theory of optimal control. After setting the problem in the frequency domain, we deal with the state equation, which is a Helmholtz partial differential equation. We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure. Two optimization problems are successively considered. The first one concerns the choice of phases and amplitudes of the actuators to minimize the noise at the sensors' location. The second one consists of determining the optimal actuators' placement. Both problems are then numerically solved. Error estimates are settled and numerical results for some tests are reported.
TL;DR: Focusing on the multidimensional wave equation with a nonlinear term, new necessary optimality conditions are derived in the form of a pointwise Pontryagin maximum principle for the state-constrained problem under consideration.
Abstract: We consider optimal control problems for hyperbolic equations with controls in Neumann boundary conditions with pointwise constraints on the control and state functions. Focusing on the multidimensional wave equation with a nonlinear term, we derive new necessary optimality conditions in the form of a pointwise Pontryagin maximum principle for the state-constrained problem under consideration. Our approach is based on modern methods of variational analysis that allow us to obtain refined necessary optimality conditions with no convexity assumptions on integrands in the minimizing cost functional.
TL;DR: When the frequency of oscillation tends to infinity it is proved that the controls converge to an approximate control for the same heat equation but on a manifold of dimension $k+1$ that is independent of time.
Abstract: We consider the linear heat equation with Dirichlet boundary conditions in a bounded domain of $\R^n$, $n\geq 1$, and with a control acting on a lower-dimensional time-dependent manifold of dimension $k\leq n-1$. We analyze the approximate controllability problem. This problem is equivalent to a suitable uniqueness or unique continuation property of solutions of the heat equation without control. More precisely, it consists of proving that the unique solution of the Dirichlet problem vanishing on the time-dependent manifold is identically zero. This uniqueness problem, however, does not fit in the class of classical Cauchy problems and therefore, the existing tools based on power series expansions, Carleman inequalities, and doubling properties do not seem to apply. We give sufficient conditions on the time-dependent manifold for this uniqueness property to hold. The techniques we employ combine the Fourier series representation and the time analyticity of solutions and allow us to reduce the problem to a uniqueness question for the eigenfunctions of the Laplacian. We then apply well-known results on the nodal sets of these eigenfunctions.
We also analyze the asymptotic behavior of the control when the time-oscillation of the manifold supporting the control increases. When the frequency of oscillation tends to infinity we prove that the controls converge to an approximate control for the same heat equation but on a manifold of dimension $k+1$ that is independent of time. This is done under suitable time-periodicity assumptions on the original manifold and confirms the fact that increasing time-oscillations of the support of the control increases the efficiency of the control mechanism.
TL;DR: The problem of determining a strategy that is efficient in the sense that it minimizes the expectation of a convex loss function of the hedging error for the case when prices change at discrete random points in time according to a geometric Poisson process is considered.
Abstract: We consider the problem of determining a strategy that is efficient in the sense that it minimizes the expectation of a convex loss function of the hedging error for the case when prices change at discrete random points in time according to a geometric Poisson process. The intensities of the jump process need not be fully known by the investor. The solution algorithm is based on dynamic programming for piecewise deterministic control problems, and its implementation is discussed as well.
TL;DR: This paper deals with optimal control problems for dynamical systems governed by constrained functional-differential inclusions of neutral type, and develops the method of discrete approximations and employs advanced tools of generalized differentiation to derive necessary optimality conditions for general optimal control issues.
Abstract: This paper deals with optimal control problems for dynamical systems governed by constrained functional-differential inclusions of neutral type. Such control systems contain time delays not only in state variables but also in velocity variables, which make them essentially more complicated than delay-differential (or differential-difference) inclusions. Our main goal is to derive necessary optimality conditions for general optimal control problems governed by neutral functional-differential inclusions with endpoint constraints. While some results are available for smooth control systems governed by neutral functional-differential equations, we are not familiar with any results for neutral functional-differential inclusions, even with smooth cost functionals in the absence of endpoint constraints. Developing the method of discrete approximations (which is certainly of independent interest) and employing advanced tools of generalized differentiation, we conduct a variational analysis of neutral functional-d...
TL;DR: A control approach for the practical and asymptotic stabilization of nonlinear driftless systems subjected to additive perturbations is proposed, and a control solution is proposed for the class of the chained systems.
Abstract: A control approach for the practical and asymptotic stabilization of nonlinear driftless systems subjected to additive perturbations is proposed. Such perturbations arise naturally, for instance, in the modeling of trajectory stabilization problems for controllable driftless systems on Lie groups. The objective of the approach is to provide practical stability of an arbitrary given point in the state space, whatever the perturbations, and asymptotic stability (resp., convergence to the point) when the perturbations are absent (resp., tend to zero). A general framework is presented in this paper, and a control solution is proposed for the class of the chained systems.