Showing papers in "Siam Journal on Control and Optimization in 2006"
TL;DR: This paper considers the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition and proves exponential stability of the solution under suitable assumptions.
Abstract: In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.
492 citations
TL;DR: This paper addresses two strategies for the stabilization of continuous-time, switched linear systems and is designed from the solution of what the authors call Lyapunov-Metzler inequalities from which the stability condition (including chattering) is expressed.
Abstract: This paper addresses two strategies for the stabilization of continuous-time, switched linear systems. The first one is of open loop nature (trajectory independent) and is based on the determination of a minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function to be uniformly decreasing at every switching time. The second one is of closed loop nature (trajectory dependent) and is designed from the solution of what we call Lyapunov-Metzler inequalities from which the stability condition (including chattering) is expressed. Being nonconvex, a more conservative but simpler-to-solve version of the Lyapunov-Metzler inequalities is provided. The theoretical results are illustrated by means of examples.
481 citations
TL;DR: This paper extends the well-known Jacobi and Gauss--Seidel iterations and presents a large family of iterative methods, which are then applied to develop iterative solutions to coupled Sylvester matrix equations and proves that the iterative algorithm always converges to the (unique) solutions for any initial values.
Abstract: In this paper we study coupled matrix equations, which are encountered in many systems and control applications. First, we extend the well-known Jacobi and Gauss--Seidel iterations and present a large family of iterative methods, which are then applied to develop iterative solutions to coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle. Next, we generalize the Sylvester equations to general coupled matrix equations, and propose a gradient-based iterative algorithm for the solutions, using a block-matrix inner product---the star $(\star)$ product; we prove that the iterative algorithm always converges to the (unique) solutions for any initial values. One advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods. Finally, we test the algorithms and show their effectiveness using numerical examples.
353 citations
TL;DR: In this article, a control problem for a class of nonlinear stochastic systems with both state and disturbance-dependent noise was discussed, and the Hamilton-Jacobi equations were developed for infinite and finite-horizon control.
Abstract: This paper discusses the $H_{\infty}$ control problem for a class of nonlinear stochastic systems with both state- and disturbance-dependent noise. By means of Hamilton--Jacobi equations, both infinite and finite horizon nonlinear stochastic $H_\infty$ control designs are developed.
Some results on nonlinear $H_\infty$ control of deterministic systems are generalized to a stochastic setting. We introduce some useful concepts such as "zero-state observability" and "zero-state detectability" which, together with the stochastic LaSalle invariance principle, yield some valuable consequences in infinite horizon nonlinear stochastic $H_\infty$ control.
271 citations
TL;DR: The role played by the observability inequalities in this context and the need of global Carleman estimates are explained and the exact controllability to the trajectories are explained.
Abstract: This paper has been conceived as an overview on the controllability properties of some relevant (linear and nonlinear) parabolic systems. Specifically, we deal with the null controllability and the exact controllability to the trajectories. We try to explain the role played by the observability inequalities in this context and the need of global Carleman estimates. We also recall the main ideas used to overcome the difficulties motivated by nonlinearities. First, we considered the classical heat equation with Dirichlet conditions and distributed controls. Then we analyze recent extensions to other linear and semilinear parabolic systems and/or boundary controls. Finally, we review the controllability properties for the Stokes and Navier-Stokes equations that are known to date. In this context, we have paid special attention to obtaining the necessary Carleman estimates. Some open questions are mentioned throughout the paper. We hope that this unified presentation will be useful for those researchers interested in the field.
200 citations
TL;DR: It is shown how approximation is allowed and the observer is modified in a way which induces a time rescaling and which follows from a forward unboundedness observability property.
Abstract: We state sufficient conditions for the existence, on a given open set, of the extension, to nonlinear systems, of the Luenberger observer as it has been proposed by Kazantzis and Kravaris. We prove it is sufficient to choose the dimension of the system, giving the observer, less than or equal to 2 + twice the dimension of the state to be observed. We show that it is sufficient to know only an approximation of the solution of a PDE, needed for the implementation. We establish a link with high gain observers. Finally we extend our results to systems satisfying an unboundedness observability property.
192 citations
TL;DR: It is proved that the error estimates are of order $O(h^{1 - 1/p}(\Gamma)$ for some $p > 2$, which is consistent with the-regularity of the optimal control.
Abstract: We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in ${\mathbb R}^2$. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order $O(h^{1 - 1/p})$ for some $p > 2$, which is consistent with the $W^{1 - 1/p,p}(\Gamma)$-regularity of the optimal control.
173 citations
TL;DR: In this paper, the authors consider the problem of steering the state of a linear time-varying system to the origin when the control is subject to magnitude constraints and give necessary and sufficient conditions for global constrained controllability.
Abstract: The paper considers the problem of steering the state of a linear time-varying system to the origin when the control is subject to magnitude constraints. Necessary and sufficient conditions are given for global constrained controllability as well as a necessary and sufficient condition for the existence of a control (satisfying the constraints) which steers the system to the origin from a specified initial epoch $(x_0 ,t_0 )$. The global result does not require zero to be an interior point of the control set $\Omega $, and the theorem for constrained controllability at $(x_0 ,t_0 )$ only requires that $\Omega $ be compact, not that it contain zero. The results are compared to those available in the literature. Furthermore, numerical aspects of the problem are discussed as is a technique for determining a steering control.
172 citations
TL;DR: A new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase-field method that yields a large-scale optimization problem with a high number of linear inequality constraints.
Abstract: We introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase-field method In the basic formulation we have a PDE-constrained optimization problem, where the finite element and design analysis are solved simultaneously The starting point of the relaxation is a reformulation of the material problem involving linear and 0-1 constraints only The 0-1 constraints are then relaxed and approximated by a Cahn-Hilliard-type penalty in the objective functional, which yields convergence of minimizers to 0-1 designs as the penalty parameter decreases to zero A major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter The relaxation scheme yields a large-scale optimization problem with a high number of linear inequality constraints We discretize the problem by finite elements and solve the arising finite-dimensional programming problems by a primal-dual interior point method Numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach
168 citations
TL;DR: This paper addresses the design problem of practical output-feedingback controls for stochastic strict-feedback nonlinear systems in observer canonical form with stable zero-dynamics under long-term average tracking risk-sensitive cost criteria by using the integrator backstepping methodology.
Abstract: This paper addresses the design problem of practical (or satisfaction) output-feedback controls for stochastic strict-feedback nonlinear systems in observer canonical form with stable zero-dynamics under long-term average tracking risk-sensitive cost criteria. The cost function adopted here is of the quadratic-integral type usually encountered in practice, rather than the quartic-integral one used to avoid difficulty in control design and performance analysis of the closed-loop system. A sequence of coordinate diffeomorphisms is introduced to separate the zero-dynamics from the entire system, so that the transformed system has an appropriate form suitable for integrator backstepping design. For any given risk-sensitivity parameter and desired cost value, by using the integrator backstepping methodology, an output-feedback control is constructively designed such that (a) the closed-loop system is bounded in probability and (b) the long-term average risk-sensitive cost is upper bounded by the desired value. In addition, this paper does not require the uniform boundedness of the gain functions of the system noise. Furthermore, an example is given to show the effectiveness of the theory.
161 citations
TL;DR: This paper applies some old results on the Wiener--Hopf factorization of Markov processes to a range of option-pricing problems for Markov-modulated models for equity prices and shows how the methodology of Rogers and Stapleton can be used to tackle finite-horizon problems.
Abstract: Markov-modulated models for equity prices have recently been extensively studied in the literature. In this paper, we apply some old results on the Wiener--Hopf factorization of Markov processes to a range of option-pricing problems for such models. The first example is the perpetual American put, where the exact (numerical) solution is obtained without discretizing any PDE. We then show how the methodology of Rogers and Stapleton [Finance Stoch., 2 (1997), pp. 3-17] can be used to tackle finite-horizon problems and illustrate the methodology by pricing European, American, single barrier, and double barrier options under Markov-modulated dynamics.
TL;DR: This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures.
Abstract: This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching.
TL;DR: The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua.
Abstract: This paper is concerned with the internal stabilization of the generalized Korteweg--de Vries equation on a bounded domain. The global well-posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval [1,4). The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua [Quart. Appl. Math., 60 (2002), pp. 111-129].
TL;DR: An extension of the velocity of the underlying Hamilton-Jacobi equation is proposed, endowed with a Hilbertian structure based on the H1 Sobolev space, for structural optimization by the level-set method.
Abstract: In the context of structural optimization by the level-set method, we propose an extension of the velocity of the underlying Hamilton-Jacobi equation. The gradient method is endowed with a Hilbertian structure based on the H1 Sobolev space. Numerical results for compliance minimization and mechanism design show a strong improvement of the rate of convergence of the level-set method. Another important application is the optimization of multiple eigenvalues.
TL;DR: These results enable us to present solutions to feedback stabilization problems for systems with delayed input by means of time-varying distributed delay feedback.
Abstract: The paper contains certain results concerning the finite-time global stabilization for triangular control systems described by retarded functional differential equations by means of time-varying distributed delay feedback. These results enable us to present solutions to feedback stabilization problems for systems with delayed input. The results are obtained by using the backstepping technique.
TL;DR: The value function of a finite horizon stochastic control problem with unbounded controls is characterized as the unique viscosity solution of the corresponding dynamic programming equation.
Abstract: In this paper, we prove a comparison result between semicontinuous viscosity sub- and supersolutions growing at most quadratically of second-order degenerate parabolic Hamilton--Jacobi--Bellman and Isaacs equations. As an application, we characterize the value function of a finite horizon stochastic control problem with unbounded controls as the unique viscosity solution of the corresponding dynamic programming equation.
TL;DR: The overall algorithmic concept of the primal-dual path-following technique is provided, and numerical tests for state constrained optimal control problems show the efficiency of the new concept.
Abstract: Primal-dual path-following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed. Regularity properties of the path are proved. The path structure allows us to define approximating models, which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. The Moreau-Yosida regularized subproblems of the new path-following technique are solved efficiently by semismooth Newton methods. The overall algorithmic concept is provided, and numerical tests (including a comparison with primal-dual path-following interior point methods) for state constrained optimal control problems show the efficiency of the new concept.
TL;DR: The main tool is the notion of a local solution of backward stochastic differential equations (BSDEs) with two reflecting barriers with an application to the American game options.
Abstract: In this paper we solve the mixed zero-sum stochastic differential game problem in the general case. The main tool is the notion of a local solution of backward stochastic differential equations (BSDEs) with two reflecting barriers. As an application we deal with the American game options.
TL;DR: It is shown that the control problem of CTL* is complete for deterministic double (resp., single) exponential time and a small model theorem for the control of C TL* is obtained.
Abstract: The supervisory control problem of discrete event systems with temporal logic specifications is studied. The full branching time logic of CTL* is used for expressing specifications of discrete event systems. The control problem of CTL* is reduced to the decision problem of CTL*. A small model theorem for the control of CTL* is obtained. It is shown that the control problem of CTL* (resp., CTL) is complete for deterministic double (resp., single) exponential time. A sound and complete supervisor synthesis algorithm for the control of CTL* is provided. Special cases of the control of computation tree logic (CTL) and linear-time temporal logic are also studied.
TL;DR: The degree of that common polynomial Lyapunov function is not uniformly bounded over all the UAS systems, and a generalization to a class of piecewise-polynomial LyAPunov functions is given.
Abstract: In this paper, we consider linear switched systems $\dot x(t)=A_{u(t)} x(t)$, $x\in\R^n$, $u\in U$, {$\{A_u: u\in U \}$ compact,} and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS). {Given a UAS system, it is always possible to build a common polynomial Lyapunov function. Our main result is that} the degree of that common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.
TL;DR: A stable nonparametric method for constructing an option pricing model of exponential Levy type, consistent with a given data set of option prices is proposed, and it is proved the existence of solutions for the regularized problem and shown that they converge to the minimum entropy least squares solution of the initial problem when the noise level in the data vanishes.
Abstract: We propose a stable nonparametric method for constructing an option pricing model of exponential Levy type, consistent with a given data set of option prices. After demonstrating the ill-posedness of the usual and least squares version of this inverse problem, we suggest to regularize the calibration problem by reformulating it as the problem of finding an exponential Levy model that minimizes the sum of the pricing error and the relative entropy with respect to a prior exponential Levy model. We prove the existence of solutions for the regularized problem and show that it yields solutions which are continuous with respect to the data, stable with respect to the choice of prior, and which converge to the minimum entropy least squares solution of the initial problem when the noise level in the data vanishes.
TL;DR: Under the notion of path-by-path optimal disturbance attenuation, finite-path-dependent controllers can outperform the usual mode-dependent ones and lead to semidefinite programming-based controller synthesis techniques.
Abstract: An exact condition for uniform stabilization and disturbance attenuation for switched linear systems is given in the discrete-time domain via the union of an increasing family of linear matrix inequality conditions. Associated with each Markovian jump linear system is a switched linear system, so we obtain a necessary and sufficient condition for almost sure uniform stabilization and disturbance attenuation for Markovian jump linear systems as well. The results lead to semidefinite programming-based controller synthesis techniques, from which optimal finite-path-dependent linear dynamic output feedback controllers arise naturally. In particular, under the notion of path-by-path optimal disturbance attenuation, finite-path-dependent controllers can outperform the usual mode-dependent ones.
TL;DR: In this article, a link between the geometric approach to linear system theory, as developed by W. M. Wonham and A. S. Morse, and the approach based on polynomial matrices is laid.
Abstract: Based on the state space model of P. Fuhrmann, a link is laid between the geometric approach to linear system theory, as developed by W. M. Wonham and A. S. Morse, and the approach based on polynomial matrices. In particular polynomial characterizations of $(A,B)$-invariant and reachability subspaces are given. These characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers.
Finally, constructive procedures and conditions are given for computing the supremal $(A,B)$-invariant subspace and reachability space and for checking the solvability of the exact model matching problem.
TL;DR: This paper completely resolve the $T$-time local observability for the bimodal conewise linear system, for finite $T$, and provides numerical examples to illustrate the difficulty associated with the long-time observability.
Abstract: Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the P-property and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: short-time and $T$-time (or finite-time) local/global observability. For the short-time observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their long-time mode-transitional behavior, which is very difficult to predict, only partial results are obtained for the $T$-time observable states. Nevertheless, we completely resolve the $T$-time local observability for the bimodal conewise linear system, for finite $T$, and provide numerical examples to illustrate the difficulty associated with the long-time observability.
TL;DR: This paper presents some global Carleman estimates for suitable adjoint problems of linearized Navier--Stokes and Boussinesq systems and deduces results concerning the local exact controllability to the trajectories of these systems.
Abstract: In this paper we deal with some controllability problems for systems of the Navier--Stokes and Boussinesq kind with distributed controls supported in small sets. Our main aim is to control $N$-dimensional systems ($N+1$ scalar unknowns in the case of the Navier--Stokes equations) with $N-1$ scalar control functions. In a first step, we present some global Carleman estimates for suitable adjoint problems of linearized Navier--Stokes and Boussinesq systems. In this way, we obtain null controllability properties for these systems. Then, we deduce results concerning the local exact controllability to the trajectories. We also present (global) null controllability results for some (truncated) approximations of the Navier--Stokes equations.
TL;DR: In this article, the adaptive observer theory is used in the design of internal models for nonlinear output regulation, which substantially enhances the existing theory of adaptive output regulation by allowing nonlinear internal models and more general classes of controlled plants.
Abstract: This paper shows how the theory of adaptive observers can be effectively used in the design of internal models for nonlinear output regulation. The main result obtained in this way is a new method for the synthesis of adaptive internal models, which substantially enhances the existing theory of adaptive output regulation by allowing nonlinear internal models and more general classes of controlled plants.
TL;DR: The results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$.
Abstract: The null-controllability property of a $1-d$ parabolic equation involving a fractional power of the Laplace operator, $(-\Delta)^\alpha$, is studied. The control is a scalar time-dependent function $g=g(t)$ acting on the system through a given space-profile $f=f(x)$ on the interior of the domain. Thus, the control $g$ determines the intensity of the space control $f$ applied to the system, the latter being given a priori. We show that, if $\alpha\leq 1/2$ and the shape function $f$ is, say, in $L^2$, no initial datum belonging to any Sobolev space of negative order may be driven to zero in any time. This is in contrast with the existing positive results for the case $\alpha >1/2$ and, in particular, for the heat equation that corresponds to $\alpha=1$. This negative result exhibits a new phenomenon that does not arise either for finite-dimensional systems or in the context of the heat equation.
On the contrary, if more regularity of the shape function $f$ is assumed, then we show that there are initial data in any Sobolev space $H^m$ that may be controlled. Once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which, when increasing the regularity of the control profile, the space of controllable data decreases.
These results show that, in order for the control properties of the heat equation to be true, the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when $\alpha=1/2$ for the fractional powers of the Dirichlet Laplacian in $1-d$. The results we present here are, in nature and with respect to techniques of proof, similar to those on the control of the heat equation in unbounded domains in [S. Micu and E. Zuazua, Trans. Amer. Math. Soc., 353 (2000), pp. 1635-1659] and [S. Micu and E. Zuazua, Portugal. Math., 58 (2001), pp. 1-24].
We also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models.
TL;DR: This method proves roughly that the norm of a u steering the system (1) from an initial state $x(0)=x_{0}$ to the final state x(T)=0$ grows at most likeexp(\alpha_{*} L^{2}/T) as the control time T tends to zero.
Abstract: In this paper, the null-controllability in any positive time T of the first-order equation (1) ${\dot{x}}(t)=e^{i\theta}Ax (t)+Bu(t)$ ($|{\theta}| < \pi/2$ fixed) is deduced from the null-controllability in some positive time L of the second-order equation (2) $\ddot{z}(t)=Az(t)+Bv(t)$. The differential equations (1) and (2) are set in a Banach space, B is an admissible unbounded control operator, and A is a generator of cosine operator function. The control transmutation method makes explicit the input function u of (1) in terms of the input function v of (2): $u(t)=\int_{\mathbb{R}} k(t,s)v(s)\, ds $, where the compactly supported kernel k depends only on T and L. This method proves roughly that the norm of a u steering the system (1) from an initial state $x(0)=x_{0}$ to the final state $x(T)=0$ grows at most like $\|{x_{0}}\|\exp(\alpha_{*} L^{2}/T)$ as the control time T tends to zero. (The rate $\alpha_{*}$ is characterized independently by a one-dimensional controllability problem.) In applications to the cost of fast controls for the heat equation, L is roughly the length of the longest ray of geometric optics which does not intersect the control region.
TL;DR: In this paper, necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input-output maps are given, and it is also shown that two minimal realizations are diffeomorphic.
Abstract: Necessary and sufficient conditions for the existence of either analytic or smooth symmetric realizations of nonlinear input-output maps are given. It is also shown that two minimal realizations are diffeomorphic.
TL;DR: An extended version of Hildreth’s iterative quadratic programming algorithm is presented, geometrically interpreted, and proved to produce a sequence of iterates that converges to the solution, and has an important intermediate optimality property.
Abstract: An extended version of Hildreth’s iterative quadratic programming algorithm is presented, geometrically interpreted, and proved to produce a sequence of iterates that (i) converges to the solution, and (ii) has an important intermediate optimality property. This extended Hildreth algorithm is cast into a new form which more pronouncedly brings out its primal-dual nature. The application of the algorithm may be governed by an index sequence which is more general than a cyclic sequence, namely, by an almost cyclic control, and a sequence of relaxation parameters is incorporated without ruining convergence. The algorithm is a row-action method which is particularly suitable for handling large (or huge) and sparse systems.