Optimal control of the primitive equations of the ocean with state constraints

We investigate in this article Pontryagin’s maximum principle for a class of control problems associated with the primitive equations (PEs) of the ocean. These optimal problems involve a state constraint similar to that considered in Wang (2002) [7] for the three-dimensional Navier-Stokes (NS) equations. The main difference between this work and Wang (2002) [7] is that the nonlinearity in the PEs is stronger than in the three-dimensional NS systems.     

Novel delay-dependent asymptotic stability criteria for neural networks with time-varying delays

The problem of delay-dependent asymptotic stability criteria for neural networks (NNs) with time-varying delays is investigated. An improved linear matrix inequality based on delay-dependent stability test is introduced to ensure a large upper bound for time-delay. A new class of Lyapunov function is constructed to derive a novel delay-dependent stability criteria. Finally, numerical examples are given to indicate significant improvement over some existing results.     

Computation of multiple Lie derivatives by algorithmic differentiation

Lie derivatives are often used in nonlinear control and system theory. In general, these Lie derivatives are computed symbolically using computer algebra software. Although this approach is well-suited for small and medium-size problems, it is difficult to apply this technique to very complicated systems. We suggest an alternative method to compute the values of iterated and mixed Lie derivatives by algorithmic differentiation.     

New filter design for linear time-delay systems

This paper develops new robust delay-dependent filter design for a class of linear systems with time-varying delays and convex-bounded parameter uncertainties. The design procedure hinges upon the constructive use of an appropriate Lyapunov functional plus a free-weighting matrices in order to exhibit the delay-dependent dynamics. The developed approach utilizes smaller number of LMI decision variables thereby leading to less conservative solutions to the delay-dependent stability and filtering problems. Subsequently, linear matrix inequalities (LMIs)-based conditions are characterized such that the linear delay system is robustly asymptotically stable with an γ-level L₂-gain. All the developed results are tested on representative examples.     

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincaré from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.Supported by grants from the Ames Research Center of NASA and the Applied Mathematics and Systems Research Programs of the National Science Foundation     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

Positive real control for 2-D discrete delayed systems via output feedback controllers

This paper considers the problem of positive real control for two-dimensional (2-D) discrete delayed systems in the Fornasini–Marchesini second local state-space model. Attention is focused on the design of dynamic output feedback controllers, which guarantee that the closed-loop system is asymptotically stable and the closed-loop transfer function is extended strictly positive real. We first present a sufficient condition for extended strictly positive realness of 2-D discrete delayed systems. Based on this, a sufficient condition for the solvability of the positive real control problem is obtained in terms of a linear matrix inequality (LMI). When the LMI is feasible, an explicit parametrization of a desired output feedback controller is presented. Finally, we provide a numerical example to demonstrate the application of the proposed method.     

On strong regular stabilizability for linear neutral type systems

The problem of strong stabilizability of linear systems of neutral type is investigated. We are interested in the case when the system has an infinite sequence of eigenvalues with vanishing real parts. This is the case when the main part of the neutral equation is not assumed to be stable in the classical sense. We discuss the notion of regular strong stabilizability and present an approach to stabilize the system by regular linear controls. The method covers the case of multivariable control and is essentially based on the idea of infinite-dimensional pole assignment proposed in [G.M. Sklyar, A.V. Rezounenko, A theorem on the strong asymptotic stability and determination of stabilizing controls, C. R. Acad. Sci. Paris Ser. I Math. 333 (8) (2001) 807-812]. Our approach is based on the recent results on the Riesz basis of invariant finite-dimensional subspaces and strong stability for neutral type systems presented in [R. Rabah, G.M. Sklyar, A.V. Rezounenko, Stability analysis of neutral type systems in Hilbert space, J. Differential Equations 214 (2) (2005) 391-428].     

Chaos synchronization between two novel different hyperchaotic systems with unknown parameters

This work is devoted to investigating the synchronization between two novel different hyperchaotic systems with fully unknown parameters, i.e., an uncertain hyperchaotic Lorenz system and an uncertain hyperchaotic Lü system. Based on the Lyapunov stability theory, a new adaptive controller with parameter update law is designed to synchronize these two hyperchaotic systems asymptotically and globally. Numerical simulations are presented to verify the effectiveness of the synchronization scheme.     

Infinite horizon disturbance attenuation for discrete-time systems. A Popov-Yakubovich approach

It is proved that for the discrete-time linear systems with time-varying coefficients the existence of a controller which simultaneously stabilizes and provides prescribed disturbance attenuation for the resultant closed-loop system, implies the existence of global solutions to several Kalman-Szegö-Popov-Yakubovich systems. It is also proved that this fact is equivalent to the existence of the positive semidefinite stabilizing solutions to corresponding game-theoretic Riccati equations. The family of all controllers with the above mentioned properties is constructed in terms of the solutions to the cited Kalman-Szegö-Popov-Yakubovich systems. The main tool is the generalized Popov-Yakubovich theory which is essentially developed in an operator-theoretic framework.     

The spectral factorization problem for multivariable distributed parameter systems

This paper studies the solution of the spectral factorization problem for multivariable distributed parameter systems with an impulse response having an infinite number of delayed impulses. A coercivity criterion for the existence of an invertible spectral factor is given for the cases that the delays are a) arbitrary (not necessarily commensurate) and b) equally spaced (commensurate); for the latter case the criterion is applied to a system consisting of two parallel transmission lines without distortion. In all cases, it is essentially shown that, under the given criterion, the spectral density matrix has a spectral factor whenever this is true for its singular atomic part, i.e. its series of delayed impulses (with almost periodic symbol). Finally, a small-gain type sufficient condition is studied for the existence of spectral factors with arbitrary delays. The latter condition is meaningful from the system theoretic point of view, since it guarantees feedback stability robustness with respect to small delays in the feedback loop. Moreover its proof contains constructive elements.     

Exact controllability of infinite dimensional systems persists under small perturbations

This paper studies the concept of controllability for infinite-dimensional linear control systems in Banach spaces. First, we prove that the set of admissible control operators for the semigroup generator is unchanged if we perturb the generator by the Desch–Schappacher perturbations. Second we show that exact controllability is not changed by such perturbations.     

New blow-up rates for fast controls of Schrödinger and heat equations

We consider the null-controllability problem for the Schrödinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates (see [L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations 204 (2004) 202-226; L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal. 172 (2004) 429-456; L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal. 218 (2005) 425-444; L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772]) on the norm of the operator associating to any initial state the minimal norm control driving the system to zero. Our main results concern the Schrödinger and heat equations in one space dimension. They yield new estimates concerning window problems for series of exponentials as described in [T.I. Seidman, The coefficient map for certain exponential sums, Nederl. Akad. Wetensch. Indag. Math. 48 (1986) 463-478] and in [T.I. Seidman, S.A. Avdonin, S.A. Ivanov, The “window problem” for series of complex exponentials, J. Fourier Anal. Appl. 6 (2000) 233-254]. These results are used, following [L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim. 45 (2006) 762-772], to deal with the case of several space dimensions.     

Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems. Research was supported by the Binational Science Foundation grant #2002038     

Infinite dimensional time varying systems with nonlinear output feedback

This paper deals with time-varying systems on Banach-spaces with unbounded input and output operators and considers nonlinear dynamical perturbations of the output feedback type. We develop an analysis which would establish existence conditions for the perturbed equations and investigate the robustness of stability for this wide class of perturbations.     

Approximate controllability of a class of semilinear degenerate systems with boundary control

This paper concerns a class of control systems governed by semilinear degenerate equations with boundary control in one-dimensional space. The control is proposed on the ‘degenerate’ part of the boundary. The control systems are shown to be approximately controllable by Kakutani's fixed point theorem.     

VARIATIONAL SYSTEMS DISCONJUGACY CRITERIA FOR WNSELFADJOINT DIFFERENTIAL EQUATIONS

Abstract

A systems representation for fourth order differential equations is used to develop variational criteria for the existence and nonexistence of systems conjugate points. The novelty of the results is due to the fact that techniques usually restricted to selfadjoint equations are extended to certain non-self adjoint problems.     

Parameter-dependent robust stability of uncertain time-delay systems

This paper presents a new delay-dependent and parameter-dependent robust stability criterion for linear continuous-time systems with polytopic parameter uncertainties and time-varying delay in the state. This criterion, expressed as a set of linear matrix inequalities, requires no matrix variable to be fixed for the entire uncertainty polytope, which produces a less conservative stability test result. Numerical examples are given to show the effectiveness of the proposed techniques.