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.     

Prediction principles for time-delay system controller synthesis

A brief review is given of the problems of time-delay controller synthesis. The class of time-delay systems known as predictor systems is discussed in detail to exemplify some of the time-delay control theory problems of interest to the applied mathematician. The review is restricted to linear systems, and disturbances are restricted to those which are in principle Laplace transformable.     

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.     

Controllability and asymptotic stabilization of the Camassa-Holm equation

We investigate the problems of exact controllability and asymptotic stabilization of the Camassa-Holm equation on the circle, by means of a distributed control. The results are global, and in particular the control prevents the solution from blowing up.     

Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters

This article is concerned with the modified projective synchronization problem for a class of four-dimensional chaotic system with uncertain parameters. By utilizing Lyapunov method, an adaptive control scheme for the synchronization has been presented. The control performances are verified by a numerical simulation.     

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.     

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.     

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.     

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     

Vanishing of the costate in pontryagin’s maximum principle and singular time optimal controls

We provide examples of time and norm optimal controls that satisfy Pontryagins maximum principle in an interval 0 t T but with a costate that vanishes in 0 t T - with < T. A refinement of this construction produces time optimal controls which do not satisfy the maximum principle, even in weak form. On the positive side, we show that when we drive to zero the costate is nonzero in the whole control interval.     

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.     

The problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures

We obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems. In order to describe the value functions, we employ an infinite dimensional dual formulation instead of using the characterization via Hamilton-Jacobi partial differential equations. In this paper we consider control problems with both infinite and finite horizons. The reflection is given by the normal cone to a proximal retract set.     

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.     

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.     

Reachability problems for a class of integro-differential equations

We study control problems for some integro-differential equations using the Hilbert Uniqueness Method. To do that we follow a harmonic analysis approach. Our results can be applied to concrete examples in viscoelasticity theory.     

Control sets and their boundaries under parameter variation

Control sets, i.e., maximal sets of approximate controllability, are described for systems with parameter-dependent control range. If the so-called inner-pair condition is satisfied, it is known that generically they change continuously under parameter variation while mergers of control sets produce discontinuous transitions. The inner-pair condition is proved for a class of control-affine systems on . Furthermore, continuity results for exit and entrance boundaries of control sets are given both for one control set that changes continuously and for two merging control sets. The results are illustrated by means of the controlled escape equation.     

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].     

The infinite-time admissibility of observation operators and operator Lyapunov equations

Let (A, –, C) be an abstract dynamical system withA being the generator of aC ₀-semigroup on a Hilbert spaceH, C:D(A)Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system (A, B, –), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.This project is supported by the NNSF of China, and the Youth Science and Technique Foundation of Shanxi Province.     

Redheffer's theorem revisited

Based on the Ben Artzi-Gohberg result concerning the equivalence between the invertibility of theL ²-operator and the exponential dichotomic evolution defined byA(t), the time-varying counterpart of the Redheffer theorem is considered under relaxed conditions.