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.     

Stability of ZakianI MN recursions for linear delay differential equations

Long sequences of linear delay differential equations (DDEs) frequently occur in the design of control systems with delays using iterative-numerical methods, such as the method of inequalities. ZakianI _MN recursions for DDEs are suitable for solving this class of problems, since they are reliable and provide results to the desired accuracy, economically even if the systems are stiff. This paper investigates the numerical stability property of theI _MN recursions with respect to Barwell's concept ofP-stability. The result shows that the recursions using full gradeI _MN approximants areP-stable if, and only if,N−2≤M≤N−1.     

Weakly Birkhoff recurrent switching signals, almost sure and partial stability of linear switched dynamical systems

Let {A₁,…,AK}⊂Cd×d be arbitrary K matrices, where K and d both ?2. For any 0<Δ<∞, we denote by the set of all switching sequences u=(λ.,t.):N→{1,…,K}×R₊ satisfying tj−tj−1?Δ and

     

Continuous and discrete characterizations for the uniform exponential stability of linear skew-evolution semiflows

In this paper, we will consider the concept “linear skew-evolution semiflows” and extend theorems of R. Datko, S. Rolewicz, Zabczyk and J.M.A.M van Neerven for this case [15].     

Stability for several type of interval matrices

The robust stability for some types of tlme-varying interval raatrices and nonlineartime-varying interval matrices is considered and some sufficient conditions for robust stability of such interval matrices are given, The main results of this paper are only related to the verticesset of a interval matrices, and therefore, can be easily applied to test robust stability of interval matrices. Finally, some examples are given to illustrate the results.     

Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization

Results of exponential/polynomial decay rates of the energy in L²-level, related to the cubic nonlinear Schrödinger equation with localized damping posed on the whole real line, will be established in this paper. We use Kato's theory and a priori estimates to obtain the result of global well-posedness and we determine exponential/polynomial stabilization combining the ideas of unique continuation due to Zhang, semigroup property and Komornik's approach.     

Stability analysis for Lotka–Volterra systems based on an algorithm of real root isolation

Based on the theory of monotone flows of solutions of systems of differential equations, the Routh–Hurwitz theorem and a real root isolation algorithm of multivariate polynomials are applied to a class of Lotka–Volterra diffusion systems. An algorithm to determine the location of equilibria and the stability of the nonlinear dynamics systems is implemented in Maple.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

Exponential stability for linear skew-product flows

We give general characterizations for uniform exponential stability of linear skew-product flows. We present a unified treatment for discrete and integral conditions for uniform exponential stability. As applications, for the particular case of evolution families, we generalize some results due to Przyluski, Rolewicz and Zabczyk.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Controllability of a quantum particle in a moving potential well

We consider a nonrelativistic charged particle in a 1D moving potential well. This quantum system is subject to a control, which is the acceleration of the well. It is represented by a wave function solution of a Schrödinger equation, the position of the well together with its velocity. We prove the following controllability result for this bilinear control system: given ψ₀ close enough to an eigenstate and ψ_f close enough to another eigenstate, the wave function can be moved exactly from ψ₀ to ψ_f in finite time. Moreover, we can control the position and the velocity of the well. Our proof uses moment theory, a Nash-Moser implicit function theorem, the return method and expansion to the second order.     

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.     

Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Admissible control and observation operators for Volterra integral equations

The admissibility of observation operators and control operators for linear Volterra systems is studied by means of the theory of composition operators on Hardy spaces.Under certain assumptions it is shown to be equivalent to admissibility for the classical Cauchy problem. A duality between control and observation operators is also established,extending known results for the Cauchy problem,which is a special case.     

Partition relations for Hurewicz-type selection hypotheses

We give a general method to reduce Hurewicz-type selection hypotheses into standard ones. The method covers the known results of this kind and gives some new ones.Building on that, we show how to derive Ramsey theoretic characterizations for these selection hypotheses.     

Flow regulation for water quality restoration in a river section: Modeling and control

This work is devoted to the numerical resolution of an optimal control problem that arises in the management of a reservoir for the remediation of a polluted river section. By using mathematical modeling and optimal control techniques we set the mathematical formulation of the problem (as a hyperbolic optimal control problem with control constraints), and obtain a fully discretized problem. Finally, we propose a gradient-free method to solve it, and present realistic numerical results.     

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.     

The converse Lyapunov’s theorem for ODEs of Carathéodory type

The converse problem of Lyapunov stability for systems of ODEs of Caratheodory type is considered. It is proved that if the right hand side of an ODE satisfies only the Osgood condition, the uniform stability of the origin is sufficient to the existence of a locally Lipschitz continuous Lyapunov function. Actually, the uniform stability is equivalent to the robust stability in this case. Moreover, as an auxiliary result, the generalization of the famous Gronwall-Bellman-Bihari inequality is also proved.     

A nonsmooth Lyapunov function and stability for ODE’s of Caratheodory type

The problem of Lyapunov stability for systems of ODE’s of Caratheodory type is considered. It is proved that without the necessity of calculating Dini or Clarke generalized gradients, the locally Lipschitz Lyapunov function can follow different kinds of stability.