On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

q-Difference linear control systems

We propose q-difference versions of some basic results of complete controllability and observability of linear systems such as criterions of controllability and observability, controllability and observability canonical forms, the q-duality theorem, the interconnection between these concepts and that of polynomials without common zeros. As an appendix, is given a simple but computationally meaningful Maple Procedure for the controllability criterion for time constant systems.     

Observability of hybrid discrete-continuous systems

We study the statement and solvability of observability problems in linear stationary hybrid discrete-continuous dynamical systems. Necessary and sufficient observability conditions expressed directly via the system parameters are derived. We consider linear observability problems and the dual controllability and reachability problems. The problem of computing the minimum number of inputs for which the system has a given observability is discussed. An example illustrating the results is presented.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

On controllability of systems of the form\dot x=A(t)x+g(t,u)

In an earlier paper, the author established a sufficient condition for controllability of systems of the form =A(t)x+g(t, u). This condition is a growth condition which generalizes the concept of an asymptotically proper system introduced by LaSalle for linear systems. The purpose of this paper is examine and apply this growth condition. We first show that the condition is also necessary for controllability. Then, we use these results to consider the controllability of perturbations of the above system. The main result of the paper is a class of systems which in many applications can be assumed to be controllable.During the writing of this paper, the author held a Junior Faculty Summer Fellowship from the Research Council of the University of Nebraska.     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

Controllability of Nonlinear Evolution Systems with Preassigned Responses

In this paper, we study the approximate controllability with preassigned responses of the nonlinear delay systems x(t)=A(t)x(t)+f(t, x(t), x((t)), u(t)) and L(x(t), x(t))=A(t)x(t)+f(t, x(t), x((t)), u(t)). The controllability is not governed by an associated linear system, but by conditions on f or A involving the domain of A(t). No compactness assumptions are imposed in the main results.     

On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.     

On asymptotic summation of potentially oscillatory difference systems

A new technique for the asymptotic summation of linear systems of difference equations Y(t+1)=(D(t)+R(t))Y(t) is derived. A fundamental solution Y(t)=Φ(t)(I+P(t)) is constructed in terms of a product of two matrix functions. The first function Φ(t) is a product of the diagonal part D(t). The second matrix I+P(t), is a perturbation of the identity matrix I. Conditions are given on the matrix D(t)+R(t) that allow us to represent I+P(t) as an absolutely convergent resolvent series without imposing stringent conditions on R(t). Our method could be applied to discretized version of singularly perturbed differential equations Y^′(t)=A(t)Y(t) that fit the setting of quantum mechanics.     

Upper bounds of a stabilizer order for linear finite-dimensional systems

The problem of stabilizing linear dynamic systems by a stabilizer (a dynamic system) is considered. The upper bounds of a stabilizer order obtained using two Hidenori Kimura results are studied. The bound k ₀ is shown to be better than the bounds k ₁ and k ₂ only in one case. In addition, all possible relations between three bounds k ₀, k ₁, and k ₂ are proven to be realized in the space of parameters of observability and controllability indices, i.e., there is a dynamic system with the respective observability and controllability indices.     

Multitime Controllability,Observability and Bang-Bang Principle

Control problems for multitime first-order PDE arise in many different contexts and ways. The obstruction of complete integrability conditions (path independent curvilinear integrals) has determined the mathematicians to study such problems only in the discrete context, though thus they loose the geometrical character which is proper to the continuous approach. In this paper, we study controllability, observability and bang-bang properties of multitime completely integrable autonomous linear PDE systems, overcoming the existent mathematical prejudices regarding the importance of a multitime evolution of m-flow type. Our geometrical arguments show that each basic theorem has a correspondent in the case of a single-time linear controlled ODE system. The main results include controllability criteria, equivalence between controllability of a PDE system and observability of the dual PDE system, geometry of the control set, extremality and multitime bang-bang principle. All of these show that the passing from controlled single-time evolution (1-flow) to the controlled multitime evolution (m-flow) is not trivial. Changing the geometrical language, the case of nonholonomic evolution can be recovered easily from our theory.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

The Nehari problem for nonexponentially stable systems

The Nehari problem and its suboptimal extension are solved under the assumption that the system (A, B, C) has bounded controllability and observability maps, an L₂-impulse response and a transfer matrix that is bounded and holomorphic on the right half-plane. Exponential stability of the semigroup is not assumed and the Hankel operator is not compact. The new contribution is an explicit parameterization of all solutions given in terms of the system parametersA, B, C.     

A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations

We introduce a generalized upper and lower solutions method for the solvability of first-order ordinary differential equations u′(t) = ƒ(t, u(t)), u(0) = u(1) in order to cover the case when the function ƒ satisfies Carathéodory conditions. This method is then applied to get multiplicity results when the nonlinearity ƒ interacts with the real eigenvalue of the linearized problem. Our proofs are based on differential inequalities and classical Leray-Schauder degree.     

Criteria for controllability and observability in complex domain models of stationary systems

We propose criteria for controllability and observability of stationary systems specified by complex domain models. Conclusions concerning controllability and observability are made on the basis of coefficients in the corresponding transfer functions. Translated fromDinamicheskie Sistemy, Vol. 12, pp. 120–126, 1993.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Hyperbolic-like solutions for singular Hamiltonian systems

We study the existence of unbounded solutions of singular Hamiltonian systems: where is a potential with a singularity. For a class of singular potentials with a strong force , we show the existence of at least one hyperbolic-like solutions. More precisely, for given and , we find a solution q(t) of (*) satisfying Received October 1998     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.