Banach State Space Research Articles

Stabilization of bilinear systems with distributed delays using the Banach state space decomposition method

Stabilization of bilinear systems with distributed delays using the Banach state space decomposition method

Asymptotic behavior of a retrial queueing system with server breakdowns

In this paper, we study the asymptotic behavior of an M/G/1 retrial queueing system with server breakdowns, which is described by infinitely many partial integro-differential equations. Through investigating the spectrum of the system operator associated with the system on the imaginary axis, we show that the time-dependent solution of the system is strongly stable in the natural Banach state space. When the server failure rate is equal to zero, we show that the system admits a unique positive time-dependent solution and the solution is strongly convergent to its steady-state solution. In addition, when the service completion rate of server is a constant, the spectrum of the system operator lies on the left real axis. Finally, the corresponding C0-semigroup generated by the system operator is uniformly exponentially stable, irreducible, uniformly mean ergodic, quasi-compact but not compact and not eventually compact.

Strong and weak-star stabilisation of bilinear systems on nonreflexive Banach spaces

This paper deals with bilinear systems evolving in a Banach state space (which is not necessarily reflexive). To the best of our knowledge, strong and weak-star stabilisation of control systems in nonreflexive state spaces has not been studied yet. This work is a first attempt towards providing sufficient conditions that guarantee the weak-star stabilisation of the system at hand. The strong stabilisation is stated as well. The approach uses results relying on convex analysis and some properties of the duality mapping and linear semigroup theory. Applications to reaction–diffusion and transport equations are further presented.

Weak and strong stabilisation of bilinear systems in a Banach space

ABSTRACTIn this paper, we study the feedback stabilisation of homogeneous bilinear systems on a Banach state-space. Sufficient conditions for weak and strong stabilisation are given. A decay rate of the stabilised state is explicitly estimated. Applications to heat and transport equations are provided.

On Control Lyapunov–Razumikhin Functions, Nonconstant Delays, Nonsmooth Feedbacks, and Nonlinear Sampled-Data Stabilization

A methodology for the design of stabilizers in the sample-and-hold sense for nonlinear retarded systems is provided. The methodology is based on control Lyapunov–Razumikhin functions. Fully nonlinear retarded systems with an arbitrary number of arbitrary time-varying time-delays of discrete and distributed type are covered by the theory developed here, as long as the map describing the dynamics is uniformly in time Lipschitz on bounded sets of the Banach state space and the Euclidean input space. The standard assumption that all involved time-delay signals are globally Lipschitz is introduced. It is assumed, for the system at hand, that there exist a control Lyapunov–Razumikhin function and a suitably defined induced steepest descent state feedback. Moreover, such state feedback has to satisfy a suitably weak Lipschitz property, uniformly in time and in any bounded subset of the Banach state space. Such weak Lipschitz property has the advantage to allow for discontinuities in the feedback, which arise, for instance, in sliding-mode control methodologies. The following fact is shown: the above steepest descent state feedback is a stabilizer in the sample-and-hold sense, that is, such feedback, if applied by suitably fast sampling and holding, guarantees practical semiglobal stability of the closed-loop system, with arbitrary small final target ball of the origin. The problem of nonavailability in the buffer device of past values of the system internal variables is solved by spline approximation methods, even in the case discontinuities arise in the state feedback.

Radius of approximate controllability of linear retarded systems under structured perturbations

In this paper we shall deal with the problem of calculation of the radius of approximate controllability in the Banach state space Kn×L2([−hk,0],Kn) for linear retarded systems of the form ẋ(t)=A0x(t)+A1x(t−h1)+⋯+Akx(t−hk)+Bu(t). By using multi-valued linear operators we are able to derive computable formulas for this radius when the system’s coefficient matrices are subjected to structured perturbations. Some examples are provided to illustrate the obtained results.

Asymptotic stability of time-varying distributed parameter semi-linear systems

The asymptotic behaviour is studied for a class of non-linear distributed parameter time-varying dissipative systems. This is achieved by using time-varying infinite-dimensional Banach state space description. Stability criteria are established, which are based on the dissipativity of the system in addition to another technical condition. The general development is applied to semi-linear systems with time varying nonlinearity. Stability criteria are extracted from the previous conditions. These theoretical results are applied to a class of transport-reaction processes. Different types of nonlinearity are studied by adapting the criteria given in the early portions of the paper.

A Pontryagin Maximum Principle for Infinite-Dimensional Problems

A basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable “needle-like control variations.” We use this idea to prove the main result of the present paper—a Pontryagin maximum principle for infinite-dimensional optimal control problems with pointwise terminal constraints in arbitrary Banach state space. By refining the classical variational technique we are able to replace the differentiability of the norm of the state space (guaranteed by the strict convexity of its dual norm, which is assumed in the known results) by a separation argument. We also drop another key assumption which is common in the existing literature on infinite-dimensional control problems—that the set of variations (in the state space) of the state trajectory's endpoint (resulting from the control variations) be finite-codimensional. Instead, we require only that it has nonempty interior in its closed affine hull. As an application of the abstract result we present an illustrative example—an optimal control problem for an age-structured system with pointwise terminal state constraints.

On asymptotic stability of semi-linear distributed parameter dissipative systems

The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description. Stability criteria are established, which are based on a weaker technical condition of the m -dissipativity. These theoretical results are applied to a class of transport–reaction processes. Different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper.

Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor

The concept of asymptotic stability is studied for a class of infinite-dimensional semilinear Banach state space (distributed parameter) systems. Asymptotic stability criteria are established, which are based on the concept of strictly m-dissipative operator. These theoretical results are applied to a nonisothermal plug flow tubular reactor model, which is described by semilinear partial differential equations, derived from mass and energy balances. In particular it is shown that, under suitable conditions on the model parameters, some equilibrium profiles are asymptotically stable equilibriums of such model.

Asymptotic stability of a nonisothermal plug flow reactor infinite-dimensional model

The asymptotic stability property is studied for a nonisothermal plug flow tubular reactor model, which is described by semi-linear partial differential equations (PDE's) derived from mass and energy balance principles. It is reported that, under some condition on the model parameters, any constant temperature equilibrium profile is an asymptotically stable equilibrium of such model. The analysis is based on an asymptotic stability criterion for a class of infinitedimensional (distributed parameter) semi-linear Banach state space systems and the concept of strictly m-dissipative operator.

Nonsmooth control problems in banach state space

Necessary conditionc are proved for optimal solutiotls in deterministic control Prlroblems where the state space is a general Banach space, and where the right hand side of the ordinary differential equation is Lipschitz continuous with respect to the stat. Directional derivatives In an ordinary or extended sense are tools used.

Constrained retarded nonlinear optimal control problems in Banach state space

Necessary conditions are proved for the optimal control of solutions of ordinary and retarded differential equations in a Banach state space, with mixed and pure state restrictions. The treatment includes the possibility of point targets. A generalization of earlier results for finite-dimensional or Hilbert state spaces is obtained.

Constrained Retarded Nonlinear Optimal Control Problems in Banach State Space

Constrained Retarded Nonlinear Optimal Control Problems in Banach State Space

Realization theory of infinite-dimensional linear systems. Part II

This paper studies the real-time behavior of constant linear systems. A function space Λ is introduced to give a precise language to discuss the working mode of systems. It is shown that a realization of a constant linear input/output map produces the desired outputs during the application of inputs. A differential equation description is derived for those systems whose weighting patterns are sufficiently smooth. The notion of topological observability in bounded time yields a necessary and sufficient condition under which the canonical realization of a constant linear input/output map has a Banach state space.