Stability of a state linear system can be identified by controllability, observability, stabilizability,
detectability, and transfer function. The approximate controllability and observability of non-autonomous
Riesz-spectral systems have been established as well as non-autonomous Sturm-Liouville systems. As a
continuation of the establishments, this paper concern on the analysis of the transfer function, stabilizability,
and detectability of the non-autonomous Riesz-spectral systems. A strongly continuous quasi semigroup
approach is implemented. The results show that the transfer function, stabilizability, and detectability can
be established comprehensively in the non-autonomous Riesz-spectral systems. In particular, sufficient and
necessary conditions for the stabilizability and detectability can be constructed. These results are parallel
with infinite dimensional of autonomous systems.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
The Laplace transform is a mathematical tool that is useful for solving differential equations. It was developed by Pierre-Simon Laplace in the late 18th century. The Laplace transform takes a function of time and transforms it into a function of complex quantities. This transformation allows differential equations to be converted into algebraic equations that are easier to solve. Some common applications of the Laplace transform include modeling problems in semiconductors, wireless networks, vibrations, and electromagnetic fields.
Classical Mechanics of a Three Spin ClusterPaul Houle
A cluster of three spins coupled by single-axis anisotropic exchange exhibits classical behaviors ranging from regular motion at low and high energies, to chaotic motion at intermediate energies. A change of variable, taking advantage of the conserved z-angular momentum, combined with a 3-d graphical presentation (described in the Appendix), produce Poincare sections that manifest all symmetries of the system and clearly illustrate the transition to chaos as energy is increased.
The three-spin system has four families of periodic orbits. Linearization around stationary points predicts orbit periods in the low energy (antiferromagnetic) and high energy (ferromagnetic) limits and also determines the stability properties of certain orbits at a special intermediate energy. The energy surface undergoes interesting changes of topology as energy is varied. We describe the similiarities of our three spin system with the Anisotropic Kepler Problem and the Henon-Heiles Hamiltonian. An appendix discusses numerical integration techniques for spin systems.
Using Laplace Transforms to Solve Differential EquationsGeorge Stevens
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
This document discusses stability analysis of control systems using transfer functions and the Routh-Hurwitz criterion. It begins by defining stability and describing different types of system responses. The key points are:
1) The Routh-Hurwitz criterion can determine stability by analyzing the signs in the first column of a constructed Routh table, with changes in sign indicating right half-plane poles and instability.
2) Special cases like a zero only in the first column or an entire row of zeros require alternative methods like the epsilon method or reversing coefficients.
3) Examples demonstrate applying the Routh-Hurwitz criterion to determine stability for different polynomials, including handling special cases. Exercises also have readers practice stability analysis using
- Isaac Newton published his theory of gravity and laws of motion in 1687 in his famous book Principia, solving the problem of explaining planetary motion.
- Newton developed calculus, which was necessary for analyzing motion, though he published the calculations in Principia using older geometrical methods due to concerns calculus would not be accepted.
- Since Newton, calculus has been essential for understanding physics and its applications in engineering and science.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
Mathematical Formulation of Quantum Mechanics rbmaitri123
This document discusses the mathematical formulation of quantum mechanics. It describes how quantum systems are represented using linear algebra concepts such as Hilbert spaces and operators. Physical states are represented by unit vectors in a Hilbert space. Observables are represented by Hermitian operators whose eigenvalues correspond to possible measurement outcomes. Dynamics are governed by Schrodinger's equation, which describes how states evolve over time.
The document discusses the Laplace transform and its applications. Specifically:
- The Laplace transform was developed by mathematicians including Euler, Lagrange, and Laplace to solve differential equations.
- It transforms a function of time to a function of complex frequencies, allowing differential equations to be written as algebraic equations.
- For a function to have a Laplace transform, it must be at least piecewise continuous and bounded above by an exponential function.
- The Laplace transform can reduce the dimension of partial differential equations and is used in applications including semiconductor mobility, wireless networks, vehicle vibrations, and electromagnetic fields.
The Laplace transform is a mathematical tool that is useful for solving differential equations. It was developed by Pierre-Simon Laplace in the late 18th century. The Laplace transform takes a function of time and transforms it into a function of complex quantities. This transformation allows differential equations to be converted into algebraic equations that are easier to solve. Some common applications of the Laplace transform include modeling problems in semiconductors, wireless networks, vibrations, and electromagnetic fields.
Classical Mechanics of a Three Spin ClusterPaul Houle
A cluster of three spins coupled by single-axis anisotropic exchange exhibits classical behaviors ranging from regular motion at low and high energies, to chaotic motion at intermediate energies. A change of variable, taking advantage of the conserved z-angular momentum, combined with a 3-d graphical presentation (described in the Appendix), produce Poincare sections that manifest all symmetries of the system and clearly illustrate the transition to chaos as energy is increased.
The three-spin system has four families of periodic orbits. Linearization around stationary points predicts orbit periods in the low energy (antiferromagnetic) and high energy (ferromagnetic) limits and also determines the stability properties of certain orbits at a special intermediate energy. The energy surface undergoes interesting changes of topology as energy is varied. We describe the similiarities of our three spin system with the Anisotropic Kepler Problem and the Henon-Heiles Hamiltonian. An appendix discusses numerical integration techniques for spin systems.
Using Laplace Transforms to Solve Differential EquationsGeorge Stevens
This document discusses using Laplace transforms to solve differential equations. It begins with an introduction to Laplace transforms and their inventor Pierre-Simon Laplace. The author then explains that Laplace transforms can be used to transform differential equations into simpler algebraic equations that are easier to solve. As an example, the document walks through using Laplace transforms to solve a first order differential equation. It concludes that Laplace transforms are an important mathematical tool for solving differential equations.
This document discusses stability analysis of control systems using transfer functions and the Routh-Hurwitz criterion. It begins by defining stability and describing different types of system responses. The key points are:
1) The Routh-Hurwitz criterion can determine stability by analyzing the signs in the first column of a constructed Routh table, with changes in sign indicating right half-plane poles and instability.
2) Special cases like a zero only in the first column or an entire row of zeros require alternative methods like the epsilon method or reversing coefficients.
3) Examples demonstrate applying the Routh-Hurwitz criterion to determine stability for different polynomials, including handling special cases. Exercises also have readers practice stability analysis using
- Isaac Newton published his theory of gravity and laws of motion in 1687 in his famous book Principia, solving the problem of explaining planetary motion.
- Newton developed calculus, which was necessary for analyzing motion, though he published the calculations in Principia using older geometrical methods due to concerns calculus would not be accepted.
- Since Newton, calculus has been essential for understanding physics and its applications in engineering and science.
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
Multi-particle Entanglement in Quantum States and EvolutionsMatthew Leifer
Slides from my first ever seminar presentation given when I was a first year Ph.D. student. University of Bristol Applied Mathematics Seminar 2001. Brief introduction to entanglement and discussion of local unitary invariants.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
Postulates of quantum mechanics & operatorsGiri dharan
1) Quantum mechanics describes the state of a micro system using a wavefunction that contains all information about the system.
2) Observables like position, momentum, and energy are represented by quantum mechanical operators.
3) The possible values of physical quantities are the eigenvalues obtained by operating the corresponding operator on the wavefunction. The average value of a physical quantity is the expectation value calculated using the wavefunction.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses the Laplace transform, which takes a function of time and transforms it into a function of complex frequency. This transformation converts differential equations into algebraic equations, simplifying solving problems involving systems. The Laplace transform has many applications in fields like engineering, physics, and astronomy by allowing analysis of linear time-invariant systems through properties like derivatives becoming multiplications in the frequency domain.
Application of Laplace transforms for solving transient equations of electrical circuits. Initial and final value theorems. Unit step, impulse and ramp inputs. Laplace transform for shifted and singular functions.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
This document provides a summary of topics covered in an advanced engineering maths course. It includes definitions and properties of the Laplace transform, formulas for taking the Laplace transform of common functions, applications of Laplace transforms to solve differential equations, and examples of using Laplace transforms to evaluate integrals. The document lists 10 students enrolled in the course and 12 topics to be covered, such as the Laplace transform of derivatives and integrals, convolution theorem, and application to chemical engineering problems.
A STATISTICAL NOTE ON SYSTEM TRANSITION INTO EQUILIBRIUM [03-21-2014]Junfeng Liu
1) The document presents a statistical perspective on how an enclosed system of irregularly moving particles evolves over time to reach an equilibrium state of uniform particle distribution.
2) It introduces the concept of "nature's law of chaotic mixture" to describe this process, where the system's temporal evolution can be modeled as a hierarchy of transition probabilities between spatial regions.
3) Numerical simulations demonstrate that after sufficient time steps allowing for chaotic particle motions, the system does achieve an overall equilibrium state where particle proportions are uniform across all regions, consistent with the proposed statistical framework.
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
The Laplace transform is a linear operator that transforms a function of time (f(t)) to a function of a complex frequency variable (F(s)). It works for functions that are at least piecewise continuous for t ≥ 0 and satisfy a boundedness criterion. The Laplace transform reduces differential equations to algebraic equations, simplifying their solution. It can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs).
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
The document summarizes the four postulates of quantum mechanics:
1) The state of a quantum system is described by a wave function. The wave function contains all information about the system and is the probability of finding the particle in a given region.
2) Every observable in classical mechanics corresponds to a linear operator in quantum mechanics. Linear combinations of degenerate eigenfunctions are also eigenfunctions.
3) The only possible measurable values of an observable are the eigenvalues of its corresponding operator. The eigenfunctions must be well-behaved.
4) If a system is in a state described by a normalized wave function, the average measured value of an observable is given by the integral of the wave function
This chapter provides complete solution of different circuits using Laplace transform method and also provides information about applications of Laplace transforms.
The document introduces second order differential equations and their solutions. It defines an initial value problem for a second order equation as consisting of the equation and two initial conditions. Linear equations are introduced, which can be written in standard form or with constant or variable coefficients. The dynamical system formulation converts a second order equation to a system of first order equations. Undamped and damped free vibrations are discussed. Examples are provided, including finding the solution to an initial value problem, and determining the quasi-frequency, quasi-period, and equilibrium crossing time.
The Laplace transform allows solving differential equations using algebra by transforming differential operators into algebraic operations. It transforms a function of time (f(t)) into a function of a complex variable (F(s)), allowing differential equations describing systems to be solved for variables of interest. Key properties include linearity, time and frequency shifting, and relationships between derivatives, integrals, and the Laplace transform that enable solving differential equations by taking the transform, performing algebra, and applying the inverse transform.
The document provides an overview of topics related to Laplace transforms and their applications. It defines the Laplace transform and discusses some of its key properties, including linearity and how it relates to derivatives, integrals, and shifting theorems. The document also outlines how Laplace transforms can be used to evaluate integrals and solve differential equations. It provides examples of calculating the Laplace transforms of common functions.
The Laplace transform transforms a function of time into a function of complex frequency by taking the integral transform. The inverse Laplace transform takes the complex frequency function and yields the original time function. The Laplace transform expresses a function as a superposition of moments, whereas the Fourier transform uses sinusoids. The Laplace transform provides an alternative functional description that often simplifies analyzing systems. It can be applied to solve switching transients in electric circuits and for load frequency control in power systems. Other applications include transient problems in flow systems and circuit analysis.
Control theorists have extensively studied stability
as well as relative stability of LTI systems. In this research
paper we attempt to answer the question. How unstable is an
unstable system? In that effort we define instability exponents
dealing with the impulse response. Using these instability
exponents, we characterize unstable systems.
1. The document describes the syllabus for the course EE1354 - Modern Control Systems. It includes 5 units that cover topics like state space analysis of continuous and discrete time systems, z-transforms, nonlinear systems, and MIMO systems.
2. Key concepts discussed include state variable representation, eigenvectors and eigenvalues, solution of state equations, controllability and observability, and deriving state space models from transfer functions.
3. Methods like pole placement, state feedback, and observer design for state estimation are also covered in the context of analysis and design of control systems.
This document discusses the inverse Laplace transform, which finds the original function given its Laplace transform. It defines the inverse Laplace transform and proves it is unique. The key points are:
1. The inverse Laplace transform of a function F(s) is the function f(t) whose Laplace transform is F(s).
2. The uniqueness theorem proves there is only one function f(t) that corresponds to a given F(s).
3. The inverse is only defined for t ≥ 0, as the Laplace transform only uses information from the positive t-axis.
Multi-particle Entanglement in Quantum States and EvolutionsMatthew Leifer
Slides from my first ever seminar presentation given when I was a first year Ph.D. student. University of Bristol Applied Mathematics Seminar 2001. Brief introduction to entanglement and discussion of local unitary invariants.
The document discusses multiple topics related to analog control systems, including:
1. Reducing multiple subsystems into a single block to simplify analysis.
2. Describing system response in terms of transient and steady state response.
3. Explaining poles, zeros and how they relate to system response.
4. Defining characteristics of second order systems and analyzing steady state error.
5. Discussing stability analysis in the complex s-plane and conditions for stable, unstable and marginally stable systems.
Postulates of quantum mechanics & operatorsGiri dharan
1) Quantum mechanics describes the state of a micro system using a wavefunction that contains all information about the system.
2) Observables like position, momentum, and energy are represented by quantum mechanical operators.
3) The possible values of physical quantities are the eigenvalues obtained by operating the corresponding operator on the wavefunction. The average value of a physical quantity is the expectation value calculated using the wavefunction.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The document discusses the Laplace transform, which takes a function of time and transforms it into a function of complex frequency. This transformation converts differential equations into algebraic equations, simplifying solving problems involving systems. The Laplace transform has many applications in fields like engineering, physics, and astronomy by allowing analysis of linear time-invariant systems through properties like derivatives becoming multiplications in the frequency domain.
Application of Laplace transforms for solving transient equations of electrical circuits. Initial and final value theorems. Unit step, impulse and ramp inputs. Laplace transform for shifted and singular functions.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
This document provides an introduction to Laplace transforms. It defines the Laplace transform, lists some of its key properties including how it transforms derivatives and functions, and demonstrates how to use Laplace transforms to solve ordinary differential equations (ODEs). The document contains examples of taking Laplace transforms, applying properties like linearity and shifting, performing inverse Laplace transforms using tables and techniques like partial fractions, and solving a sample ODE using Laplace transforms. It also introduces concepts like the step function, Dirac delta function, and convolution as related topics.
This document provides a summary of topics covered in an advanced engineering maths course. It includes definitions and properties of the Laplace transform, formulas for taking the Laplace transform of common functions, applications of Laplace transforms to solve differential equations, and examples of using Laplace transforms to evaluate integrals. The document lists 10 students enrolled in the course and 12 topics to be covered, such as the Laplace transform of derivatives and integrals, convolution theorem, and application to chemical engineering problems.
A STATISTICAL NOTE ON SYSTEM TRANSITION INTO EQUILIBRIUM [03-21-2014]Junfeng Liu
1) The document presents a statistical perspective on how an enclosed system of irregularly moving particles evolves over time to reach an equilibrium state of uniform particle distribution.
2) It introduces the concept of "nature's law of chaotic mixture" to describe this process, where the system's temporal evolution can be modeled as a hierarchy of transition probabilities between spatial regions.
3) Numerical simulations demonstrate that after sufficient time steps allowing for chaotic particle motions, the system does achieve an overall equilibrium state where particle proportions are uniform across all regions, consistent with the proposed statistical framework.
this slide is introduce the postulates of quantum mechanics in which has all important definable objects is defined. so that presentation is helpful for the undergraduate students
The Laplace transform is a linear operator that transforms a function of time (f(t)) to a function of a complex frequency variable (F(s)). It works for functions that are at least piecewise continuous for t ≥ 0 and satisfy a boundedness criterion. The Laplace transform reduces differential equations to algebraic equations, simplifying their solution. It can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs).
The Laplace transform is an integral transform that converts a function of time into a function of complex frequency. It is defined as the integral of the function multiplied by e-st from 0 to infinity. The Laplace transform is used to solve differential equations by converting them to algebraic equations. Some key properties of the Laplace transform include linearity, shifting theorems, differentiation and integration formulas, and methods for periodic and anti-periodic functions.
The document summarizes the four postulates of quantum mechanics:
1) The state of a quantum system is described by a wave function. The wave function contains all information about the system and is the probability of finding the particle in a given region.
2) Every observable in classical mechanics corresponds to a linear operator in quantum mechanics. Linear combinations of degenerate eigenfunctions are also eigenfunctions.
3) The only possible measurable values of an observable are the eigenvalues of its corresponding operator. The eigenfunctions must be well-behaved.
4) If a system is in a state described by a normalized wave function, the average measured value of an observable is given by the integral of the wave function
This chapter provides complete solution of different circuits using Laplace transform method and also provides information about applications of Laplace transforms.
The document introduces second order differential equations and their solutions. It defines an initial value problem for a second order equation as consisting of the equation and two initial conditions. Linear equations are introduced, which can be written in standard form or with constant or variable coefficients. The dynamical system formulation converts a second order equation to a system of first order equations. Undamped and damped free vibrations are discussed. Examples are provided, including finding the solution to an initial value problem, and determining the quasi-frequency, quasi-period, and equilibrium crossing time.
The Laplace transform allows solving differential equations using algebra by transforming differential operators into algebraic operations. It transforms a function of time (f(t)) into a function of a complex variable (F(s)), allowing differential equations describing systems to be solved for variables of interest. Key properties include linearity, time and frequency shifting, and relationships between derivatives, integrals, and the Laplace transform that enable solving differential equations by taking the transform, performing algebra, and applying the inverse transform.
The document provides an overview of topics related to Laplace transforms and their applications. It defines the Laplace transform and discusses some of its key properties, including linearity and how it relates to derivatives, integrals, and shifting theorems. The document also outlines how Laplace transforms can be used to evaluate integrals and solve differential equations. It provides examples of calculating the Laplace transforms of common functions.
The Laplace transform transforms a function of time into a function of complex frequency by taking the integral transform. The inverse Laplace transform takes the complex frequency function and yields the original time function. The Laplace transform expresses a function as a superposition of moments, whereas the Fourier transform uses sinusoids. The Laplace transform provides an alternative functional description that often simplifies analyzing systems. It can be applied to solve switching transients in electric circuits and for load frequency control in power systems. Other applications include transient problems in flow systems and circuit analysis.
Control theorists have extensively studied stability
as well as relative stability of LTI systems. In this research
paper we attempt to answer the question. How unstable is an
unstable system? In that effort we define instability exponents
dealing with the impulse response. Using these instability
exponents, we characterize unstable systems.
1. The document describes the syllabus for the course EE1354 - Modern Control Systems. It includes 5 units that cover topics like state space analysis of continuous and discrete time systems, z-transforms, nonlinear systems, and MIMO systems.
2. Key concepts discussed include state variable representation, eigenvectors and eigenvalues, solution of state equations, controllability and observability, and deriving state space models from transfer functions.
3. Methods like pole placement, state feedback, and observer design for state estimation are also covered in the context of analysis and design of control systems.
This document presents an internship report on studying atomic dynamics out of thermal equilibrium using a three-level atom model. It first derives a Markovian master equation to describe the dynamics of open quantum systems interacting with environments. It then introduces the model of a system interacting with electromagnetic fields at and out of thermal equilibrium. Finally, it examines three-level atom configurations (ladder, Λ, and V) and how population inversion can be achieved when the environment is out of thermal equilibrium.
Modified Projective Synchronization of Chaotic Systems with Noise Disturbance,...IJECEIAES
The synchronization problem of chaotic systems using active modified projective non- linear control method is rarely addressed. Thus the concentration of this study is to derive a modified projective controller to synchronize the two chaotic systems. Since, the parameter of the master and follower systems are considered known, so active methods are employed instead of adaptive methods. The validity of the proposed controller is studied by means of the Lyapunov stability theorem. Furthermore, some numerical simulations are shown to verify the validity of the theoretical discussions. The results demonstrate the effectiveness of the proposed method in both speed and accuracy points of views.
Hybrid Chaos Synchronization of Hyperchaotic Newton-Leipnik Systems by Slidin...ijctcm
This paper investigates the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems (Ghosh and Bhattacharya, 2010) by sliding mode control. The stability results derived in this paper for the hybrid chaos synchronization of identical hyperchaotic Newton-Leipnik systems are established using Lyapunov stability theory. Hybrid synchronization of hyperchaotic Newton-Leipnik systems is achieved through the complete synchronization of first and third states of the systems and the anti-synchronization of second and fourth states of the master and slave systems. Since the Lyapunov exponents are not required for these calculations, the sliding mode control is very effective and convenient to achieve hybrid chaos synchronization of the identical hyperchaotic Newton-Leipnik systems. Numerical simulations are shown to validate and demonstrate the effectiveness of the synchronization schemes derived in this paper.
Sliding Mode Controller Design for Hybrid Synchronization of Hyperchaotic Che...ijcsa
This paper derives new results for the design of sliding mode controller for the hybrid synchronization of identical hyperchaotic Chen systems (Jia, Dai and Hui, 2010). The synchronizer results derived in this paper for the hybrid synchronization of identical hyperchaotic Chen systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve hybrid synchronization of the
identical hyperchaotic Chen systems. Numerical simulations are shown to illustrate and validate the hybrid synchronization schemes derived in this paper for the identical hyperchaotic Chen systems.
This document discusses state-space representation of linear time-invariant (LTI) systems. It defines system state, state equations, and output equations. The key points are:
1) State equations describe the dynamics of a system using first-order differential equations relating state variables. Output equations relate outputs to state variables and inputs.
2) For LTI systems, the state equations can be written in matrix form as dx/dt = Ax + Bu, and output equations as y = Cx + Du.
3) Block diagrams can be constructed from the state-space model, with integrators for each state variable and blocks representing the A, B, C, and D matrices.
The document introduces complex systems and defines a system as a set of interacting parts within a single structure. It provides examples of real-world systems and discusses describing a system from different points of view using models. The document outlines functional and oriented models, abstraction in models, and analyzing dynamic and stationary systems. It discusses representing stationary systems using transition and transformation functions and modeling economic systems with relevant variables and equations. Finally, it presents a case study modeling a school system as a linear, stationary system to calculate total student population over time under changing retention rates.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
A new two-scroll chaotic system with two nonlinearities: dynamical analysis a...TELKOMNIKA JOURNAL
Chaos theory has several applications in science and engineering. In this work, we announce a
new two-scroll chaotic system with two nonlinearities. The dynamical properties of the system such as
dissipativity, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension and bifurcation diagram are
explored in detail. The presence of coexisting chaotic attractors, coexisting chaotic and periodic attractors
in the system is also investigated. In addition, the offset boosting of a variable in the new chaotic system is
achieved by adding a single controlled constant. It is shown that the new chaotic system has rotation
symmetry about the z-axis. An electronic circuit simulation of the new two-scroll chaotic system is built
using Multisim to check the feasibility of the theoretical model.
The document discusses distributivity of fuzzy implications over t-norms and t-conorms. Specifically, it aims to solve the open problem of characterizing functions I that satisfy the equation I(x, S1(y, z)) = S2(I(x, y), I(x, z)), where S1 and S2 are strict or nilpotent t-conorms and I is an R-implication from a strict t-norm. It provides background on t-norms, t-conorms, and fuzzy implications. It then characterizes solutions I when S1 and S2 are strict or nilpotent t-conorms, showing there are no
This document summarizes a study of nonlinear nonequilibrium statistical thermodynamics for systems that are far from equilibrium. The authors propose a method using Zubarev's nonequilibrium distribution function as the mathematical basis. They derive an expression for mean nonequilibrium fluxes to second order, including second derivatives and squares of first derivatives of thermodynamic parameters. A successive approximations method is constructed to eliminate time derivatives of parameters in expressions for mean nonequilibrium fluxes.
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEM...ijcseit
This paper investigates the hybrid chaos synchronization of identical 4-D hyperchaotic Liu systems
(2006), 4-D identical hyperchaotic Chen systems (2005) and hybrid synchronization of 4-D hyperchaotic
Liu and hyperchaotic Chen systems. The hyperchaotic Liu system (Wang and Liu, 2005) and hyperchaotic
Chen system (Li, Tang and Chen, 2006) are important models of new hyperchaotic systems. Hybrid
synchronization of the 4-dimensional hyperchaotic systems addressed in this paper is achieved through
complete synchronization of two pairs of states and anti-synchronization of the other two pairs of states
of the underlying systems. Active nonlinear control is the method used for the hybrid synchronization of
identical and different hyperchaotic Liu and hyperchaotic Chen and the stability results have been
established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these
calculations, the proposed nonlinear control method is effective and convenient to achieve hybrid
synchronization of the hyperchaotic Liu and hyperchaotic Chen systems. Numerical simulations are
shown to demonstrate the effectiveness of the proposed chaos synchronization schemes.
This document provides an overview of geometrical optimal control theory for dynamical systems. It discusses several problems in optimal control theory where geometrical ideas can provide insights, including singular optimal control, implicit optimal control, integrability of optimal control problems, and feedback linearizability. For singular optimal control problems, the document analyzes the behavior at both regular and singular points, and describes how singular problems can be treated as singularly perturbed systems.
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Transfer Function, Stabilizability, and Detectability of Non-Autonomous Riesz-spectral Systems
1. TELKOMNIKA, Vol.16, No.6, December 2018, pp. 3024–3033
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v16i6.10112 3024
Transfer Function, Stabilizability, and Detectability
of Non-Autonomous Riesz-spectral Systems
Sutrima*1
, Christiana Rini Indrati2
, and Lina Aryati3
1
Universitas Sebelas Maret, Ir.Sutami St no.36 A Kentingan Surakarta, Indonesia
1,2,3
Universitas Gadjah Mada, Sekip Utara Kotak Pos: BLS 21, Yogyakarta, Indonesia
*
Corresponding author, e-mail: sutrima@mipa.uns.ac.id, zutrima@yahoo.co.id
Abstract
Stability of a state linear system can be identified by controllability, observability, stabilizability,
detectability, and transfer function. The approximate controllability and observability of non-autonomous
Riesz-spectral systems have been established as well as non-autonomous Sturm-Liouville systems. As a
continuation of the establishments, this paper concern on the analysis of the transfer function, stabilizability,
and detectability of the non-autonomous Riesz-spectral systems. A strongly continuous quasi semigroup
approach is implemented. The results show that the transfer function, stabilizability, and detectability can
be established comprehensively in the non-autonomous Riesz-spectral systems. In particular, sufficient and
necessary conditions for the stabilizability and detectability can be constructed. These results are parallel
with infinite dimensional of autonomous systems.
Keywords: detectability, non-autonomous Riesz-spectral system, stabilizability, transfer function
Copyright c 2018 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Let X, U, and Y be complex Hilbert spaces. This paper concerns on the linear non-
autonomous control systems with state x, input u, and output y:
˙x(t) = A(t)x(t) + B(t)u(t), x(0) = x0,
y(t) = C(t)x(t), t ≥ 0,
(1)
where A(t) is a linear closed operator in X with domain D(A(t)) = D, independent
of t and dense in X; B(t) : U → X and C(t) : X → Y are bounded operators such that
B(·) ∈ L∞(R+
, Ls(U, X)) and C(·) ∈ L∞(R+
, Ls(X, Y )), where Ls(V, W) and L∞(Ω, W) denote
the space of bounded operators from V to W equipped with strong operator topology and the
space of bounded measurable functions from Ω to W provided with essential supremum norm,
respectively. The state linear system (1) is denoted by (A(t), B(t), C(t)). Symbol (A(t), B(t), −)
and (A(t), −, C(t)) denote the state linear system (1) if C(t) = 0 and B(t) = 0, respectively. In
particular, this paper shall investigate stability of the state linear system (1) where each A(t) is
a generalized Riesz-spectral operator [1] using transfer function, stabilizability, and detectability
. The following explanations give some reasons why these investigations are important for the
systems. In the autonomous control system (A, B, C), where A is an infinitesimal generator of a
C0-semigroup T(t), there exist special relationships among input, state, and output [2]. Controlla-
bility map specifies the relationship between the input and the state, observability map specifies
the relationship between an initial state and the output. The relationship between the input and the
output can be characterized by transfer function, a linear map that specifies relationship between
the Laplace transform of the inputs and outputs. The following studies show how urgency of the
transfer function is in the linear system. In the infinite-dimensional autonomous system, external
stability is indicated by boundedness of its transfer function [3]. Weiss [4] have proved a formula
for the transfer function of a regular linear system, which is similar to the formula in the finite di-
mensions. As an extension of results of [4], Staffans and Weiss [5] have generalized the results
Received May 31, 2018; Revised October 5, 2018; Accepted November 3, 2018
2. 3025 ISSN: 1693-6930
of regular linear systems to well-posed linear systems. They have also introduced the Lax-Phillips
semigroup induced by a well-posed linear system. Weiss [6] have studied four transformations
which lead from one well-posed linear system to another: time-inversion, flow-inversion, time-
flow-inversion, and duality. In particular, a well-posed linear system is flow-invertible if and only
if the transfer function of the system has a uniformly bounded inverse on some right half-plane.
Finally, Partington [7] have given simple sufficient conditions for a space of functions on (0, ∞)
such that all shift-invariant operators defined on the space are represented by transfer functions.
Controllability and stability are qualitative control problems which are important aspects of the
theory of control systems. Kalman et al. [8] had initiated the theory for the finite dimensional of
autonomous systems. Recently, the theory was generalized into controllability and stabilizability
of the non-autonomous control systems of various applications, see; e.g. [9, 10, 11], and the
references therein. The concept of the stabilizability is to find an admissible control u(t) such that
the corresponding solution x(t) of the system has some required properties. If the stabilizabil-
ity is identified by null controllability, system (1) is said to be stabilizable if there exists a control
u(t) = F(t)x(t) such that the zero solution of the closed-loop system
˙x(t) = [A(t) + B(t)F(t)]x(t), t ≥ 0,
is asymptotically stable in the Lyapunov sense. In this case, u(t) = F(t)x(t) is called the stabili-
zing feedback control. In particular for autonomous system (A, B, −), where A is an infinitesimal
generator of an exponentially stable C0-semigroup T(t), (A, B, −) is stabilizable if operator A+BF
is an infinitesimal generator of an exponentially stable C0-semigroup TBF (t) [2]. In the finite-
dimensional autonomous control system, Kalman et al. [8] and Wonham [12] had shown that
the system is stabilizable if it is null controllable in a finite time. But, it does not hold for the
converse. Furthermore, if the system is completely stabilizable, then it is null controllable in
a finite time. For finite-dimensional non-autonomous control systems, Ikeda et al. [13] proved
that the system is completely stabilizable whence it is null controllable. Generalizations of the
results of the stabilizability for the finite-dimensional systems into infinite-dimensional systems
have been successfully done. Extending the Lyapunov equation in Banach spaces, Phat and Kiet
[14] specified the relationship between stability and exact null controllability in the autonomous
systems. Guo et al. [15] proved the existence of the infinitesimal generator of the perturbation
semigroup. For neutral type linear systems in Hilbert spaces, Rabah et al. [16] proved that exact
null controllability implies the complete stabilizability. In the paper, unbounded feedback is also
investigated. In the non-autonomous systems, Hinrichsen and Pritchard [17] investigated radius
stability for the systems under structured non-autonomous perturbations. Niamsup and Phat [18]
proved that exact null controllability implies the complete stabilizability for linear non-autonomous
systems in Hilbert spaces. Leiva and Barcenas [19] have introduced a C0-quasi semigroup as a
new approach to investigate the non-autonomous systems. In this context, A(t) is an infinitesimal
generator of a C0-quasi semigroup on a Banach space. Sutrima et al. [20] and Sutrima et al. [21]
investigated the advanced properties and some types of stabilities of the C0-quasi semigroups
in Banach spaces, respectively. Even Barcenas et al. [22] have characterized the controllability
of the non-autonomous control systems using the quasi semigroup approach, although it is still
limited to the autonomous controls. In particular, Sutrima et al. [1] characterized the controllability
and observability of non-autonomous Riesz-spectral systems. The references explain that the
transfer function, stabilizability, and detectability of the control systems are important indicators
for the stability. Unfortunately, there are no studies of these concepts in the non-autonomous
Riesz-spectral systems. Therefore, this paper concerns on investigations of the transfer function,
stabilizability, and detectability of the non-autonomous Riesz-spectral systems that have not been
investgated at this time yet. These investigations use the C0-quasi semigroup approach.
2. Proposed Methods and Discussion
2.1. Transfer Function
We recall the definition of a non-autonomous Riesz-spectral system that refers to Sutrima
et al. [1]. The definition of a Riesz-spectral operator follows [2].
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3. TELKOMNIKA ISSN: 1693-6930 3026
Definition 1. The linear non-autonomous system (A(t), B(t), C(t)) is called a non-autonomous
Riesz-spectral system if A(t) is the infinitesimal generator of a C0-quasi semigroup which has an
expression
A(t) = a(t)A (2)
where A is a Riesz-spectral operator on X and a is a bounded continuous function such that
a(t) > 0 for all t ≥ 0.
By expression (2), for every t ≥ 0, we see that A(t) and A have the common domain
and eigenvectors. In fact, if λn, n ∈ N, is an eigenvalue of A, then a(t)λn is the eigenvalue of
A(t) of (2). Hence, in general A(t) may have the non-simple eigenvalues. In the non-autonomous
Riesz-spectral system (A(t), B(t), C(t)), where A(t) is an infinitesimal generator of a C0-quasi
semigroup R(t, s), relationships among input, state, and output are determined by R(t, s). The
relationships of state-input and state-output of the system have been discussed by Sutrima et al.
[1]. In this subsection, we focus on characterizing the relationship between the input and output of
the system using the transfer function, where B(t) ∈ L2(U, X) and C(t) ∈ L2(X, Y ) for all t ≥ 0.
We define multiplicative operators B : L2(R+
, U) → L2(R+
, X) and C : L2(R+
, X) → L2(R+
, Y )
by:
(Bu)(t) = B(t)u(t) and (Cx)(t) = C(t)x(t), t ≥ 0,
respectively. We see that the operators B and C are bounded. The definition of the transfer
function of the system (A(t), B(t), C(t)) follows the definition for the autonomous systems of [2].
Definition 2. Let (A(t), B(t), C(t)) be the non-autonomous Riesz-spectral system with zero initial
state. If there exists a real α such that ˆy(s) = G(s)ˆu(s) for Re (s) > α, where ˆu(s) and ˆy(s) denote
the Laplace transforms of u and y, respectively, and G(s) is a L(U, Y )-valued function of complex
variable defined for Re (s) > α, then G is called transfer function of the system (A(t), B(t), C(t)).
The impulse response h of (A(t), B(t), C(t)) is defined as the Laplace inverse transform of G.
The transfer function of the non-autonomous Riesz-spectral systems with finite-rank in-
puts and outputs can be stated in eigenvalues and eigenvectors of the Riesz-spectral operator.
Theorem 3. The transfer function G and impulse response h of the non-autonomous Riesz-
spectral system (A(t), B(t), C(t)) exist and are given by
G(s) = Ca(·) (sI − a(·)A)
−1
B on ρ(A(·)),
and
h(t) =
CT(t)B, t ≥ 0
0, t < 0,
where T(t) is a C0-semigroup with the infinitesimal generator A.
Proof. By definition of Riesz-spectral operator and Theorem 3 of [1], we have that the resolvent
set of A is connected, so ρ∞(A) = ρ(A). We shall verify the existences of the transfer function
and impulse response. Let R(t, s) be a C0-quasi semigroup with infinitesimal generator A(t) of
the form (2). For Re (s) > ω0a1, where ω0 is the growth bound of a C0-semigroup T(t) with
infinitesimal generator A and a1 := sup
t≥0
a(t). By Lemma 2.1.11 of [2], we have
Ca(·) (sI − a(·)A)
−1
Bu = C
∞
0
e
−
s
a(·) σ
T(σ)Budσ =
∞
0
e
−
s
a(·) σ
CT(σ)Budσ
for all u ∈ L2(R+
, U) and Re (s) > ω0a1. As in the proof of Lemma 2.5.6 of [2], these s can be
extended to all s ∈ ρ(a(·)A).
Corollary 4. Let A(t) be an operator of the form (2), where A is a Riesz-spectral operator with
eigenvalues {λn ∈ C : n ∈ N}. If B : R+
→ L(Cm
, X) and C : R+
→ L(X, Ck
) such that
Transfer Function, Stabilizability, and Detectability of Non-Autonomous... (Sutrima)
4. 3027 ISSN: 1693-6930
B(t) ∈ L(Cm
, X) and C(t) ∈ L(X, Ck
), then the transfer function and impulse response of the
system (A(t), B(t), C(t)) are given by
G(s) =
∞
n=1
a(·)
s − a(·)λn
Cφn B∗ψn
tr
for s ∈ ρ(A(·)) (3)
h(t) =
∞
n=1
e
λn
a(·) t
Cφn B∗ψn
tr
, t ≥ 0
0, t < 0,
(4)
where φn and ψn are the corresponding eigenvectors of A and A∗
, respectively. Symbol Wtr
denotes transpose of W.
Proof. By representation of (sI − a(·)A)
−1
of Theorem 3 of [1] and facts that operators B(t) and
C(t) are bounded for all t ≥ 0, then from Theorem 3 for s ∈ ρ(A(·)) we have
G(s)u = Ca(·) (sI − a(·)A)
−1
Bu
= C lim
N→∞
N
n=1
a(·)
s − a(·)λn
Bu, ψn φn =
∞
n=1
a(·)
s − a(·)λn
Cφn B∗ψn
tr
.
Here, we use the property v, w Cm = wtr
v. Thus, expression (3) is proved. By a similar argument
and condition (c) of Theorem 3 of [1] we have expression (4) for h(t). The following example illus-
trates the transfer function of a non-autonomous Riesz-spectral system. The example is modified
from Example 4.3.11 of [2].
Example 5. Consider the controlled non-autonomous heat equation on the interval [0, 1],
∂x
∂t
(t, ξ) = a(t)
∂2
x
∂ξ2
(t, ξ) + 2b(t)u(t)χ[ 1
2 ,1](ξ), 0 < ξ < 1, t ≥ 0,
∂x
∂ξ
(t, 0) =
∂x
∂ξ
(t, 1) = 0,
y(t) = 2c(t)
1/2
0
x(t, ξ)dξ,
(5)
where a : R+
→ R is a boundedly uniformly continuous positive function and b : R+
→ C is a
bounded continuous function.
We verify the transfer function and impulse response of the governed system. Setting
X = L2[0, 1] and U = Y = C, the problem (5) is a non-autonomous Riesz-spectral system:
˙x(t) = A(t)x(t) + B(t)u(t),
y(t) = C(t)x(t), t ≥ 0,
(6)
where A(t) = a(t)A with Ax =
d2
x
dξ2
on domain
D = D(A) = {x ∈ X : x, dx
dξ are absolutely continuous, d2
x
dξ2 ∈ X, dx
dξ (0) = dx
dξ x(1) = 0},
B(t)u(t) = 2b(t)u(t)χ[ 1
2 ,1](ξ), and C(t)x(t) = 2c(t)
1/2
0
x(t, ξ)dξ.
The eigenvalues and eigenvectors of A are {0, −n2
π2
: n ∈ N} and {1,
√
2 cos(nπξ) : n ∈ N},
respectively. It is easy to show that A is a self-adjoint Riesz-spectral operator with its Riesz basis
{1,
√
2 cos(nπξ) : n ∈ N}. In this case we have
B∗
(t)x =
1
0
xχ
[
1
2 ,1]
(ξ)dξ,
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5. TELKOMNIKA ISSN: 1693-6930 3028
for all ∈ X and t ≥ 0. In virtue of (3) evaluating at t, the transfer function of the system (6) is given
by
G(s)(t) =
1
s
+
∞
n=1
2a(t)[cos(nπ) − 1]
[s + a(t)n2π2](nπ)2
,
and evaluating at s in (4), we have the impulse response
h(t)(s) = 1 +
∞
n=1
2[cos(nπ) − 1]
(nπ)2
e−
(nπ)2
a(s)
t
,
on t ≥ 0 and zero elsewhere, respectively.
2.2. Stabilizability
We first recall the definition of uniformly exponentially stable C0-quasi semigroup that
refers to [21, 23]. The concept play an important role in characterizing stabilizability of the non-
autonomous Riesz-spectral systems.
Definition 6. A C0-quasi semigroup R(t, s) is said to be:
(a) uniformly exponentially stable on a Banach space X if there exist constants α > 0 and
N ≥ 1 such that
R(t, s)x ≤ Ne−αs
x , (7)
for all t, s ≥ 0 and x ∈ X;
(b) β-uniformly exponentially stable on a Banach space X if (7) holds for −α < β.
A constant α is called decay rate and the supremum over all possible values of α is called
stability margin of R(t, s). Indeed, the stability margin is minus of the uniform growth bound ω0(R)
defined
ω0(R) = inf
t≥0
ω0(t),
where ω0(t) = inf
s>0
1
s log R(t, s) . We see that R(t, s) is β-uniformly exponentially stable if its
stability margin is at least −β. We give two preliminary results which are urgent in discussing the
stabilizability.
Theorem 7. Let R(t, s) be a C0-quasi semigroup on a Banach space X. The R(t, s) is uniformly
exponentially stable on X if and only if ω0(R) < 0.
Proof. By taking log on (7), we have the assertion.
Theorem 8. Let A(t) be an infinitesimal generator of C0-quasi semigroup R(t, s) on a Banach
space X. If B(·) ∈ L∞(R+
, Ls(X)), then there exists a uniquely C0-quasi semigroup RB(t, s) with
its infinitesimal generator A(t) + B(t) such that
RB(r, t)x = R(r, t)x +
t
0
R(r + s, t − s)B(r + s)RB(r, s)xds, (8)
for all t, r, s ≥ 0 with t ≥ s and x ∈ X. Moreover, if R(r, t) ≤ M(t), then
RB(r, t) ≤ M(t)e B M(t)t
.
Proof. We define
R0(r, t)x = R(r, t)x,
Rn(r, t)x =
t
0
R(r + s, t − s)B(r + s)Rn−1(r, s)xds, (9)
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6. 3029 ISSN: 1693-6930
for all t, r, s ≥ 0 with t ≥ s, x ∈ X, and n ∈ N, and
RB(r, t) =
∞
n=0
Rn(r, t), (10)
Following the proof of Theorem 2.4 of [19], we obtain the assertions. The investigations of stabiliz-
ability and detectability of the non-autonomous Riesz-spectral systems are generalizations of the
concepts of stabilizability and detectability for the autonomous systems that had been developed
by Curtain and Zwart [2].
Definition 9. The non-autonomous Riesz-spectral system (A(t), B(t), C(t)) is said to be:
(a) stabilizable if there exsits an operator F ∈ L∞(R+
, Ls(X, U)) such that A(t) + B(t)F(t),
t ≥ 0, is an infinitesimal generator of a uniformly exponentially stable C0-quasi semigroup
RBF (t, s). The operator F is called a stabilizing feedback operator;
(b) detectable if there exists an operator K ∈ L∞(R+
, Ls(Y, X)) such that A(t) + K(t)C(t),
t ≥ 0, is an infinitesimal generator of a uniformly exponentially stable C0-quasi semigroup
RKC(t, s). The operator K is called an output injection operator.
If the quasi semigroup RBF (t, s) is β-uniformly exponentially stable, we say that the system
(A(t), B(t), −) is β-stablizable. If RKC(t, s) is β-uniformly exponentially stable, we say that the
system (A(t), −, C(t)) is β-detectable.
For δ ∈ R, we can decompose the spectrum of A in complex plane into two distinct parts:
σ+
δ (A) := σ(A) ∩ C+
δ ; C+
δ = {λ ∈ C : Re (λ) > δ}
σ−
δ (A) := σ(A) ∩ C−
δ ; C−
δ = {λ ∈ C : Re (λ) < δ}.
In the autonomous case, if B has finite-rank and the system (A, B, −) is stabilizable, then
we can decompose the spectrum of A into a δ-stable part and a δ-unstable part which comprises
eigenvalues with finite multiplicity. In other word, A has at most finitely many eigenvalues in C+
δ .
We shall apply the decomposition of the spectrum to the non-autonomous Riesz-spectral systems.
Definition 10. An operator A is said to be satisfying the spectrum decomposition assumption at
δ if σ+
δ (A) is bounded and separated from σ−
δ (A) in such way that a rectifiable, simple, closed
curve, Γδ, can be drawn so as to enclose an open set containing σ+
δ (A) in its interior and σ−
δ (A)
in its exterior.
According to Definition 10, we have that classes of Riesz-spectral operators with a pure
point spectrum and only finitely many eigenvalues in σ+
δ (A) satisfy the spectrum decomposition
assumption. For every t ≥ 0 we define the spectral projection Pδ(t) on X by
Pδ(t)x =
1
2πi Γδ
(λI − A(t))−1
xdλ, (11)
for all x ∈ X, where Γδ is traversed once in the positive direction (counterclockwise). By this
operator, we can decompose any Hilbert space X to be:
X = X+
δ ⊕ X−
δ , where X+
δ := Pδ(t)X and X−
δ := (I − Pδ(t))X. (12)
By this decomposition, we denote
A(t) =
A+
δ (t) 0
0 A−
δ (t)
, R(t, s) =
R+
δ (t, s) 0
0 R−
δ (t, s)
(13)
B(t) =
B+
δ (t)
B−
δ (t)
, C(t) = C+
δ (t) C−
δ (t) , (14)
TELKOMNIKA Vol. 16, No. 6, December 2018 : 3024 − 3033
7. TELKOMNIKA ISSN: 1693-6930 3030
where B+
δ (t) = Pδ(t)B(t) ∈ L(U, X+
δ ), B−
δ (t) = (I − Pδ(t))B(t) ∈ L(U, X−
δ ), C+
δ (t) = C(t)Pδ(t) ∈
L(X+
δ , Y ), and C−
δ (t) = C(t)(I − Pδ(t)) ∈ L(X−
δ , Y ). In virtue of the decomposition, we can ex-
press the system (A(t), B(t), C(t)) as the vector sum of the two subsystems: (A+
δ (t), B+
δ (t), C+
δ (t))
on X+
δ and (A−
δ (t), B−
δ (t), C−
δ (t)) on X−
δ . In particular, if the input operator B(·) has finite-rank,
then the subsystem (A+
δ (t), B+
δ (t), −) has a finite dimension.
Theorem 11. Let (A(t), B(t), −) be a non-autonomous Riesz-spectral system on the state space
X where B(·) has a finite rank. If A satisfies the spectrum decomposition assumption at β, X+
β has
a finite dimension, R−
β (t, s) is β-uniformly exponentially stable, and subsystem (A+
β (t), B+
β (t), −)
is controllable, then the system (A(t), B(t), −) β-stabilizable. In this case, a β-stabilizing feed-
back operator is given by F(t) = F0(t)Pβ(t), where F0 is a β-stabilizing feedback operator for
(A+
β (t), B+
β (t), −).
Proof. Since (A+
β (t), B+
β (t), −) is controllable, there exists a feedback operator balik F0(·) ∈
L∞(R+
, Ls(X+
β , U) such that the spectrum of A+
β (t)+B+
β (t)F0(t) in C−
β for all t ≥ 0. We choose a
feedback operator F(·) = (F0(·), 0) ∈ L∞(R+
, Ls(X, U)) for the system (A(t), B(t), −). According
to Theorem 8, the perturbed operator A(t) + B(t)F(t) =
A+
β (t) + B+
β (t)F0(t) 0
B−
β (t)F0(t) A−
β (t)
is the
infinitesimal generator of a C0-quasi semigroup R(t, s). Moreover, if β1 is the growth bound of
R(t, s), then β1 is the maximum of that of the quasi semigroups generated by A+
β (t) + B+
β (t)F0(t)
and A−
β (t). Therefore, β1 < β, i.e. the system (A(t), B(t), −) is β-stabilizable. By the dual
property between the stabilizability and detectability, Theorem 11 provides the sufficiency for the
β-detectability.
Theorem 12. Let (A(t), −, C(t)) be the non-autonomous Riesz-spectral system on the state space
X where C(·) has a finite rank. If A satisfies the spectrum decomposition assumption at β,
X+
β has a finite dimension, R−
β (t, s) is β-uniformly exponentially stable, and (A+
β (t), −, C+
β (t)) is
observable, then (A(t), −, C(t)) is β-detectable. In this case, β-stabilizing output injection operator
is given by K(t) = iβ(t)K0(t), where K0 is an operator such that A+
β (t) + K0(t)C+
β (t) is the
infinitesimal generator of the β-uniformly exponentially stable C0-quasi semigroup and iβ(t) is an
injection operator from X+
β to X.
Proof. By the dual concept, we have that the system (A(t), −, C(t)) is β-detectability if and only
if (A∗
(t), C∗
(t), −) is β-stabilizability. From Theorem 11, A∗
satisfies the spectrum decomposition
assumption at β. The corresponding spectral projection is given by
P∗
β (t)x =
1
2πi Γβ
(λI − A∗
(t))−1
xdλ,
for all x ∈ X. We can choose such that Γβ is symmetric with respect to the real axis. Hence, the
decomposition of (A∗
(t), C∗
(t), −) is the adjoint of the decomposition of (A(t), −, C(t)). By the
dual argument, we have the required results.
Example 13. Consider the non-autonomous Riesz-spectral system in Example 5. Using all of the
agreement there, we show that the system is stabilizable and detectable.
We have the spectral decomposition of A(t):
A(t)x = a(t)
∞
n=1
−(nπ)2
x, φn φn for x ∈ D,
where φn(ξ) =
√
2 cos(nπξ), and the family of the operators generates a C0-quasi semigroup
R(t, s) given by
R(t, s)x = x, 1 1 +
∞
n=1
e−(nπ)2
(g(t+s)−g(t))
x, φn φn,
Transfer Function, Stabilizability, and Detectability of Non-Autonomous... (Sutrima)
8. 3031 ISSN: 1693-6930
where g(t) =
t
0
a(s)ds. Since the set of eigenvalues of A has an upper bound, then A satisfies
the spectrum assumption at any real β. Suppose we choose β = −2. In this case we have
σ+
−2(A) = {0} and so there exists a closed simple curve Γ−2 that encloses the eigenvalue 0. For
any x ∈ X and t ≥ 0, Cauchy’s Theorem gives
P−2(t)x =
1
2πi Γ−2
(λI − a(t)A)−1
xdλ =
1
2πi Γ−2
1
λ
x, 1 dλ = x, 1 .
In virtue of (12), we have that X+
−2 has dimension 1. Moreover, in the subspace we have
(A+
−2(t), B+
−2(t), C+
−2(t)) = (0, I, I) which are controllable and observable. There exists F0(·) ∈
L∞(R+
, Ls(X+
−2, U)) such that the spectrum of A+
−2(t) + B+
−2(t)F0(t) in C−
−2 for all t ≥ 0. We can
choose F0(t)x = −3 x, 1 for all x ∈ X+
−2. Theorem 11 concludes that (A(t), B(t), C(t)) is (−2)-
stabilizable with the feedback operator u = F(t)z, where F(t) = F0(t) 0 = −3I 0 .
By the duality, Theorem 12 shows that the system (A(t), −, C(t)) is (−2)-detectable with output
injection operator K(t) =
−3I
0
such that K(t)y = −3y1. The decay constants of the quasi
semigroup generated by A(t) + B(t)F(t) and A(t) + L(t)C(t) are 3. The following theorem is a
similar result with Theorem 13 of [1] for controllability and observability of the non-autonomous
Riesz-spectral systems on Hilbert spaces.
Theorem 14. Let (A(t), B(t), C(t)) be a non-autonomous Riesz-spectral system. Necessary and
sufficient conditions for (A(t), B(t), −) is β-stabilizable are that there exists an > 0 such that
σ+
β− (A) comprises at most finitely many eigenvalues and
rank ( b1(t), ϕn , . . . , bm(t), ϕn ) = 1, (15)
for all n such that λn ∈ σ+
β− (A) and t ≥ 0. Necessary and sufficient conditions for (A(t), −, C(t))
is β-detectable are that there exists an > 0 such that σ+
β− (A) comprises at most finitely many
eigenvalues and
rank ( φn, c1(t) , . . . , φn, ck(t) ) = 1, (16)
for all n such that λn ∈ σ+
β− (A) and t ≥ 0.
Proof. We only need to prove necessary and sufficient conditions for β-stabilizability. For β-
detectability follows the dual argument of the system. Sufficiency for β-stabilizability. Since
σ+
β− (A) only contains at most finitely many eigenvalues of A, then X+
β− has a finite dimension
and A satisfies the spectrum decomposition assumption at β − . By condition (a) of Theorem 3
of [1] and Cauchy’s Theorem, for any t ≥ 0 we have
Pβ− (t)x =
1
a(t)
λn∈σ+
β−
x, ψn φn.
Since A(t) and A have common eigenvectors for every t ≥ 0, again Theorem 3 of [1] gives
X+
β− = span
λn∈σ+
β−
{φn}, X−
β− = span
λn∈σ−
β−
{φn}, R−
β− (t, s) =
λn∈σ−
β−
eλn(g(t+s)−g(t)
x, ψn φn,
A+
β− (t)x = a(t)
λn∈σ+
β−
λn x, ψn φn, and B+
β− (t)u =
λn∈σ+
β−
B(t)u, ψn φn, (17)
where g(t) =
t
0
a(s)ds. This result shows that R−
β− (t, s) is a C0-quasi semigroup correspond-
ing to the Riesz-spectral operator A−
β− on X−
β− . Consequently, ω0(R−
β− ) < 0 and Theorem 7
states that R−
β− (t, s) is β-uniformly exponentially stable. We need prove that (A+
β− (t), B+
β− (t), −)
is controllable. The reachibility subspace of (A+
β− (t), B+
β− (t), −) is the smallest A+
β− -invariant
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9. TELKOMNIKA ISSN: 1693-6930 3032
subspace of X+
β− which contains ran B+
β− (t). In virtue of Lemma 2.5.6 of [2], this subspace is
spanned by the eigenvectors of A+
β− . Hence, if this subspace does not equal with the state space
X+
β− , then there exists a λj ∈ σ+
β− (A) such that φj /∈ X+
β− . Therefore, its biorthogonal element
ψj is orthogonal to the reachibility subspace, in particular B(t)u, ψj = 0 for every u ∈ Cm
and
t ≥ 0. This contradicts to (15), and so (A+
β− (t), B+
β− (t), −) is controllable. Thus, Theorem 11
shows that the system (A(t), B(t), −) is β-stabilizable. Necessity for β-stabilizability. From Def-
inition 6 and Definition 9, if (A(t), B(t), −) is β-stabilizable, then there exists > 0 such that the
system is also (β − )-stabilizable. In virtue of Theorem 11, A satisfies the spectrum decompo-
sition assumption at β − . Moreover, the subspace X+
β− is R(t, s)-invariant. Lemma 2.5.8 of [2]
gives
X+
β− = spann∈J{φn}.
Since X+
β− has a finite dimension, then J contains at most finitely many elements. So the spec-
trum of A+
β− = A|X+
β−
is contained in C+
β− and the spectrum of A−
β− = A|X−
β−
is contained in
C−
β− . This concludes that the index set J equals with the set {n ∈ N : λn ∈ σ+
β− (A)}. Thus,
σ+
β− (A)} comprises at most finitely many eigenvalues. By (17) and Theorem 11, we conclude
that (A+
β− (t), B+
β− (t), −) is controllable. Suppose that the condition (15) does not hold. There
exists λn ∈ σ+
β− (A) such that B(t)u, ψn = 0 for all u ∈ Cm
. This states that the reachibility
subspace of (A+
β− (t), B+
β− (t), −) does not equal to X+
β− , that is (A+
β− (t), B+
β− (t), −) is not con-
trollable. This gives a contradiction. In practice, Theorem 14 is more applicable than Theorem 11
and 12 in characterizing the stabilizability and detectability of the non-autonomous Riesz-spectral
systems. We return to Example 13. For β = −2, we can choose = 1 such that σ+
−3(A) = {0}
and the corresponding eigenvector is φ0(ξ) = 1. In this case, it is obvious that
1
0
b(t)φ0(ξ)dξ = 0 and
1
0
c(t)φ0(ξ)dξ = 0,
for all t ≥ 0, i.e. the conditions (15) and (16) are confirmed. Hence, the system (A(t), B(t), C(t))
are stabilizable and detectable.
3. Conclusion
The concepts of the transfer function, stabilizability, and detectability of the autonomous
systems can be generalized to the non-autonomous Riesz-spectral systems. The results are
alternative considerations in analyzing the related control problems. There are opportunities to
generalize these results to the generally non-autonomous systems including the time-dependent
domain.
Acknowledgement
The authors are grateful to Research Institution and Community Service of Universitas
Sebelas Maret Surakarta for funding and to the reviewers for helpful comments.
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