Local constrained controllability of second order dynamical systems with delay

In the present paper finite-dimensional dynamical control systems described by linear higherorder ordinary differential state equations with multiple point delays in control are considered. Using algebraic methods, necessary and sufficient condition for relative controllability in a given time interval for linear dynamical system with multiple point delays in control is formulated and proved. This condition is generalization to relative controllability case some previous results concerning controllability of linear dynamical systems without multiple point delays in the control. Simple numerical example, which illustrates theoretical result is also given. Finally, some remarks and comments on the existing results for controllability of dynamical systems with delays in control are also presented.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

In this article, we study the controllability of finite-dimensional dynamical control systems modelled by semilinear impulsive ordinary differential equations with multiple constant time delays in the control function. Initially, we recall a necessary and sufficient condition for the controllability of the corresponding linear system without impulses, with multiple constant time delays in the control function in terms of a matrix rank condition. Then under some sufficient conditions, we show that the actual system is also controllable for certain classes of non-linearities and impulse functions. We employ Schauder fixed-point theorem and Banach contraction mapping principle to establish the results. Our obtained results are applicable for both autonomous and non-autonomous systems. An example is given to illustrate the theoretical results.

Constrained exact controllability of semilinear systems

Finite-dimensional dynamical control systems described by semilinear ordinary differential state equations with a single point delay in control are considered. Using a generalized open mapping theorem, sufficient conditions for constrained local relative controllability are formulated. It is generally assumed, that the values of admissible controls are in a convex and closed cone with vertex at zero.

Controllability of dynamical systems is studied. A dynamical control system is defined by a family of vector fields. Dynamical control systems are considered on two-dimensional manifolds. First of all, dynamical control systems on a sphere are analyzed. Some new notions are introduced for dynamical control systems: a skeleton of a state space, irreducible components of a skeleton, a linking property for irreducible components, a cycle of irreducible components, a bifurcation graph. A piece of a path on the bifurcation graph is called a scenario of control. Some pieces of paths on the bifurcation graph are selected. They correspond to controllable systems.

In this paper, we discuss the controllability of fractional Langevin delay dynamical systems represented by the fractional delay differential equations of order 0 < α,β ≤ 1. Necessary and sufficient conditions for the controllability of linear fractional Langevin delay dynamical system are obtained by using the Grammian matrix. Sufficient conditions for the controllability of the nonlinear delay dynamical systems are established by using the Schauders fixed-point theorem. The problem of controllability of linear and nonlinear fractional Langevin delay dynamical systems with multiple delays and distributed delays in control are studied by using the same technique. Examples are provided to illustrate the theory.

The paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function $f$. The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered. The results are illustrated by an example.

We present a necessary optimality conditions for a class of optimal control problems. The dynamical control system involves integer and fractional order derivatives and the final time is free. Optimality conditions are obtained. Feedback control laws for linear dynamic system are obtained.

Chapter 1: Introduction to dynamic modeling 1. Modeling drugs in the bloodstream 2. Terminology 3. Equilibrium values 4. Dynamic economic applications 5. Applications of dynamics using spreadsheets Chapter 2: First order dynamical systems 1. Solutions to linear dynamical systems with applications 2. Solutions to an affine dynamical system 3. An introduction to genetics 4. Solution to affine dynamical systems with applications 5. Applications to finance Chapter 3: Introduction to probability 1. The multiplication to probability 2. Introduction to probability 3. Multistage tasks 4. An introduction to Markov chains Chapter 4: Nonhomogeneous dynamical systems 1. Exponential terms 2. Exponential terms, a special case 3. Fractal geometry 4. Polynomial terms 5. Polynomial terms, a special case Chapter 5: Higher order linear dynamical systems 183 1. An introduction to second order linear equations 2. Multiple roots 3. The gambler's ruin 4. Sex-linked genes 5. Stability for second order affine equations 6. Modeling a vibrating string 7. Second order nonhomogeneous equations 8. Gambler's ruin revisited 9. A model of a national economy 10. Dynamical systems with order greater than two 11. Solutions involving trigonometric functions Chapter 6: Introduction to nonlinear dynamical systems 1. A model of population growth 2. Using linearization to study stability 3. Harvesting strategies 4. More linearization Chapter 7: Vectors and matrices 1. Introduction to vectors and matrices 2. Rules of linear algebra 3. Gauss-Jordan elimination 4. Determinants 5. Inverse matrices Chapter 8: Dynamical systems of several equations 1. Introduction to dynamical systems of several equations 2. Characteristic values 3. First order dynamical systems of several equations 4. Regular Markov chains 5. Absorbing Markov chains 6. Applications of absorbing Markov chains 7. Long term behavior of solutions 8. The heat equation 9. Bibliography

The paper presents finite-dimensional dynamical control systems described by semilinear fractional-order state equations with multiple delays in the control and nonlinear function $f$. The relative controllability of the presented semilinear system is discussed. Rothe's fixed point theorem is applied to study the controllability of the fractional-order semilinear system. A control that steers the semilinear system from an initial complete state to a final state at time $t>0$ is presented. A numerical example is provided to illustrate the obtained theoretical results and a practical example is given to show a possible application of the study.

INTRODUCTION Assessing the sustainability of dynamic, open and complex systems with many stakeholders, interrelated components and interactions and forecasting with traditional study methods is complicated and has its limitations. Therefore, often researchers, when forecasting the sustainability of a dynamic system, rely on subjective judgment without references to assessment standards. The aim of the paper is to create an imitation model for the sustainability of a dynamic system in order to assess and forecast the sustainability of the system under alternative development scenarios. It includes 3 main aspects - how sustainable is the dynamic system, what is the level of sustainability of a dynamic system under alternative development scenarios and what additions are needed to improve the functioning of the imitation model. The research question of the paper is: what imitation model can effectively analyse and forecast a dynamic system in the case of the tourism object Cesis Palace. Sustainable development researchers offer to build on traditional principles and interlinked dimensions of sustainable development: environment, economic and social, and adapt them to the dynamic system, which is characterized by the interactions between components (Tanguay et.al., 2011; Mai, Smith, 2018). One type of study that helps explain such systems is simulation modelling, which is often used when researching the interaction of dynamic systems (Johnson, 2011). MATERIALS AND METHODS The Cesis Palace complex as a tourism site was used in the paper for an example of a dynamic system, since tourism is both a dynamic system with many interlinked components and equally important are the three dimensions of sustainability for its long-term development. During the study, multiple data acquisition methods were used: a structured interview, analysis of statistics, case study analysis and an expert interview on the created imitation model. To achieve the goal of the research, a model of a tourism sustainability imitation model was created using the STELLA dynamic system modelling environment. The model and the selection of indicators were based on the three key sustainability dimensions: economy, environment and society/culture. RESULTS The result of the work is a computer model that helps to assess the sustainability of a real-life system and its dimensions by entering data generated during the study. It concludes that the tourism object under consideration is potentially sustainable. Simulating alternative development scenarios, it can be concluded that the elements of one group of indicators can affect both the sustainability level of their own dimension, as well as the indicators of other dimensions and their sustainability level, as well the sustainability of the system overall. Significant changes in the system take place in a situation where a number of indicator groups are affected by the changes. DISCUSION AND CONCLUSIONS In order to use this model further, it would be necessary to develop an improved methodology for evaluating model indicators. To ensure a more efficient model performance and data quality, data security and integrity should be taken into account. Creation of an imitation model requires the acquisition, processing and issuing of large amounts of data that are exposed to data security and integrity risks, which may have a negative impact, not only on the functioning of the model, but also on the system itself. To significantly improve the definition, selection, value assignment, and to more accurately identify the importance of interfacing elements and to express future forecasts, the author proposes to evaluate the use of machine-learning in imitation modelling. Machine-learning algorithms are increasingly used by researchers in mechanical engineering applications.

Controllability problems studied in this Chapter concern semilinear fractional discrete-time control systems with multiple delays in control. More precisely, in the present paper unconstrained local controllability problem of finite-dimensional fractional-discrete time semilinear systems with multiple delays in control is addressed.

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete‐time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

The realization problem that we will state here can be divided into the following three problems A, B and C. Where I/O is the set of input/output maps that may be the input/output relation of a given black-box. CD is the category of dynamical systems which may have the same behavior (equivalently, input/output relation) as the black-box. A. The existence and uniqueness in algebraic sense. For any input/output map a I/O , find out at least one dynamical system CD such that the behavior of it is a . Also prove that any two dynamical systems that have the same behavior a are isomorphic in the sense of the category CD. B. The finite dimensionality of the dynamical systems. Clarify when a dynamical system CD is finite dimensional. Because finite dimensional dynamical systems are actually appearing by linear (or non-linear) circuits or computer programs, it is very important that these conditions become clear. C. Deriving the dynamical systems from finite data. Partial realization problems are to find the minimal dynamical system fit to a given finite input/output’s data and to clarify when the minimal dynamical systems are isomorphic.