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Optimization of Delayed Differential Systems by Lyapunov's Direct Method
Demchenko, Hanna ; Růžičková, Miroslava (oponent) ; Shatyrko,, Andriy (oponent) ; Diblík, Josef (vedoucí práce)
The present thesis deals with processes controlled by systems of delayed differential equations $$x'(t) =f(t,x_t,u),\,\,\,\, t\ge t_{0}$$ where $t_0 \in \mathbb{R}$, $f$ is defined on a subspace of $[t_0,\infty)\times {C}_{\tau}^{m}\times {\mathbb{R}}^r$, $m,r \in \mathbb{N}$, ${C}_{\tau}^{m}=C([-\tau,0],{\mathbb{R}}^{m})$, $\tau>0$, $x_t(\theta):=x(t+\theta)$, $\theta\in[-\tau,0]$, $x\colon [t_0-\tau,\infty)\to \mathbb{R}^{m}$. Under the assumption $f(t,\theta_m^*,\theta_r)=\theta_m$, where ${\theta}_m^*\in {C}_{\tau}^{m}$ is a zero vector-function, $\theta_r$ and $\theta_m$ are $r$ and $m$-dimensional zero vectors, a control function $u=u(t,x_t)$, $u\colon[t_0,\infty)\times {C}_{\tau}^{m}\to \mathbb{R}^{r}$, $u(t,{\theta}_m^*)=\theta_r$ is determined such that the zero solution $x(t)=\theta_m$, $t\ge t_{0}-\tau$ of the system is asymptotically stable and, for an arbitrary solution $x=x(t)$, the integral $$\int _{t_{0}}^{\infty}\omega \left(t,x_t,u(t,x_t)\right)\diff t,$$ where $\omega$ is a positive-definite functional, exists and attains its minimum value in a given sense. To solve this problem, Malkin's approach to ordinary differential systems is extended to delayed functional differential equations and Lyapunov's second method is applied. The results are illustrated by examples and applied to some classes of delayed linear differential equations.
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Representation of Solutions of Linear Discrete Systems with Delay
Morávková, Blanka ; Růžičková, Miroslava (oponent) ; Khusainov, Denys (oponent) ; Diblík, Josef (vedoucí práce)
The dissertation thesis is concerned with linear discrete systems with constant matrices of linear terms with a single or two delays. The main objective is to obtain formulas analytically describing exact solutions of initial Cauchy problems. To this end, some matrix special functions called discrete matrix delayed exponentials are defined and used. Their basic properties are proved. Such special matrix functions are used to derive analytical formulas representing the solutions of initial Cauchy problems. First is discussed the initial problem with impulses are acting at some prescribed points and formulas describing the solutions of this problem are derived. In the next part of the dissertation, two definitions of discrete matrix delayed exponentials for two delays are given and their basic properties are proved. Such discrete special matrix functions make it possible to find representations of solutions of linear systems with two delays. This is done in the last part of dissertation thesis where two different formulas giving the analytical solution of this problem are derived.
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Existence and Properties of Global Solutions of Mixed-Type Functional Differential Equations
Vážanová, Gabriela ; Růžičková, Miroslava (oponent) ; Fečkan,, Michal (oponent) ; Diblík, Josef (vedoucí práce)
This thesis focuses on functional differential equations of mixed type also referred to as advance-delay equations. It gives sufficient conditions for the existence of global and semi-global solutions to nonlinear mixed differential systems. The methods used in this thesis consist of building suitable operators for differential equations and proving the existence of their fixed points. These fixed points are then used to construct the solutions of advance-delay equations. The monotone iterative method and Schauder-Tychonoff fixed point theorems are used in the proofs. In both cases, we also provide solution estimates. Moreover, with the monotone iterative method, these estimates may be improved by iterations. In addition, criteria for linear equations and systems are derived and series of examples are provided. The results obtained are also applicable to ordinary, delayed or advanced differential equations.
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Weakly Delayed Systems of Linear Discrete Equations in R^3
Šafařík, Jan ; Khusainov, Denys (oponent) ; Růžičková, Miroslava (oponent) ; Diblík, Josef (vedoucí práce)
The present thesis deals with the construction of a general solution of weakly delayed systems of linear discrete equations in ${\mathbb R}^3$ of the form \begin{equation*} x(k+1)=Ax(k)+Bx(k-m) \end{equation*} where $m>0$ is a positive integer, $x\colon \bZ_{-m}^{\infty}\to\bR^3$, $\bZ_{-m}^{\infty} := \{-m, -m+1, \dots, \infty\}$, $k\in\bZ_0^{\infty}$, $A=(a_{ij})$ and $B=(b_{ij})$ are constant $3\times 3$ matrices. The characteristic equations of weakly delayed systems are identical with those of the same systems but without delayed terms. The criteria ensuring that a system is weakly delayed are developed and then specified for every possible case of the Jordan form of matrix $A$. The system is solved by transforming it into a higher-dimensional system but without delays \begin{equation*} y(k+1)=\mathcal{A}y(k), \end{equation*} where ${\mathrm{dim}}\ y = 3(m+1)$. Using methods of linear algebra, it is possible to find the Jordan forms of $\mathcal{A}$ depending on the eigenvalues of matrices $A$ and $B$. Therefore, general the solution of the new system can be found and, consequently, the general solution of the initial system deduced.
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Linear Matrix Differential Equation with Delay
Piddubna, Ganna Konstantinivna ; Růžičková, Miroslava (oponent) ; Dzhalladova, Irada (oponent) ; Baštinec, Jaromír (vedoucí práce)
This work is devoted to computing the solution, stability of the solution and controllability of respective system of linear matrix differential equation with delay x'(t)=A0x(t)+A1 x(t-tau), where A0, A1 are constant matrices and tau>0 is the constant delay. To solve this equation, the "step by step" method was used. The solution was found in recurrent form and in general form. Stability and the asymptotic stability of the solution of the equation was investigated. Conditions for stability were defined. The Lyapunov’s functional theory is basic for the investigation. Necessary and sufficient condition for controllability in same matrices case was defined and the control was built. Sufficient conditions for controllability in communicative matrices case and general case were defined and controls were built. All results were illustrated with non-trivial examples.
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Weakly Delayed Linear Planar Systems of Discrete Equations
Halfarová, Hana ; Růžičková, Miroslava (oponent) ; Khusainov, Denys (oponent) ; Diblík, Josef (vedoucí práce)
The present thesis deals with planar weakly delayed linear discrete systems. The characteristic equations of weakly delayed systems are identical with those of the same systems but without delayed terms. In this case, after several steps, the space of solutions with a given starting dimension is pasted into a space with a dimension less than the starting one. In a sense, this situation is analogous to one known in the theory of linear differential systems with constant coefficients and special delays when the initially infinite dimensional space of solutions on the initial interval turns (after several steps) into a finite dimensional set of solutions. For every possible case, explicit general solutions are constructed and, finally, results on the dimensionality of the space of solutions are obtained. The stability of solutions is investigated as well.
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Estimation of Solutions of Differential Systems with Delayed Argument of Neutral Type
Baštincová, Alena ; Růžičková, Miroslava (oponent) ; Dzhalladova,, Irada (oponent) ; Diblík, Josef (vedoucí práce)
This dissertation discusses the solutions to the differential equation and to systems of differential equations. The main attention is paid to study of asymptotical properties of equations with delay and systems of equations with delay. In the first chapter are given physical and technical examples described by differential equations with delay and their systems. The classification of equations with delay is given and basic notions of theory of stability are formulated (mainly with the emphasis on the Lyapunov second method). In the second chapter estimates of solutions of equations of neutral type are studied. The third chapter deals with systems of differential equations of neutral type. Asymptotic estimates for solutions and their derivatives are proved. At the end of the chapter examples and comparisons of our results and of other authors are given. The calculation were performed with the MATLAB software. Last, the fourth chapter deals with asymptotical properties of systems having a special type of nonlinearities, so called ``sector nonlinearities''. Properties and estimations of solutions and derivatives are derived. The basic tools used in the dissertation are the Lyapunov second method and functionals of Lyapunov-Krasovskii type.
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Stochastic Calculus and Its Applications in Biomedical Practice
Klimešová, Marie ; Růžičková, Miroslava (oponent) ; Dzhalladova, Irada (oponent) ; Baštinec, Jaromír (vedoucí práce)
In the presented dissertation is defined the stochastic differential equation and its basic properties are listed. Stochastic differential equations are used to describe physical phenomena, which are also influenced by random effects. Solution of the stochastic model is a random process. Objective of the analysis of random processes is the construction of an appropriate model, which allows understanding the mechanisms. On their basis observed data are generated. Knowledge of the model also allows forecasting the future and it is possible to control and optimize the activity of the applicable system. In this dissertation is at first defined probability space and Wiener process. On this basis is defined the stochastic differential equation and the basic properties are indicated. The final part contains biology model illustrating the use of the stochastic differential equations in practice.
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Asymptotic Properties of Solutions of the Second-Order Discrete Emden-Fowler Equation
Korobko, Evgeniya ; Galewski, Marek (oponent) ; Růžičková, Miroslava (oponent) ; Diblík, Josef (vedoucí práce)
In the literature a differential second--order nonlinear Emden--Fowler equation $$ y'' \pm x^\alpha y^m = 0, $$ where $\alpha$ and $m$ are constants, is often investigated. This thesis deals with a discrete equivalent of the second--order Emden-Fowler differential equation $$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0, $$ where $k\in \mathbb{N}(k_0):= \{k_0, k_0+1, ....\}$ is an independent variable, $k_0$ is an integer and $u \colon \mathbb{N}(k_0) \to \mathbb{R}$ is an unknown solution. In this equation, $\Delta^2u(k)=\Delta(\Delta u(k))$, $\Delta u(k)$ is the the first-order forward difference of $u(k)$, i.e., $\Delta u(k) = u(k+1)-u(k)$, and $\Delta^2 (k)$ is its second--order forward difference, i.e., $\Delta^2u(k) = u(k+2)-2u(k+1)+u(k)$, $\alpha$, $m$ are real numbers. The asymptotic behaviour of the solutions to this equation is discussed and the conditions are found such that there exists a power-type asymptotic: $u(k) \sim {1}/{k^s}$, where $s$ is some constant. We also discuss a discrete analogy of so-called ``blow-up'' solutions in the classical theory of differential equations, i.e., the solutions for which there exists a point $x^*$ such that $\lim_{x \to x^*} y(x) = \infty$, where $y(x)$ is a solution of the Emden-Fowler differential equation $$ y''(x) = y^s(x), $$ with $s \ne 1$ being a real number. The results obtained are compared to those already known and illustrated with examples.
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