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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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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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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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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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