Leonid Shaikhet
Ariel University, Department of Higher Mathematics, Department Member
- I am Mathematicianedit
The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic... more
The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic perturbations that are of the type of white noise and are proportional to the deviation of the system state from its equilibrium point. Sufficient conditions of stability in probability for each from the five equilibria of the considered model are obtained by virtue of the Routh–Hurwitz criterion and the method of linear matrix inequalities (LMIs). The obtained results are illustrated by numerical analysis of appropriate LMIs and numerical simulations of solutions of the considered system of stochastic differential equations. The research method can also be used in other applications for similar nonlinear models with the order of nonlinearity higher than one.
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We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The model is presented as a system of differential... more
We use a system biology approach to translate the interaction of Bacillus Calmette-Gurin (BCG) + interleukin 2 (IL-2) for the treatment of bladder cancer into a mathematical model. The model is presented as a system of differential equations with the following variables: number of tumor cells, bacterial cells, immune cells, and cytokines involved in the tumor-immune response. This work investigates the delay effect induced by the proliferation of tumor antigen-specific effector cells after the immune system destroys BCG-infected urothelium cells following BCG and IL-2 immunotherapy in the treatment of bladder cancer. For the proposed model, three equilibrium states are found analytically. The stability of all equilibria is analyzed using the method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs).
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In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have... more
In spite of the fact that the theory of stability and optimal control for different types of stochastic systems is well developed and very popular in research, there are some simply and clearly formulated problems, solutions of which have not been found so far. To the readers’ attention six open stability problems for stochastic differential equations with delay, for stochastic difference equations with discrete and continuous time and one open optimal control problem for stochastic hyperbolic equation with two-parameter white noise are offered.
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In this paper we study optimal control problems involving stochastic integral equations with an integral cost functional, and obtain necessary conditions for optimality of controls in this type of problem.The optimal... more
In this paper we study optimal control problems involving stochastic integral equations with an integral cost functional, and obtain necessary conditions for optimality of controls in this type of problem.The optimal control for a linear equation with a quadratic cost functional is also given.
Shaikhet L.E. Optimal control of stochastic integral equations. Lecture Notes in control and information science. 1986. V.81, p.178-187.
Shaikhet L.E. Optimal control of stochastic integral equations. Lecture Notes in control and information science. 1986. V.81, p.178-187.
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Stochastic LTI system is considered with a delay term described by Stieitjes integral. This includes systems with discrete or distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability are presented that lead... more
Stochastic LTI system is considered with a delay term described by Stieitjes integral. This includes systems with discrete or distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability are presented that lead to sufficient conditions in the form of linear matrix inequalities (LMIs). The first one employs neutral type model transformation and augmented Lyapunov functionals. Differently from the existing LMI stability conditions based on neutral type transformation, the proposed conditions do not require the stability of the corresponding integral equations. Moreover, it is shown that in the existing LMI stability conditions based on simple (non-augmented) Lyapunov functionals, the stability analysis of the integral equation can be omitted. The second method is based on a stochastic extension of simple Lyapunov functionals depending on the state derivative. Numerical examples give comparison of results via different methods.
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16th IMACS World Congress 2000 On Scientific Computation, Applied Mathematics a
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A nonlinear stochastic differential equation with delay and logarithmic nonlinearity is considered. Some properties of asymptotic behavior of the solution of this equation are discussed. In particular, the asymptotic behavior of the... more
A nonlinear stochastic differential equation with delay and logarithmic nonlinearity is considered. Some properties of asymptotic behavior of the solution of this equation are discussed. In particular, the asymptotic behavior of the solution in the neighborhood of the zero and positive equilibria is described.
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Abstract The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved. The suggestion of the replacement of Liapunov functions... more
Abstract The theorem on existence of the Liapunov functionals and the theorem on stability in first approximation for a stochastic differential equation with aftereffect are proved. The suggestion of the replacement of Liapunov functions by functionals [1] in the investigation of the stability of ordinary differential equations with lag, has been widely utilized in dealing with determinate systems, as well as in the case of linear and nonlinear stochastic systems (see e. g. [2 – 11]). Results concerning the stability in the first approximation were obtained for stochastic systems in [12 – 18] and others. Use of Liapunov functionals for the differential equations with aftereffect was first encountered in [1, 19, 20] where the inversion theorems were proved and conditions for the stability in first approximation were obtained. Below a stochastic differential equation with aftereffect is investigated where the random perturbations represent an arbitrary process with independent increments.
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It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear... more
It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear stochastic differential equations with delay and a high order of nonlinearity can be extended to nonlinear mathematical models of a much more general form. An important feature is the combination of the method of Lyapunov functionals with the method of Linear Matrix Inequalities (LMIs). Some examples of applications of the proposed method of stability research to known mathematical models are given.
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For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results... more
For the example of one nonlinear mathematical model in food engineering with several equilibria and stochastic perturbations, a simple criterion for determining a stable or unstable equilibrium is reported. The obtained analytical results are illustrated by detailed numerical simulations of solutions of the considered Ito stochastic differential equations. The proposed criterion can be used for a wide class of nonlinear mathematical models in different applications.
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We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for... more
We present a new analytical method to find the asymptotic stable equilibria states based on the Markov chain technique. We reveal this method on the Susceptible-Infectious-Recovered (SIR)-type epidemiological model that we developed for viral diseases with long-term immunity memory. This is a large-scale model containing 15 nonlinear ordinary differential equations (ODEs), and classical methods have failed to analytically obtain its equilibria. The proposed method is used to conduct a comprehensive analysis by a stochastic representation of the dynamics of the model, followed by finding all asymptotic stable equilibrium states of the model for any values of parameters and initial conditions thanks to the symmetry of the population size over time.
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Stability and Control: Theory and Applications, 2000. V.3.N.l,p.78-87.
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In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to... more
In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to the current population size. Using the direct Lyapunov method, we established the global properties of this stochastic differential equation. In particular, we found that solutions of the equation oscillate around an interval, and explicitly found the end points of this interval. Moreover, we found that, if the magnitude of the noise exceeds a certain critical level (which is also explicitly found), then the stochastic stabilisation ("stabilisation by noise") of the zero solution occurs. In this case, (i) the origin is the lower boundary of the interval, and (ii) the extinction of the population due to stochasticity occurs almost sure (a.s.) for a finite time.
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Compared to the replication of double-stranded RNA and DNA viruses, the replication of single-stranded viruses requires the production of a number of intermediate strands that serve as templates for the synthesis of genomic-sense strands.... more
Compared to the replication of double-stranded RNA and DNA viruses, the replication of single-stranded viruses requires the production of a number of intermediate strands that serve as templates for the synthesis of genomic-sense strands. Two theoretical extreme mechanisms for replication for such single-stranded viruses have been proposed; one extreme being represented by the so-called linear stamping machine and the opposite extreme by the exponential growth. Of course, real systems are more complex and examples have been described in which a combination of such extreme mechanisms can also occur: a fraction of the produced progeny resulting from a stamping-machine type of replication that uses the parental genome as template, whereas other fraction of the progeny results from the replication of other progeny genomes. Martínez et al. 1 , Sardanyés et al. 2 and Fornés et al. 3 suggested and analyzed a deterministic model of single-stranded RNA (ssRNA) virus intracellular replication...
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Theorem 1 The equation x n 1 + ... + x n k = z n has no solution for non-zero integers x 1 , ... , x k and z if n is an integer greater than the fixed integer k ≥ 2. Remark. It is clear that if k = 2, x 1 = x, x 2 = y then Theorem 1 turns... more
Theorem 1 The equation x n 1 + ... + x n k = z n has no solution for non-zero integers x 1 , ... , x k and z if n is an integer greater than the fixed integer k ≥ 2. Remark. It is clear that if k = 2, x 1 = x, x 2 = y then Theorem 1 turns into Fermat's Last Theorem.
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Some new Lyapunov type theorems for stochastic difference equations with continuous time are proven. It is shown that these theorems simplify an application of Lyapunov functionals construction method.
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Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically... more
Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.
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Many stability results in the theory of stochastic hereditary systems and their applications were obtained by construction of appropriate Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and... more
Many stability results in the theory of stochastic hereditary systems and their applications were obtained by construction of appropriate Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and developed by the authors during last decade for stability investigation of deterministic and stochastic functional-differential and difference equations. In this paper a survey of some typical examples of this method application and at the same time some new features of this method for stochastic functional differential equations of neutral type are shown, which allow to use the method more effectively. The considered method is illustrated by a lot of figures of stability regions obtained by numerical calculations.
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Abstract This article continues stability investigation of systems with fading stochastic perturbations. In recent results for systems with the continuous time, it was shown that if stochastic perturbations fade on the infinity quickly... more
Abstract This article continues stability investigation of systems with fading stochastic perturbations. In recent results for systems with the continuous time, it was shown that if stochastic perturbations fade on the infinity quickly enough then asymptotically stable deterministic system remains to be an asymptotically mean square stable independently of the magnitude of the intensity maximum of these stochastic perturbations. Here similar statements are obtained for systems with the discrete time by the condition that the level of stochastic perturbations is given by a square summable sequence. Besides the unsolved problem is proposed: is it possible to get analogous results with not so quickly fading stochastic perturbations. This problem is an open problem and for systems with the continuous time too.
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Abstract A system of two connected nonlinear delay differential neoclassical growth models is considered. Conditions for stability of the zero and positive equilibria of this system under stochastic perturbations of the white noise type... more
Abstract A system of two connected nonlinear delay differential neoclassical growth models is considered. Conditions for stability of the zero and positive equilibria of this system under stochastic perturbations of the white noise type that are directly proportional to the deviation of the system state from the equilibrium are obtained. Numerical examples and figures illustrate the obtained theoretical results.
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Stabilization by noise for stochastic delay differential equation.
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Authors: Kolmanovskii V.B., Shaikhet L.E.
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Some unsolved mathematical problem for discussion
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Theorem 1 The equation x n 1 + ... + x n k = z n has no solution for non-zero integers x 1 , ... , x k and z if n is an integer greater than the fixed integer k ≥ 2. Remark. It is clear that if k = 2, x 1 = x, x 2 = y then Theorem 1 turns... more
Theorem 1 The equation x n 1 + ... + x n k = z n has no solution for non-zero integers x 1 , ... , x k and z if n is an integer greater than the fixed integer k ≥ 2. Remark. It is clear that if k = 2, x 1 = x, x 2 = y then Theorem 1 turns into Fermat's Last Theorem.