By Leonid Shaikhet
Brief advent to balance thought of Deterministic useful Differential Equations -- Stochastic practical Differential Equations and technique of making Lyapunov Functionals -- balance of Linear Scalar Equations -- balance of Linear structures of 2 Equations -- balance of platforms with Nonlinearities -- Matrix Riccati Equations in balance of Linear Stochastic Differential Equations with Delays -- Stochastic structures with Markovian Switching -- Stabilization of the managed Inverted Pendulum via a keep watch over with hold up -- balance of Equilibrium issues of Nicholson's Blowflies Equation with Stochastic Perturbations -- balance of optimistic Equilibrium element of Nonlinear method of kind of Predator-Prey with Aftereffect and Stochastic Perturbations -- balance of SIR Epidemic version Equilibrium issues -- balance of a few Social Mathematical types with hold up less than Stochastic Perturbations
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Extra resources for Lyapunov functionals and stability of stochastic functional differential equations
L, Qi (t), i = 1, . . , k, Rj (t), j = 0, . . , m, R(s, t), s ≥ t ≥ 0, are nonnegative definite matrices, τi (t), i = 1, . . , k, t ≥ 0, are differentiable nonnegative functions with τ˙i (t) ≤ τˆi < 1, Ki (s), i = 1, . . , n, νi (s), i = 0, . . , l, μj (s), j = 0, . . 37) t where K is the diagonal matrix with elements ki , i = 1, . . , x G(t)x ≤ −c|x|2 , c > 0, x ∈ Rn . 9) is asymptotically mean-square stable. 38) 42 2 Stochastic Functional Differential Equations and Procedure Proof Put ∞ l V2 (t, xt ) = t dνi (s) x (θ )Si (θ )x(θ ) dθ i=1 0 t−s ∞ n + k t dKi (s) i=1 0 t−s ∞ m + xi2 (θ ) dθ + t dμj (s) j =0 0 t−s ∞ + i=1 t−s x (s)Qi (s)x(s) ds t−τi (t) x (τ )R(τ + s, θ )x(τ ) dθ dτ.
0 ⎟ ⎜ ⎟ ⎜ 0 −S . . 0 ⎟ −S 1 3 ⎜ ⎟. ⎜ 0 1 S2 . . 0 ⎟ ⎜ ⎟ ⎝ ... ... ... ⎠ ... . . . 24) have negative real parts if and only if 1 = −S1 > 0, −S1 1 0 = 3 −S1 1 0 n= 0 ... −S3 S2 −S1 1 ... 2 = −S3 S2 −S1 −S5 S4 −S3 S2 ... −S1 1 −S3 > 0, S2 −S5 S4 > 0, −S3 ... ... ... 26) 0 0 0 > 0. 0 ... 2 A symmetric n × n matrix Q is called a positive (negative) definite matrix (Q > 0) (Q < 0) if there exists c > 0 such that x Qx ≥ c|x|2 (x Qx ≤ −c|x|2 ) for all x ∈ Rn . 27) has a positive definite solution P .
In Figs. 18) goes to constant on the 2 bound a = b of stability region, is a periodical on the bound a + b = πh2 , goes to zero between these bounds, and goes to infinity out of these bounds. 01. 4 The Influence of Small Delays on Stability 21 Fig. 9 Fig. 4 The Influence of Small Delays on Stability The effect of time delay is very essential in different real processes [46, 109, 225]. Many actual phenomena involve small delays, which are often neglected in the process of mathematical modeling. Sometimes this leads to false conclusions.