By Marat Akhmet (auth.)
The booklet is principally approximately hybrid platforms with continuous/discrete-time dynamics. the main a part of the publication involves the idea of equations with piece-wise consistent argument of generalized style. The structures in addition to means of research have been brought by means of the writer very lately. They either generalized identified conception approximately differential equations with piece-wise consistent argument, brought through ok. cook dinner and J. Wiener within the Nineteen Eighties. furthermore, differential equations with fastened and variable moments of impulses are used to version actual global difficulties. We contemplate versions of neural networks, blood strain distribution and a generalized version of the cardiac pacemaker. all of the result of the manuscript haven't been released in any booklet, but. they're very fresh and united with the presence of the continuous/discrete dynamics of time. it really is of massive curiosity for experts in biology, drugs, engineering sciences, electronics. Theoretical features of the ebook meet very powerful expectancies of mathematicians who examine differential equations with discontinuities of any type.
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1) has the structure of a continuous dynamical system within intervals [θi , θi+1 ), i ∈ Z. In this chapter, we assume that the solutions of the equation are continuous functions. But the deviating function γ (t) is discontinuous. 1) has discontinuities at moments θi . As a result, we consider the solutions of the equation as functions, which are continuous and continuously differentiable within intervals [θi , θi+1 ), i ∈ Z. The theory of differential equations with piecewise constant arguments necessitates a more careful discussion of the initial value problem.
2. Suppose that conditions (C1)–(C5) are fulfilled. 1) with z(ζi ) = (c, F(ζi , c)), c ∈ Rk , is defined on R and satisfies 2K c e−α (t−ζi ) , t z(t) ζi . 12) Proof. 6). Denote z0 (t) = (0, 0)T , zm = (um , vm )T , m 0, um+1 (t) = eB− (t−ζi ) c + vm+1 (t) = − ∞ t t ζi eB− (t−s) f− (s, zm (s), zm (γ (s)))ds, eB+ (t−s) f+ (s, zm (s), zm (γ (s)))ds. First we need to show that zm (t) 2K c e−α (t−ζi ) , t ζi , m 0. 13) Indeed, z0 satisfies the relation. 13). 9) is given. Similarly, one can establish the following inequality zm+1 (t) − zm (t) K c (2p )m e−α (t−ζi ) .
2) such that X(0) = I , where I is an n × n identity matrix. 2). Let κ = supt∈R A(t) < ∞. 1 ( [148]). Assume (H1) is satisfied. 3) holds. 2 ( [148]). Assume (H1) is satisfied. Then, the inequality m X(t, s) M, where m = exp(−κθ ), M = exp(κθ ), holds for |t − s| θ. 1). 2. 3. Suppose (H1) is satisfied. 1) on R if and only if it is a solution on R of the following integral equation x(t) = X(t,t0 )x0 + t t0 X(t, s)[ f (s) + μ g(s, x(s), x(β (s)), μ )] ds. 4. Suppose (H1) is satisfied. 1) on R if and only if it is a solution on R of the following integral equation x(t) = x0 + t t0 [A(s)x(s) + f (s) + μ g(s, x(s), x(β (s)), μ )] ds.