Controllability of Nonlinear Deterministic Chaotic Systems with Control-Induced Delay

This paper presents the analysis of a class of retarded nonlinear chaotic systems with control-induced delay. The existence and uniqueness of the mild solution are obtained by using the local Lipschitz condition on nonlinearity and Banach contraction principle. The approximate controllability for linear and nonlinear control delay systems has been established by sequence method and using the Nemytskii operator. The application of results is explained through an example of a parabolic partial differential equation.

Existence, uniqueness and regularity of the projection onto differentiable manifolds

We investigate the maximal open domain $${\mathscr {E}}(M)$$ E ( M ) on which the orthogonal projection map p onto a subset $$M\subseteq {{\mathbb {R}}}^d$$ M ⊆ R d can be defined and study essential properties of p. We prove that if M is a $$C^1$$ C 1 submanifold of $${{\mathbb {R}}}^d$$ R d satisfying a Lipschitz condition on the tangent spaces, then $${\mathscr {E}}(M)$$ E ( M ) can be described by a lower semi-continuous function, named frontier function. We show that this frontier function is continuous if M is $$C^2$$ C 2 or if the topological skeleton of $$M^c$$ M c is closed and we provide an example showing that the frontier function need not be continuous in general. We demonstrate that, for a $$C^k$$ C k -submanifold M with $$k\ge 2$$ k ≥ 2 , the projection map is $$C^{k-1}$$ C k - 1 on $${\mathscr {E}}(M)$$ E ( M ) , and we obtain a differentiation formula for the projection map which is used to discuss boundedness of its higher order differentials on tubular neighborhoods. A sufficient condition for the inclusion $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) is that M is a $$C^1$$ C 1 submanifold whose tangent spaces satisfy a local Lipschitz condition. We prove in a new way that this condition is also necessary. More precisely, if M is a topological submanifold with $$M\subseteq {\mathscr {E}}(M)$$ M ⊆ E ( M ) , then M must be $$C^1$$ C 1 and its tangent spaces satisfy the same local Lipschitz condition. A final section is devoted to highlighting some relations between $${\mathscr {E}}(M)$$ E ( M ) and the topological skeleton of $$M^c$$ M c .

Asymptotic behavior of random coupled Ginzburg–Landau equation driven by colored noise on unbounded domains

In this paper, a random coupled Ginzburg–Landau equation driven by colored noise on unbounded domains is considered, in which the nonlinear term satisfies a local Lipschitz condition. It is shown that the random attractor of such a coupled Ginzburg–Landau equation is a singleton set, and the components of solutions are very close when the coupling parameter becomes large enough.

Locally Lipschitz Composition Operators in Space of the Functions of BoundedκΦ-Variation

We give a necessary and sufficient condition on a functionh:R→Runder which the nonlinear composition operatorH, associated with the functionh,Hu(t)=h(u(t)), acts in the spaceκΦBV[a,b]and satisfies a local Lipschitz condition.

Numerical Solutions to Neutral Stochastic Delay Differential Equations with Poisson Jumps under Local Lipschitz Condition

Recently, Liu et al. (2011) studied the stability of a class of neutral stochastic delay differential equations with Poisson jumps (NSDDEwPJs) by fixed points theory. To the best of our knowledge to date, there are not any numerical methods that have been established for NSDDEwPJs yet. In this paper, we will develop the Euler-Maruyama method for NSDDEwPJs, and the main aim is to prove the convergence of the numerical method. It is proved that the proposed method is convergent with strong order 1/2 under the local Lipschitz condition. Finally, some numerical examples are simulated to verify the results obtained from theory.

Positive Solutions of Advanced Differential Systems

We study asymptotic behavior of solutions of general advanced differential systemsy˙(t)=F(t,yt), whereF:Ω→ℝnis a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument andΩis a subset inℝ×Crn,Crn:=C([0,r],ℝn),yt∈Crn, andyt(θ)=y(t+θ),θ∈[0,r]. A monotone iterative method is proposed to prove the existence of a solution defined fort→∞with the graph coordinates lying between graph coordinates of two (lower and upper) auxiliary vector functions. This result is applied to scalar advanced linear differential equations. Criteria of existence of positive solutions are given and their asymptotic behavior is discussed.