Nonlinear nonautonomous functional differential equations in Lp spaces

Webb, Glenn; Badii, Maurizio

doi:10.1016/0362-546x(81)90045-6

Cited by 11 publications

(7 citation statements)

References 22 publications

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“…where G(t) = (g(t), 0) ∈ R × H. Problem (5.5) gives rise to the σ-periodic cocycle (ψ, ϑ) in H, where ϑ t = ϑ t σ is acting on Q = S 1 σ = R/σZ and ψ t (q, u 0 ) := u(t + q, q, u 0 ), where u(s, q, u 0 ), s ≥ q, is a solution (in a generalized sense) to (5.5) with u(q, q, u 0 ) = u 0 (see [36] or [12]).…”

Section: Delayed Feedback Systemsmentioning

confidence: 99%

“…Let us identify elements of C([−τ, 0]; R) with their images in H under the embedding φ → (φ(0), φ). Since cocycle (ψ, ϑ) given by the generalized solutions of (5.5) agrees on C([−τ, 0]; R) with the cocycle given by the classical solutions and ψ τ (q, H) ⊂ C([−τ, 0]; R) (see [36]), it is clear that S 0 := {u ∈ H | |u| ≤ R √ τ + 1} is a sink for (ψ, ϑ) with G := H. Now the existence of a Lyapunov stable σ-periodic trajectory follows from Theorem 3.…”

Section: 19)mentioning

confidence: 99%

“…In order to apply our results to delay equations we consider them in the L 2 -setting. For checking the well-posedness of delay equations the theory presented in [36,12,7,35] is useful.…”

Section: Introductionmentioning

confidence: 99%

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A non-local reduction principle for cocycles in Hilbert spaces

Anikushin

2019

Preprint

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We study cocycles (non-autonomous dynamical systems) satisfying a certain squeezing condition with respect to the quadratic form of a bounded selfadjoint operator acting in a Hilbert space. We prove that (under additional assumptions) the orthogonal projector maps the fibres of some invariant set, containing bounded trajectories, in a one-to-one manner onto the negative subspace of the operator. This allows to reduce interesting dynamics onto this invariant set, which in some cases can be considered as a kind of inertial manifold for the cocycle. We consider applications of the reduction principle for periodic cocycles. For such cocycles we give an extension of the Massera second theorem, obtain the conditions for the existence of a Lyapunov stable periodic trajectory and prove convergence-type results, which we apply to study nonlinear periodic in time delayed-feedback equations posed in a proper Hilbert space and parabolic problems with a nonlinear periodic in time boundary control. The required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for C 0 -semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part of the problem.

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Section: Delayed Feedback Systemsmentioning

confidence: 99%

Section: 19)mentioning

confidence: 99%

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A non-local reduction principle for cocycles in Hilbert spaces

Anikushin

2019

Preprint

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“…From this and (2.31, we conclude that A E p(A,(t)) and Q,(t) = R(A, A,(t)). -r I 7 1 0, a strongly continuous semigroup is associated with the functional differential equation (see, e.g., [6,7,10,15,[18][19][20][21]). Usually, a corresponding semigroup is built on a suitable space of functions from [ -r , 01 to E and it is then shown that under some conditions, this semigroup is the solution semigroup of the functional differential equations.…”

Section: Pro08mentioning

confidence: 99%

On Nonautonomous Functional Differential Equations

Lan

1999

Journal of Mathematical Analysis and Applications

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In this paper, we prove a theorem on boundary perturbation of nonautonomous Cauchy problerris arid then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous, Banach space valued functional differential equation0 1999 Academic Press where each A(t) is a linear operator on X . Supposing dense domains D ( A ( t ) ) , one can characterize well-posedness of ( wlCP) by the following definition (see 191). DEFINITION 1.1. The problem ( d C P ) is called wellposed if there exists a unique, strongly continuous evolutionfamily {U(t, s)}, ~ ,~ ~ ~ of bounded, NONAUTONOMOUS FDE'S 159 linear operators on X satisfying 1. U(t, t ) = Id for all 0 I t I T , 2. U(t, s)U(s, r ) = U(t, r ) for all 0 I r I s I t I T , 3. ( t , s) -U(t, s) is strongly continuous, and for every s E [0, TI there exists a dense subspace Y, L X satisfying U(t, 3)x L D ( A ( t ) ) for every 0 I s I t I T such that d -U ( t , S ) X = A ( t ) U ( T , S ) X d t for all x E Y,.

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“…This is very natural for linear equations [6] and some of these aspects are reflected in the classical studies [18], where they were used implicitly. Well-posedness of general non-linear delay equations in Hilbert spaces was studied by G. F. Webb [31], G. F. Webb and M. Badii [32], M. Faheem and M. R. M. Rao [16]. However, these studies (which are based on the theory of accretive operators) have strong limitations and in many situations it is easier to act in a more concrete way to obtain the well-posedness (for example, if the delay part in the nonlinear term is given by a bounded in L 2 operator, one can use a standard fixed-point argument).…”

Section: Introductionmentioning

confidence: 99%

Frequency theorem and inertial manifolds for neutral delay equations

Anikushin¹

2020

Preprint

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We study the quadratic regulator problem with an unbounded cost functional of general type. The motivation comes from delay equations, which has the feedback part with discrete delays (or, in other words, delta-like measurements, which are unbounded in L 2 ). We treat the problem in an abstract context of a certain Hilbert space, which is rigged by a Banach space. We obtain a version of the non-singular frequency theorem, which guarantees the existence of a unique optimal process, starting in the Banach space. We show that the optimal cost (that is the value of the quadratic functional on the optimal process) is given by the "quadratic form" of a bounded linear operator from the Banach space to its dual and this form can be used as a Lyapunov-like functional. For a large class of non-autonomous nonlinear delay equations in feedback form we obtain an analog of the circle criterion, which is a natural extension of the corresponding criterion for ODEs.

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Nonlinear nonautonomous functional differential equations in Lp spaces

Cited by 11 publications

References 22 publications

A non-local reduction principle for cocycles in Hilbert spaces

A non-local reduction principle for cocycles in Hilbert spaces

On Nonautonomous Functional Differential Equations

Frequency theorem and inertial manifolds for neutral delay equations

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