Volterra equations in Banach spaces with completely monotone kernels

Abstract

Abstract. We consider a class of infinite delay equations in Banach spaces that models arising in the theory of viscoelasticity, for instance. The equation involves a completely monotone convolution kernel with a singularity at t = 0 and a sectorial linear spatial operator. Our main goal here is the construction of a semigroup formulation for the integral equation; in the last part of the paper, we illustrate the potentiality of the approach by considering a stochastic perturbation of the problem. Existence and uniqueness of a weak solution is established. The corresponding evolutionary solution process is Markovian, and the tools of linear analytic semigroup theory can be utilized.Mathematics Subject Classification. 60H15, 60H20, 45K05.

Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

“…Therefore, we use another method of proof using the centered Steklov average. 3 In order to prove the integration-by-parts formula on (0, t), we first have to extend the functions considered to the interval (−η, 0). We then prove the formula on an arbitrary interval (α, β) with −η + h 0 < α < β < T − h 0 .…”

Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

“…Further, the optimal control problem, reformulated into the state setting X, consists in minimizing the cost functional J(x, γ) = E ∞ 0 e −λt ℓ(Jx(t), γ(t))dt (where the initial condition u 0 is substituted by x and the process u is substituted by Jx). It follows that γ is an optimal control for the original Volterra equation if and only if it is an optimal control for that state equation (3). To our knowledge, this paper is the first attempt to study optimal control problems with infinite horizon for stochastic Volterra equations.…”

“…where ℓ : H × U → R is a given real bounded function and λ is any positive number. Our first task is to provide a semigroup setting for the problem, by the state space setting first introduced by Miller in [36] and Desch and Miller [17] and recently revised, for the stochastic case, in Homan [29], Bonaccorsi and Desch [3], Bonaccorsi and Mastrogiacomo [4]. Within this approach, equation (1) is reformulated into an abstract evolution equation without memory on a different Hilbert space X. Namely, we rewrite equation (1)…”

“…For more details, we refer to the original papers [3,29]. Further, the optimal control problem, reformulated into the state setting X, consists in minimizing the cost functional J(x, γ) = E ∞ 0 e −λt ℓ(Jx(t), γ(t))dt (where the initial condition u 0 is substituted by x and the process u is substituted by Jx).…”

Regularity of stochastic Volterra equations by functional calculus methods

“…In [31,35] it is used to derive maximal estimates for stochastic convolutions by a dilation argument. Solutions with paths in D((−A 1/2 )) almost surely are obtained via square function estimates in [3,13,15,26,27,33,34]. More indirectly, characterizations of the H ∞ -calculus have already been employed in Theorem 6.14 of [11] and in [4], in the form that D A (θ, 2) = D((−A) θ ) for some θ ∈ (0, 1) and that (−A) is is bounded for all s ∈ R, respectively.…”