Approximate Controllability of Nonlocal Stochastic Integrodifferential System in Hilbert Spaces

This project investigates the approximate controllability of a class of stochastic integrodifferential equations in Hilbert space with non-local beginning conditions. In a departure from the conventional concerns expressed in the literature, we will not consider compactness or the Lipschitz criteria concerning the nonlocal term. We use the fact that the resolvent operator is compact. We first prove the controllability of the nonlinear system using Schauder’s fixed point theorem, a method known for its robustness; as well, we also use Grimmer’s resolvent operator theory. Subsequently, we employ the reliable approximation methods and the powerful diagonal argument to determine the approximate controllability of the stochastic system. To conclude, we present an example that validates our theoretical statement.

💡 Research Summary

The paper addresses the approximate controllability of a class of stochastic integrodifferential equations in a Hilbert space with non‑local initial conditions. The state equation (1.1) contains a linear part generated by a densely defined operator A, a Volterra‑type convolution term with a family of closed operators Π(t), a control term C u(t), a deterministic nonlinear drift f(t, x), and a stochastic diffusion term g(t, x) dW(t). The non‑local condition is given by h(x)=∫₀ᶜ ζ(s, x(s)) ds, meaning the initial value depends on the whole past trajectory.

Unlike most works, the authors do not impose Lipschitz continuity or compactness on the non‑local term. Instead, they rely on the compactness of the Grimmer resolvent operator ℛ(t), which exists under the standard assumptions (R1)–(R2) that A generates a C₀‑semigroup and Π(t) is suitably regular. Lemmas show that ℛ(t) is compact for t>0 and norm‑continuous, which is crucial for the subsequent analysis.

The nonlinearities f, g, and the kernel ζ satisfy only Carathéodory conditions together with integrable growth functions τ_f, τ_g, τ_ζ and non‑decreasing functions Ω_f, Ω_g, Ω_ζ (assumptions H1–H3). Under these mild hypotheses, the authors construct a control map u_μ (formula (2.5)) and prove, via Schauder’s fixed‑point theorem, that the associated operator is continuous and maps a bounded ball into itself. Lemma 3.2 establishes uniform L²‑bounds for u_μ, which leads to the existence of a mild solution of the stochastic system.

The controllability analysis proceeds by first examining the linearized system (2.4). They introduce the controllability operator Δ_{c0}=∫₀ᶜ ℛ(c−s) C C* ℛ*(c−s) ds and the regularized inverse S(μ,Δ_{c0})=(μI+Δ_{c0})^{-1}. Lemma 2.12 states that the linear system is approximately controllable iff μ S(μ,Δ_{c0})→0 strongly as μ→0. This condition is verified using the compactness of ℛ(t).

For the full nonlinear stochastic system, the authors combine the linear controllability condition with the properties of u_μ, employing an approximation scheme and a diagonal argument. They show that for any target state x₁∈H and any ε>0, one can choose μ sufficiently small and a corresponding control u_μ such that the mild solution satisfies ‖x(c)−x₁‖<ε almost surely. Hence the system is approximately controllable despite the weakened assumptions on the non‑local term.

An illustrative example is provided where A is a Laplacian operator, Π(t) and C are concrete bounded operators, and f, g, ζ are chosen to meet the growth conditions. The example confirms that ℛ(t) is compact, the hypotheses hold, and the approximate controllability result applies.

Overall, the paper contributes a new approximate controllability theorem for stochastic integrodifferential equations with fully non‑local initial data, relaxing traditional Lipschitz and compactness requirements on the non‑local term, and leveraging the compactness of the resolvent operator together with Schauder’s fixed‑point theorem and a diagonal approximation technique.