Kernel sensitivity and robust optimal control for Banach-valued weakly singular Volterra systems

We study kernel sensitivity and robust optimal control for Banach-space-valued Volterra systems whose memory kernels may have integrable diagonal singularities. A kernel space based on continuous causal sections provides the topology for perturbing the entire memory law. We establish well-posedness of the state equation and, under a compact control action, show that weak convergence of controls yields strong convergence of states. The state mapping and reduced cost are Fréchet differentiable with respect to the kernel; their derivatives are characterized through a linearized Volterra equation, and the corresponding first-order expansions are uniform over the admissible controls. These properties yield existence and stability of nominal minimizers and a Hadamard envelope formula for the optimal value without assuming uniqueness. For bounded kernel uncertainty, we derive a first-order min-sup expansion of the robust value and a selection principle for robust minimizers. The framework is finally specialized to Caputo-Hadamard systems, giving sensitivity and robust expansions with respect to the fractional order.