Pontryagin’s Maximum Principle for Optimal Control Problems Governed by Integral Equations with State and Control Constraints

This work presents a new characterization of the controllability of linear control systems governed by Volterra integral equations. This result, established through a novel lemma, constitutes a fundamental contribution to the theory of integral equations and opens new avenues for future research. As an application, we prove that the classical assumption regarding the controllability of the variational linear equation around an optimal pair is, in fact, superfluous. Building upon this result, we extend Pontryagin’s Maximum Principle to a broad class of optimal control problems governed by Volterra-type integral equations. The formulation incorporates general constraints on both control and state variables, including fixed terminal constraints and time-dependent state restrictions. The cost functional consists of a terminal term and an integral term that may depend on the state. Using the Dubovitskii–Milyutin framework, we construct conic approximations to the cost functional, the integral dynamics, and the constraints, deriving necessary optimality conditions under minimal regularity assumptions. Two types of optimality system are established: one involving a classical adjoint equation when only terminal constraints are imposed, and another involving a Stieltjes integral with respect to a non-negative Borel measure in the presence of time-dependent state constraints. In the particular case where the Volterra system reduces to a differential system, our results recover the classical Pontryagin Maximum Principle. To illustrate the practical implications of our findings, we present an example related to the optimal control of an epidemic SIR model.