Pontryagin's maximum principle and indirect descent method for optimal impulsive control of nonlocal transport equation

We study a singular problem of optimal control of a nonlocal transport equation in the space of probability measures, in which the structure of the drivng vector field with respect to the control variable is somewhat equivalent to the affine one, while the set of controls is norm-unbounded and constrained in the integral sense only. We show that the problem at hand admits an impulse-trajectory relaxation in terms of discontinuous time reparameterization. This relaxation provides a correct statement of the variational problem in the class of control inputs constrained in both pointwise and integral senses. For the relaxed problem, we derive a new form of the Pontryagin's maximum principle (PMP) with a separate adjoint system of linear balance laws on the space of signed measures. In contrast to the canonical formulation of the PMP in terms of the Hamiltonian equation on the cotangent bundle of the state space, our form allows one to formulate an indirect descent method for optimal impulsive control analogous to classical gradient descent. We expose a version of this method, namely, an algorithm of the steepest descent with an internal line search of the Lagrange multiplier associated with the integral bound on control. The algorithm is proven to monotonically converge to a PMP-extremal up to a subsequence.