Krzysztof Nowak

Krzysztof Nowak completed his doctoral studies in the Department of Theoretical Physics at the Moscow Institute of Physics and Technology (Phystech), where his research developed a rigorous analytical connection between the variational principles of classical mechanics and the mathematics of optimal control. His thesis began from the observation that the Hamilton–Jacobi equation of classical mechanics and the Hamilton–Jacobi–Bellman equation of dynamic programming are formally identical — the latter arising when the former is extended from deterministic trajectory optimisation to stochastic control problems over continuous state spaces. Working within the Phystech tradition of grounding applied mathematics in physical first principles, he established a path-integral formulation of stochastic optimal control in which the value function emerges as the saddle-point of an action functional defined over a space of admissible trajectories, importing techniques from quantum field theory — in particular, functional analysis on infinite-dimensional path spaces — into the optimal control setting to obtain new existence and regularity results.

The technical core of his thesis addressed HJB equations posed in separable Hilbert spaces, which arise when the controlled system is governed by a delay-differential equation or an abstract evolution equation with infinite-dimensional state. Classical viscosity solution theory, developed for finite-dimensional equations, does not extend to this setting without substantial reformulation. Nowak established a comparison principle for the infinite-dimensional HJB equation and derived regularity conditions that guarantee the existence of smooth value functions under physically motivated growth assumptions on the Hamiltonian — closing a gap between the Pontryagin maximum principle, which applies directly in infinite dimensions, and dynamic programming, which had lacked a fully rigorous formulation in this setting. A complementary line of work examined the degenerate case in which the Hamiltonian fails to be strictly convex in the control variable, establishing conditions under which viscosity solutions of the HJB equation and adjoint-state solutions of the maximum principle characterise the same optimal policy.

At IADU, Nowak applies this framework to optimal control problems in quantitative policy analysis, where the state process is typically high-dimensional and may carry memory — treasury debt dynamics, reserve levels with path-dependent depletion rules, and central bank balance sheet evolution. His research underpins the HJB-based computational methods used across IADU's policy-facing work, and he maintains an active line of research on the convergence of policy and value iteration algorithms for problems in which the Hamilton–Jacobi structure is inherited from an underlying physical variational principle.