**November 30, 2023 @ 17h00**

**Title:** Exact increment formulas for dynamic optimization: From classical to mean-field control problems**Location:** FEUP – DEEC – room i-105**Speakers:** M. Staritsyn and R. Chertovskih (SYSTEC/ARISE)

**Abstract:**

In the classical Calculus of Variations, there is a formula that allows one to express the difference between two values of the payoff functional (cost increment) as an integral of a certain function. This formula is known as the Weierstrass formula, and the integral is called the invariant Hilbert integral. It turns out that the same can be done (in different ways) for optimal control problems, yielding exact representations of the cost increment, i.e., increment formulas devoid of residual terms. These formulas allow us, in turn, to develop new necessary optimality conditions, not equivalent to the paradigmatic Pontryagin maximum principle, and descent algorithms without free parameters. Our talk will start with an explosion of some basic ideas behind the promoted approach for the classical nonlinear optimal control of ODEs. Next, we will show how similar results can be developed for more complex problems in the space of probability measures, when the dynamics are represented by the continuity/Fokker-Planck-Kolmogorov equations. In the final part, we will discuss the numerical aspects of the algorithm, and demonstrate its application to several prominent optimization tasks such as optimal control of Bloch equations and synchronization of neural ensembles