Conference Agenda

We show that the derivatives in the sense of Fréchet and Gâteaux can be viewed as derivatives oriented towards a star convex set with the origin as center. The resulting oriented differential calculus extends the mean value theorem, the chain rule and the Taylor formula in Banach spaces. As applications in stochastic control, we consider functionals and operators of stochastic processes.

In this talk we consider Nash equilibria in two player continuous time stochastic differential games with diffusion control, and where the Brownian motions driving the state processes are correlated. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players' state processes. We explicitly compute the players' equilibrium strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the equilibrium strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the optimal controls are mixed strategies. To do so, we rely on a relaxed formulation of the game based on solutions to martingale problems, allowing the players to randomize their actions.

In this talk, we consider a domestic micro-grid equipped with a local renewable energy production unit such as photovoltaic panels, consumption units, and a battery storage to balance supply and demand and investigate the stochastic optimal control problem for its cost-optimal management. Such systems are complex to control because of uncertainties in the weather and environmental conditions which affect the production and demand of energy. As a special feature, the manager has no access to the grid but has access to a local generator, which makes it possible to produce energy using fuel when needed. Further, we assume that the battery and the fuel tank have limited capacities and the fuel tank can only be filled once at the beginning of the planning period. This leads us to the so-called finite fuel problem. In addition, we assume that the energy demand is not always satisfied and we impose penalties on unsatisfied demand, the so-called inconvenience cost. The main goal is to minimize the expected aggregated cost for generating power using the generator and operating the system. This leads to a challenging mathematical optimization problem. The optimization problem is formulated first as a continuous-time stochastic optimal control problem for a controlled multi-dimensional diffusion process. Then, we transform the continuous-time optimal control problem into a discrete-time control problem and solve it numerically using methods from the theory of Markov decision processes.

We consider stochastic optimal control problems arising in the mathematical modeling of decision-making processes for the cost-optimal management and containment of epidemics. We focus on the impact of uncertainties such as dark figures and study the resulting optimal control problems which are under partial information since some components of the state process are hidden.

Working with diffusion approximations for the population dynamics and the associated Kalman filter estimates of non-observable state variables leads to control problems for controlled diffusion processes. Applying time-discretization, the latter is transformed into a Markov Decision Process that we solve numerically using a backward recursion algorithm. We use the results of optimal quantization and model reduction techniques to overcome the curse of dimensionality. Numerical results are presented.