Risk-Averse Approach to Control-Affine Systems of ODEs

  • Scagliotti, Alessandro (Technical University of Munich, and MCML)
  • Surowiec, Thomas (Simula Research Laboratory)

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In this talk, we consider an ensemble of control systems in $\mathbb{R}^n$ of the form \[ \dot x^\theta(t) = G_0^\theta\big(x^\theta(t)\big) + G^\theta\big(x^\theta(t)\big)\cdot u(t), \qquad x^\theta(0)=x_0^\theta, \] where $\theta \in \Theta$ is an unknown quantity ranging in a compact subset of an Euclidean space and parametrizing the systems. We remark that all the ODEs of the ensemble are \emph{simultaneously} driven on the time horizon $[0,T]$ by the same control $u \in \mathcal{U} := L^2([0,T],\mathbb{R}^m)$. In the case where $\theta$ is the realization of a random vector with law $\mu$, we define the functional \[ J_{\mathrm{r.a.}}(u) := \mathrm{CVaR}_{\beta}^\mu\!\left(\ell\big(x_u^\theta(T)\big)\right) + \alpha \int_0^T f(t,u(t))\,\mathrm{d}t \;\to\; \min, \] where $\ell\colon \mathbb{R}^n \to \mathbb{R}$ is a $C^1$ function prescribing the terminal cost, $f\colon [0,T]\times \mathbb{R}^m \to \mathbb{R}$ is the running cost, and $\alpha>0$. Here, $\mathrm{CVaR}_{\beta}^\mu$ denotes the conditional value at risk, defined for $X\in L^1_\mu(\Theta)$ as \[ \mathrm{CVaR}_{\beta}^\mu(X) := \inf_{t\in \mathbb{R}} \left( t + \frac{1}{1-\beta}\,\mathbb{E}_\mu[(X-t)^+] \right), \qquad \beta\in(0,1). \] We discuss the existence of minimizers for $J_{\mathrm{r.a.}}$ and study their stability under perturbations of the measure $\mu$. Finally, we investigate the limiting problems as $\beta$ tends to $0$ and to $1$.