## Solution Concepts for Optimal Feedback Control of Nonlinear Partial Differential Equations

#### Karl KunischUniversität Graz

Feedback control of nonlinear systems in practice is still frequently based on linearisation and subsequent treatment by efficient Riccati solvers.
I concentrate on three solution strategies which aim at the nonlinear control system directly. The first one is based on higher order Taylor expansions of the value function and leads to controls which rely on generalized Ljapunov equations.
The second approach is based on Newton steps applied to the HJB equation. Combined with spectral techniques and tensor calculus this allows to solve HJB equations up to dimension 100. The results are demonstrated for the control of discretized Fokker Planck equations.
The third technique circumvents the direct solution of the HJB equation. Rather a neural network is trained by means of a succinctly chosen ansatz and it is proven that it approximates the solution to the HJB equation as the dimension of the network is increased.
This work relies on collaborations with T.Breiten, S.Dolgov, D.Kalise, L.Pfeiffer, and D.Walter.

Back to Workshop I: High Dimensional Hamilton-Jacobi Methods in Control and Differential Games