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by Prof. Dr. Karl Kunisch (University of Graz)

Abstract:

I consider controlled dynamical systems governed by nonlinear partial differential equations Feedback – or closed loop – control allows to express the control as a function of the state. Due to its robustness against perturbations it has significant advantages over open loop control. Optimal closed loop controls are characterized by means of solutions to Hamilton-Jacobi-Bellman (HJB) equations. These equations represent a formidable task to solve in practice. For this reason, solution strategies to date are frequently based on linearisation and subsequent treatment by efficient Riccati solvers.
I concentrate on the nonlinear case and present three solution strategies. The first one is based on Taylor expansions of the value function and leads to controls which rely on generalized Ljapunov equations. This contains the Riccati synthesis as a special case. The second approach is based on value iteration or alternatively Newton steps applied to the HJB equation. Combined with spectral techniques and tensor calculus this allowed to solve HJB equations up to dimension 100.
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
Everybody is welcome!