Greedy optimal control for elliptic problems and its application to turnpike problems

Hernández-Santamaría V., Lazar M., Zuazua E. , DOI: 10.1007/s00211-018-1005-z

Abstract: In this paper, we deal with the approximation of optimal controls for parameter-dependent elliptic and 7 parabolic equations. We adapt well-known results on greedy algorithms to approximate in an efficient 8 way the optimal controls for parameterized elliptic control problems. Our results yield an optimal 9 approximation procedure that, in particular, performs better than simply sampling the parameter-space 10 to compute controls for each parameter value. The same method can be adapted for parabolic control 11 problems, but this leads to greedy selections of the realizations of the parameters that depend on the 12 initial datum under consideration. To avoid this difficulty we employ the turnpike property for time
13 evolution control problems that ensures the asymptotic simplification of optimal control problems for 14 evolution equations towards the elliptic steady-state ones in long time horizons [0, T]. The combination 15 of the turnpike property and greedy methods allows us to develop efficient methods for the approximation 16 of the parameter-dependent parabolic optimals too. We present various numerical experiments discussing 17 the efficiency of our methodology and its application to turnpike control problems.

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