Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917936996810752 |
|---|---|
| author | de Feo, Filippo Święch, Andrzej Wessels, Lukas |
| author_facet | de Feo, Filippo Święch, Andrzej Wessels, Lukas |
| contents | We study optimal control problems governed by abstract infinite dimensional stochastic differential equations using the dynamic programming approach. In the first part, we prove Lipschitz continuity, semiconcavity and semiconvexity of the value function under several sets of assumptions, and thus derive its $C^{1,1}$ regularity in the space variable. Based on this regularity result, we construct optimal feedback controls using the notion of the $B$-continuous viscosity solutions for the associated Hamilton--Jacobi--Bellman equation. This is done in the case when the noise coefficient is independent of the control variable. We also discuss applications of our results to optimal control problems governed by stochastic reaction-diffusion equations and, under economic motivations, stochastic delay differential equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03181 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions de Feo, Filippo Święch, Andrzej Wessels, Lukas Optimization and Control Analysis of PDEs Probability 93E20, 49L25, 49L12, 49K45, 60H15, 49L20, 49N35, 35R15, 35K57, 34K50 We study optimal control problems governed by abstract infinite dimensional stochastic differential equations using the dynamic programming approach. In the first part, we prove Lipschitz continuity, semiconcavity and semiconvexity of the value function under several sets of assumptions, and thus derive its $C^{1,1}$ regularity in the space variable. Based on this regularity result, we construct optimal feedback controls using the notion of the $B$-continuous viscosity solutions for the associated Hamilton--Jacobi--Bellman equation. This is done in the case when the noise coefficient is independent of the control variable. We also discuss applications of our results to optimal control problems governed by stochastic reaction-diffusion equations and, under economic motivations, stochastic delay differential equations. |
| title | Stochastic optimal control in Hilbert spaces: $C^{1,1}$ regularity of the value function and optimal synthesis via viscosity solutions |
| topic | Optimization and Control Analysis of PDEs Probability 93E20, 49L25, 49L12, 49K45, 60H15, 49L20, 49N35, 35R15, 35K57, 34K50 |
| url | https://arxiv.org/abs/2310.03181 |