Abstract
In the present paper, we investigate stability of solutions of Hamilton–Jacobi–Bellman equations under state constraints by studying stability of value functions of a suitable family of Bolza optimal control problems under state constraints. The stability is guaranteed by the classical assumptions imposed on Hamiltonians and an inward-pointing condition on state constraints.
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References
Crandall, M.G., Lions, P.L.: Viscosity solution of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 277, 1–42 (1983)
Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations and Optimal Control. Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser, Boston (2004)
Frankowska, H.: Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 31, 257–272 (1993)
Capuzzo-Dolcetta, I., Lions, P.L.: Hamilton–Jacobi equations with state constraints. Trans. Am. Math. Soc. 318, 643–685 (1990)
Frankowska, H., Vinter, R.B.: Existence of neighbouring feasible trajectories: applications to dynamic programming for state constrained optimal control problems. J. Optim. Theory Appl. 104, 21–40 (2000)
Soner, H.M.: Optimal control with state-space constraint. SIAM J. Control Optim. 24, 552–561 (1986)
Frankowska, H., Plaskacz, S.: Semicontinuous solutions of Hamilton–Jacobi–Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251, 818–838 (2000)
Bettiol, P., Frankowska, H., Vinter, R.B.: \(L^{\infty }\)estimates on trajectories confined to a closed subset. J. Differ. Equ. 252, 1912–1933 (2012)
Frankowska H., Mazzola M.: On relations of the adjoint state to the value function for optimal control problems with state constraints. NODEA. doi:10.1007/s00030-012-0183-0
Ishii, H., Koike, S.: A new formulation of state constraint problems for the first order PDE’s. SIAM J. Control Optim. 34, 554–571 (1996)
Frankowska H., Plaskacz S.: Hamilton–Jacobi equations for infinite horizon control problems with state constraints. In: Proceedings of International Conference, Calculus of Variations and Related Topics, Haifa (1999)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
Frankowska, H., Sedrakyan, H.: Stable representation of convex Hamiltonians. J. Nonlinear Anal. TMA 100, 30–42 (2014)
Aubin, J.-P.: Viability Theory. Birkhäuser, Boston (1991)
Acknowledgments
This study was funded by the European Union under the Seventh Framework Programme ”FP7-PEOPLE-2010-ITN,” grant agreement number 264735-SADCO
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Communicated by Lars Grüne.
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Sedrakyan, H. Stability of Solutions to Hamilton–Jacobi Equations Under State Constraints. J Optim Theory Appl 168, 63–91 (2016). https://doi.org/10.1007/s10957-015-0765-1
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DOI: https://doi.org/10.1007/s10957-015-0765-1
Keywords
- Hamilton–Jacobi equation
- Optimal control
- Bolza problem
- Viscosity solution
- State constraints
- Stability of solutions