Abstract
This paper concerns the characterization of the value function associated with a state constrained Bolza problem in which the data are allowed to be discontinuous w.r.t. the time variable on a set of zero measure and have everywhere left and right limits. Using techniques coming from viability theory and nonsmooth analysis, we provide a characterization of the value function as the unique solution to the Hamilton-Jacobi equation, in a generalized sense which employs the lower Dini derivative and the proximal normal vectors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Springer, Heidelberg (2009). https://doi.org/10.1007/978-0-8176-4848-0
Bernis, J., Bettiol, P.: Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data. ESAIM Control Optim. Calc. Var. https://doi.org/10.1051/cocv/2019041
Bettiol, P., Vinter, R.B.: The Hamilton Jacobi equation for optimal control problems with discontinuous time dependence. SIAM J. Control Optim. 55(2), 1199–1225 (2017)
Bettiol, P., Bressan, A., Vinter, R.B.: On trajectories satisfying a state constraint: \(W^{1,1}\) estimates and counterexamples. SIAM J. Control Optim. 48(7), 4664–4679 (2010)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, vol. 178. Springer, Heidelberg (1998). https://doi.org/10.1007/b97650
Frankowska, H., Mazzola, M.: Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints. Calc. Var. Partial Differ. Equ. 46, 725–747 (2013)
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(1), 21–40 (2000)
Frankowska, H.: Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31, 257–272 (1993)
Vinter, R.B.: Optimal Control. Springer, Heidelberg (2010). https://doi.org/10.1007/978-0-8176-8086-2
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bernis, J., Bettiol, P. (2020). Solutions to the Hamilton-Jacobi Equation for Bolza Problems with State Constraints and Discontinuous Time Dependent Data. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2019. Lecture Notes in Computer Science(), vol 11958. Springer, Cham. https://doi.org/10.1007/978-3-030-41032-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-41032-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-41031-5
Online ISBN: 978-3-030-41032-2
eBook Packages: Computer ScienceComputer Science (R0)