Abstract
In the present paper, it is provided a representation result for the weak solutions of a class of evolutionary Hamilton–Jacobi–Bellman equations on infinite horizon, with Hamiltonians measurable in time and fiber convex. Such Hamiltonians are associated with a—faithful—representation, namely involving two functions measurable in time and locally Lipschitz in the state and control. Our results concern the recovering of a representation of convex Hamiltonians under a relaxed assumption on the Fenchel transform of the Hamiltonian with respect to the fiber. We apply them to investigate the uniqueness of weak solutions, vanishing at infinity, of a class of time-dependent Hamilton–Jacobi–Bellman equations. Assuming a viability condition on the domain of the aforementioned Fenchel transform, these weak solutions are regarded as an appropriate value function of an infinite horizon control problem under state constraints.
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Notes
We denote \(\int _{a}^{\infty }w(s)\,\mathrm{d}s:=\lim _{b\rightarrow \infty }\int _{a}^{b}w(s)\,\mathrm{d}s\), provided this limit exists.
References
Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans. Am. Math. Soc. 282(2), 487–502 (1984)
Lions, P.-L.: Generalized Solutions of Hamilton–Jacobi Equations. Research Notes in Mathematics, vol. 69. Pitman, Boston (1982)
Barles, G.: Existence results for first-order Hamilton–Jacobi equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(5), 325–340 (1984)
Souganidis, P.E.: Existence of viscosity solutions of Hamilton–Jacobi equations. J. Differ. Equ. 56(3), 345–390 (1985)
Vinter, R.B.: Optimal Control. Birkhäuser, Boston (2000)
Frankowska, H., Plaskacz, S., Rzeżuchowski, T.: Measurable viability theorems and the Hamilton–Jacobi–Bellman equation. J. Differ. Equ. 116(2), 265–305 (1995)
Ishii, H.: Hamilton–Jacobi equations with discontinuous Hamiltonians on arbitrary open sets. Bull. Fac. Sci. Eng. Chuo Univ. 28, 33–77 (1985)
Ishii, H.: On representation of solutions of Hamilton–Jacobi equations with convex hamiltonians. In: Masuda, K., Mimura, M. (eds.) Recent Topics in Nonlinear PDE II, vol. 128, pp. 15–52. North-Holland, Amsterdam (1985)
Ishii, H.: Representation of solutions of Hamilton–Jacobi equations. Nonlinear Anal. Theory Methods Appl. 12(2), 121–146 (1988)
Rampazzo, F.: Faithful representations for convex Hamilton–Jacobi equations. SIAM J. Control Optim. 44(3), 867–884 (2005)
Frankowska, H., Sedrakyan, H.: Stable representation of convex Hamiltonians. Nonlinear Anal. Theory Methods Appl. 100, 30–42 (2014)
Misztela, A.: Representation of Hamilton–Jacobi equation in optimal control theory with compact control set. SIAM J. Control Optim. 57(1), 53–77 (2019)
Soner, H.M.: Optimal control problems with state-space constraints I. SIAM J. Control Optim. 24, 552–562 (1986)
Basco, V., Frankowska, H.: Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints. In: Alabau-Boussouira F., et al. (eds.) Trends in Control Theory and Partial Differential Equation, vol. 32, pp. 17–38. Springer INdAM Series (2019)
Ishii, H.: Perron’s method for monotone systems of second-order elliptic partial differential equations. Differ. Integral Equ. 5(1), 1–24 (1992)
Barron, E.N., Jensen, R.: Optimal control and semicontinuous viscosity solutions. Proc. Am. Math. Soc. 113(2), 397–402 (1991)
Frankowska, H.: Lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations. SIAM J. Control Optim. 31(1), 257–272 (1993)
Basco, V., Frankowska, H.: Hamilton–Jacobi–Bellman equations for infinite horizon control problems under state constraints with time-measurable data. NoDEA-Nonlinear Differ. Equ. Appl. 26(1), 7 (2019)
Rockafellar, R.T.: Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization. Math. Oper. Res. 6(3), 424–436 (1981)
Rockafellar, R.T., Wets, R.J.B.: Variational analysis. In: van der Waerden, B.L., Hurwitz, A., Ross, K.A., Bieberbach, L. (eds.) Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)
Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (2009)
Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997)
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Communicated by Hélène Frankowska.
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Basco, V. Representation of Weak Solutions of Convex Hamilton–Jacobi–Bellman Equations on Infinite Horizon. J Optim Theory Appl 187, 370–390 (2020). https://doi.org/10.1007/s10957-020-01763-1
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DOI: https://doi.org/10.1007/s10957-020-01763-1
Keywords
- Hamilton–Jacobi–Bellman equations
- Weak solutions
- Infinite horizon
- State constraints
- Representation of Hamiltonians